From f0868e65515dc2f319a6b75ff1df5b8e1b2079c4 Mon Sep 17 00:00:00 2001
From:  <>
Date: Fri, 23 Feb 2024 19:57:04 +0000
Subject: [PATCH] Update documentation

---
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 _sources/chapter0/bezier-curves.md            |  556 +++
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 _sources/chapter0/bsplines-surfaces.md        |  255 ++
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 _sources/chapter0/performance-acceleration.md |  316 ++
 _sources/chapter0/poisson-1d.md               |  303 ++
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 _sources/chapter1/install.md                  |   31 +
 _sources/chapter1/poisson.ipynb               |  330 ++
 _sources/chapter1/rules.md                    |   60 +
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 _sources/chapter1/topology.md                 |  225 ++
 .../advection-diffusion-stabilized.ipynb      |  356 ++
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 .../bingham_plastic_flow_in_a_pipe.ipynb      |   78 +
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 ...tion-diffusion_equation_in_a_channel.ipynb |   78 +
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HcmV?d00001

diff --git a/_images/eval_field_2d.png b/_images/eval_field_2d.png
new file mode 100644
index 0000000000000000000000000000000000000000..5ce6056cae322effb9201299ddcb2445abe36b64
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diff --git a/_sources/chapter0/bezier-curves.md b/_sources/chapter0/bezier-curves.md
new file mode 100644
index 0000000..b0fae2f
--- /dev/null
+++ b/_sources/chapter0/bezier-curves.md
@@ -0,0 +1,556 @@
+# Bézier curves
+
+We recall the definition of a Bézier curve:
+
+$$
+\begin{align}
+  \mathcal{C}(t) = \sum_{k=0}^n \mathbf{P}_k B_k^n(t)
+\end{align}
+$$
+
+where $\left( \mathbf{P} \right)_{0 \le k \le n}$ are the control points or Bézier points.
+
+## Evaluation of a point on Bézier curve
+
+The following function evaluates a Bézier curve, given the control points $P$ at $x$. The degree of the Bernstein polynomials is computed from the length of $P$.
+
+
+```python
+def point_on_bezier_curve(P,x):
+    n = len(P) - 1
+    b = all_bernstein(n, x)
+    c = 0.
+    for k in range(0, n+1):
+        c += b[k]*P[k]
+    return c
+```
+
+## Examples
+
+````{prf:example}
+:label: bezier-curve-ex-1
+
+```python
+nt = 200
+ts = np.linspace(0., 1., nt)
+
+P = np.zeros((2, 2))
+P[:, 0] = [0., 1.]
+P[:, 1] = [1., 0.]
+
+Q = np.zeros((nt, 2))
+for i,t in enumerate(ts):
+    Q[i,:] = point_on_bezier_curve(P,t)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, 2):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-0.2, 1.2, -0.2, 1.2])
+```
+    
+![png](images/bezier/output_16_0.png)
+````    
+
+````{prf:example}
+:label: bezier-curve-ex-2
+
+```python
+nt = 200
+ts = np.linspace(0., 1., nt)
+
+P = np.zeros((3, 2))
+P[:, 0] = [1., 1., 0.]
+P[:, 1] = [0., 1., 1.]
+
+Q = np.zeros((nt, 2))
+for i,t in enumerate(ts):
+    Q[i,:] = point_on_bezier_curve(P,t)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, 3):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-0.2, 1.2, -0.2, 1.2])
+```
+    
+![png](images/bezier/output_17_0.png)
+````    
+    
+````{prf:example}
+:label: bezier-curve-ex-3
+
+```python
+nt = 200
+ts = np.linspace(0., 1., nt)
+
+P = np.zeros((4, 2))
+P[:, 0] = [1., 1., 0., -1.]
+P[:, 1] = [0., 1., 1.,  0.]
+
+Q = np.zeros((nt, 2))
+for i,t in enumerate(ts):
+    Q[i,:] = point_on_bezier_curve(P,t)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, 3):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+i = 3
+x,y = P[i,:]
+plt.text(x-0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-1.2, 1.2, -0.2, 1.2])
+```
+ 
+![png](images/bezier/output_18_0.png)
+````    
+    
+````{prf:example}
+:label: bezier-curve-ex-4
+
+```python
+nt = 200
+ts = np.linspace(0., 1., nt)
+
+P = np.zeros((4, 2))
+P[:, 0] = [1., 1.,  0., -1.]
+P[:, 1] = [0., 1., -0.5,  0.]
+
+Q = np.zeros((nt, 2))
+for i,t in enumerate(ts):
+    Q[i,:] = point_on_bezier_curve(P,t)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, 3):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+i = 3
+x,y = P[i,:]
+plt.text(x-0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-1.2, 1.2, -0.7, 1.2])
+```
+    
+![png](images/bezier/output_19_0.png)
+````    
+    
+````{prf:example}
+:label: bezier-curve-ex-5
+
+```python
+nt = 200
+ts = np.linspace(0., 1., nt)
+
+P = np.zeros((4, 2))
+P[:, 0] = [1., 0., 1., -1.]
+P[:, 1] = [0., 1., 1.,  0.]
+
+Q = np.zeros((nt, 2))
+for i,t in enumerate(ts):
+    Q[i,:] = point_on_bezier_curve(P,t)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, 3):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+i = 3
+x,y = P[i,:]
+plt.text(x-0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-1.2, 1.2, -0.2, 1.2])
+```
+    
+![png](images/bezier/output_20_0.png)
+````    
+
+## Derivatives of a Bezier curve
+
+Using the formulae \ref{eq:bernstein-der}, we have
+
+$$
+\begin{align}
+  \mathcal{C}^\prime(t) = \sum_{k=0}^n \mathbf{P}_k {B_k^n}^\prime(t) 
+                        = \sum_{k=0}^n n \mathbf{P}_k \left(B_{k-1}^{n-1}(x) - B_k^{n-1}(x) \right)  
+\end{align}
+$$
+
+by reordering the indices, we get \ref{eq:bezier-curve-der}
+
+$$
+\begin{align}
+  \mathcal{C}^\prime(t) &= \sum_{k=0}^{n-1} n \left( \mathbf{P}_{k+1} - \mathbf{P}_k \right) B_k^{n-1}(t) 
+  \label{eq:bezier-curve-der}
+\end{align}
+$$
+
+Therefor, we get the direct acces to the first order derivatives at the extremeties of a Bézier curve using the formulae \ref{eq:bezier-curve-der-1-ext}
+
+$$
+\begin{align}
+  \begin{cases}
+    \mathcal{C}^\prime(0) = n \left( \mathbf{P}_{1} - \mathbf{P}_0 \right)  
+    \\ 
+    \mathcal{C}^\prime(1) = n \left( \mathbf{P}_{n} - \mathbf{P}_{n-1} \right)  
+  \end{cases}
+  \label{eq:bezier-curve-der-1-ext}
+\end{align}
+$$
+
+Second derivatives can also be computed directly from the control points using the formulae  \ref{eq:bezier-curve-der-2-ext}
+
+$$
+\begin{align}
+  \begin{cases}
+    \mathcal{C}^{\prime\prime}(0) = n(n-1) \left( \mathbf{P}_{0} - 2 \mathbf{P}_1 + \mathbf{P}_{2} \right)  
+    \\ 
+    \mathcal{C}^{\prime\prime}(1) = n(n-1) \left( \mathbf{P}_{n} - 2 \mathbf{P}_{n-1} + \mathbf{P}_{n-2} \right)  
+  \end{cases}
+  \label{eq:bezier-curve-der-2-ext}
+\end{align}
+$$
+
+
+````{prf:definition} $r^{th}$ forward difference
+
+  Let us consider a set of (control) points $\mathbf{P}$.
+  The $r^{th}$ forward difference of $\mathbf{P}$ is defined as  
+$$
+  \begin{align}
+    \triangle^r \mathbf{P}_i := \triangle^{r-1} \mathbf{P}_{i+1} - \triangle^{r-1} \mathbf{P}_i  
+  \end{align}
+$$
+  with 
+$$
+  \begin{align}
+    \triangle \mathbf{P}_i = \triangle^1 \mathbf{P}_i := \mathbf{P}_{i+1} - \mathbf{P}_i  
+  \end{align}
+$$
+````
+
+````{prf:proposition} High order dirivatives
+$$
+  \begin{align}
+    \mathcal{C}^{(r)}(t) = \frac{n!}{\left( n-r \right)!} \sum\limits_{i=0}^{n-r} \triangle^r \mathbf{P}_i B_i^{n-r}(t) 
+    \label{eq:bezier-high-deriv}
+  \end{align}
+$$
+````
+
+Eqs (\ref{eq:bezier-curve-der-1-ext}) and (\ref{eq:bezier-curve-der-2-ext}) can be generelaized for high order derivatives. We have in fact the following result:
+
+```{prf:remark}
+The derivatives of a Bézier curve at its extremeties up to order $r$ depend only on the first (or last) $r+1$ control points, and vice versa.
+```
+
+````{prf:proposition}
+  \label{prop:bezier-forward-difference}
+  $$\triangle^r \mathbf{P}_0 = \sum\limits_{i=0}^{r} \left( -1 \right)^{r-i} \binom{r}{i} \mathbf{P}_i $$
+````
+
+## Integration of Bezier curves
+
+````{prf:proposition}
+A primitive of A Bézier curve $\mathcal{C}(t) = \sum_{k=0}^n \mathbf{P}_k B_k^n(t)$ has the Bézier representation
+
+$$
+\begin{align}
+  \left( \int \mathcal{C} \right) (t) = \sum_{k=0}^{n+1} \mathbf{Q}_k B_k^{n+1}(t)
+  \label{eq:bezier-primitive}
+\end{align}
+$$
+
+where
+
+$$
+\begin{align}
+  \mathbf{Q}_k := \mathbf{Q}_{k-1} + \frac{1}{n+1} \mathbf{P}_{k-1} 
+                = \mathbf{Q}_{0}  + \frac{1}{n+1} \left( \sum\limits_{i=0}^{k-1} \mathbf{P}_{i}  \right)
+\end{align}
+$$
+````
+
+Here, $\mathbf{Q}_{0}$ denotes an arbitrary integration constant.
+
+```{prf:remark}
+  Using Eq. (\ref{eq:bezier-primitive}), we have $\int_0^1 \mathcal{C}(t)~dt = \frac{1}{n+1}\left( \sum\limits_{i=0}^n \mathbf{P}_i \right)$.
+```
+
+## Properties of Bezier curves
+Because of the *symmetry* of the Bernstein polynomials, we have
+
+- $\sum_{k=0}^n \mathbf{P}_k B_k^n = \sum_{k=0}^n \mathbf{P}_k B_{n-k}^n$
+
+Thanks to the interpolation property, for $B_0^n$ and $B_n^n$, of the Bernstein polynomials at the endpoints, we get
+
+- $\mathcal{C}(0) = \mathbf{P}_0$  and $\mathcal{C}(1) = \mathbf{P}_n$ 
+
+Using the partition unity property of the Bernstein polynomials, we get
+
+- any point $\mathcal{C}(t)$ is an *affine combination* of the control points 
+
+Therefor,
+
+- Bézier curves are affinely invariant; *i.e.* the image curve $\Phi(\sum_{k=0}^n \mathbf{P}_k B_k^n)$ of a Bézier curve, by an affine mapping $\Phi$, is the Bézier curve having $\left( \Phi(\mathbf{P}_i) \right)_{0 \le i \le n}$ as control points. 
+
+due to the partition unity of the Bernstein polynomials and their non-negativity, 
+
+- any point $\mathcal{C}(t)$ is a *convex combination* of the control points 
+
+finally, we get the *convex hull property*,
+
+- A Bézier curve lies in the convex hull of its control points 
+
+## DeCasteljau Algorithm
+In this section, we introduce the deCasteljau algorithm Algo. (\ref{algo:decasteljau}). The basic idea comes from the following remark;
+using the reccurence formulae Eq. (\ref{eq:bernstein-rec}), we have
+
+$$
+\begin{align}
+  \mathcal{C}(t) &= \sum_{k=0}^n \mathbf{P}_k B_k^n(t) = \sum_{k=0}^n \mathbf{P}_k \left( (1-t) B_k^{n-1}(t) + t B_{k-1}^{n-1}(t) \right) \\ 
+                 &= \sum_{k=0}^{n-1} \mathbf{P}_k^1  B_k^{n-1}(t) \quad\quad\quad\quad\quad [\mathbf{P}_k^{1} := (1-t)\mathbf{P}_k + t \mathbf{P}_{k+1}] \\ 
+                 &= \sum_{k=0}^{n-2} \mathbf{P}_k^2  B_k^{n-2}(t) \quad\quad\quad\quad\quad [\mathbf{P}_k^{2} := (1-t)\mathbf{P}_k^1 + t \mathbf{P}_{k+1}^1] \\ 
+                 &= \ldots \\
+                 &= \sum_{k=0}^{0} \mathbf{P}_k^n  B_k^{0}(t) \quad\quad\quad\quad\quad [\mathbf{P}_k^{n} := (1-t)\mathbf{P}_k^{n-1} + t \mathbf{P}_{k+1}^{n-1}] \\ 
+                 &= \mathbf{P}_0^n 
+\end{align}
+$$
+
+```{prf:algorithm} Evaluation of a Bézier curve, defined by its control points $\mathbf{P}$ at $x$
+:label: algo-decasteljau
+**Input** $\mathbf{P}, x$
+**Output** $value$
+
+$n \gets { len } \left( \mathbf{P} \right) - 1$
+
+$\mathbf{Q} \gets \mathbf{P}$
+
+For $j \gets 0$ : $n$ 
+
+  For $i \gets 0$ : $n-j$
+
+    $Q[i] \gets (1-x) Q[i] + x Q[i+1]$
+
+Return $Q[0]$
+```
+
+```python
+def decasteljau(P, x):
+    n = len(P) - 1
+    Q = P.copy()
+    x1 = 1.-x
+    for j in range(1, n+1):
+        for i in range(0, n-j+1):
+            Q[i] = x1*Q[i] + x*Q[i+1]
+    c = Q[0]
+    return c
+```
+
+## Conversion from monomial form
+Let us consider a monomial representation of a curve 
+
+$$
+\begin{align}
+\mathcal{C}(t) = \sum_{k=0}^n \mathbf{Q}_k t^k 
+\end{align}
+$$
+
+Since
+
+$$
+\begin{align}
+  t^k &= \frac{1}{\binom{n}{k}} \binom{n}{k} t^k \left(1-t+t \right)^{n-k} 
+      = \frac{1}{\binom{n}{k}} \left( \sum\limits_{i=0}^{n-k} \binom{n}{k} \binom{n-k}{n-k-i} t^{i+k} \left( 1-t \right)^{n-k-i} \right) \\
+      &= \frac{1}{\binom{n}{k}} \left(  \sum\limits_{i=0}^{n-k} \binom{k+i}{k} B_{k+i}^n(t) \right)
+      = \frac{1}{\binom{n}{k}} \left(  \sum\limits_{j=0}^{n} \binom{j}{k} B_{j}^n(t) \right) 
+\end{align}
+$$
+
+we get,
+
+$$
+\begin{align}
+\mathcal{C}(t) = \sum_{k=0}^n \mathbf{Q}_k t^k = \sum_{k=0}^n \mathbf{Q}_k \frac{1}{\binom{n}{k}} \left(  \sum\limits_{j=0}^{n} \binom{j}{k} B_{j}^n(t) \right)  
+\end{align}
+$$
+
+hence,
+
+$$
+\begin{align}
+\mathcal{C}(t) = \sum_{j=0}^n \left(\sum\limits_{k=0}^{n} \frac{\binom{j}{k}}{\binom{n}{k}} \mathbf{Q}_k \right) B_{j}^n(t)  
+\end{align}
+$$
+
+which is a Bézier curve of the form
+
+$$
+\begin{align}
+\mathcal{C}(t) = \sum_{k=0}^n \mathbf{P}_k B_k^n(t)
+\end{align}
+$$
+
+with 
+
+$$
+\begin{align}
+\mathbf{P}_k =  \sum\limits_{k=0}^{n} \frac{\binom{j}{k}}{\binom{n}{k}} \mathbf{Q}_k 
+\end{align}
+$$
+
+```{prf:remark}
+  If $\mathbf{Q}_k = 0$ for all $k \geq 2$, then the control points are given by $\mathbf{P}_k = \mathbf{Q}_0 + k \mathbf{Q}_1$.
+```
+
+````{prf:proposition} Linear Precision
+  Conversly, if the $n+1$ control points $\mathbf{P}$ lie equidistantly on a line, then $\mathcal{C}$ is a linear polynomial, which can be written as $\mathcal{C}(t) = \left(1-t\right) \mathbf{P}_0 + t \mathbf{P}_n$. 
+````
+
+## Conversion to monomial form
+Given a Bézier curve $\mathcal{C}(t) = \sum_{k=0}^n \mathbf{P}_k B_k^n(t)$, we can derive its monomial representation using the Taylor expansion,
+
+$$
+\begin{align}
+  \mathcal{C}(t) &= \sum\limits_{k=0}^n \frac{1}{k!}\mathcal{C}^{(k)}(0) t^k \\
+                 &= \sum\limits_{k=0}^n \binom{n}{k} \triangle^k \mathbf{P}_0 t^k  
+\end{align}
+$$
+
+## Rational Bézier curves
+We first introduce the notion of Rational Bernstein polynomials defined as,  and $w_i > 0, \forall i \in \left[0, n \right]$ 
+
+$$
+\begin{align}
+  R_k^n(t) := \frac{w_i B_k^n(t)}{\sum_{i=0}^n w_i B_i^n(t)}
+  \label{eq:rational-bernstein}
+\end{align}
+$$
+
+````{prf:proposition} Properties of Rational Bernstein polynomials
+- $R_k^n(x) \ge 0$, for all $k \in \left[0, n \right]$ and $x \in \left[ 0, 1 \right]$ \hfill [positivity] 
+- $\sum_{k=0}^n R_k^n(x) = 1$, for all $x \in \left[ 0, 1 \right]$ \hfill [partition of unity] 
+- $R_0^n(0) = R_n^n(1) = 1$ 
+- $R_k^n$ has exactly one maximum on the interval $\left[ 0, 1 \right]$, 
+- Bernstein polynomials are Rational Bernstein polynomials when all weights are equal
+````
+
+````{prf:definition} Rational Bézier curve
+
+$$
+\begin{align}
+  \mathcal{C}(t) = \sum_{k=0}^n \mathbf{P}_k R_k^n(t) 
+  \label{eq:bezier-curve}
+\end{align}
+$$
+
+where $R_k^n(t) := \frac{w_i B_k^n(t)}{\sum_{i=0}^n w_i B_i^n(t)}$ and $w_i > 0, \forall i \in \left[0, n \right]$ 
+````  
+
+### Example
+
+````{prf:example}
+:label: bezier-curve-ex-7
+
+```python
+nt = 200
+ts = np.linspace(0., 1., nt)
+
+P = np.zeros((3, 2))
+P[:, 0] = [1., 1., 0.]
+P[:, 1] = [0., 1., 1.]
+
+# weights
+W = np.asarray([1., 1., 2.])
+
+# weithed control points in 3D
+Pw = np.zeros((3,3))
+for i in range(0, 3):
+    Pw[i,:2] = W[i]*P[i,:]
+    Pw[i,2]  = W[i]
+
+Qw = np.zeros((nt, 3))
+for i,t in enumerate(ts):
+    Qw[i,:] = point_on_bezier_curve(Pw,t)
+
+Q = np.zeros((nt, 2))
+Q[:,0] = Qw[:,0]/Qw[:,2]
+Q[:,1] = Qw[:,1]/Qw[:,2]
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, 3):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-0.2, 1.2, -0.2, 1.2])
+```
+    
+![png](images/bezier/output_25_0.png)
+````    
+    
+## Composite Bézier curve
+
+````{prf:example}
+:label: bezier-curve-ex-6
+
+```python
+n = 2
+
+nt = 200
+ts = np.linspace(0., 1., nt)
+
+# ...
+P = np.zeros((3, 2))
+P[:, 0] = [1., 1., 0.]
+P[:, 1] = [0., 1., 1.]
+
+Q = np.zeros((nt, 2))
+for i,t in enumerate(ts):
+    Q[i,:] = point_on_bezier_curve(P,t)
+
+plt.plot(Q[:,0], Q[:,1], '-r')
+plt.plot(P[:,0], P[:,1], '--or', linewidth=0.7)
+# ...
+
+# ...
+P = np.zeros((3, 2))
+P[:, 0] = [0., -1., 0.]
+P[:, 1] = [1., -1., -1.]
+
+Q = np.zeros((nt, 2))
+for i,t in enumerate(ts):
+    Q[i,:] = point_on_bezier_curve(P,t)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ob', linewidth=0.7)
+# ...
+
+# ...
+P = np.zeros((3, 2))
+P[:, 0] = [1., 1., 0.]
+P[:, 1] = [0., -1., -1.]
+
+Q = np.zeros((nt, 2))
+for i,t in enumerate(ts):
+    Q[i,:] = point_on_bezier_curve(P,t)
+
+plt.plot(Q[:,0], Q[:,1], '-g')
+plt.plot(P[:,0], P[:,1], '--og', linewidth=0.7)
+# ...
+```
+    
+![png](images/bezier/output_22_0.png)
+````    
diff --git a/_sources/chapter0/bezier.md b/_sources/chapter0/bezier.md
new file mode 100644
index 0000000..e999a10
--- /dev/null
+++ b/_sources/chapter0/bezier.md
@@ -0,0 +1,74 @@
+# Bernstein polynomials
+
+Without loss of generality, we restrict to the case of the unit interval, namely $a=0$ and $b=1$.
+In figure (Fig. \ref{fig:bernstein-polynomials}), we plot the first sixth Bernstein polynomials.
+
+
+````{prf:definition}
+
+Fro $n \in \mathbb{N}$, the $n$-th degree Bernstein polynomials $B_{j,n}\,:\, [0,1] \longrightarrow \mathbb{R}$ are defined by
+
+$$
+B_{j,n}(x)=\frac{n!}{(n-j)!j!} x^j (1-x)^{n-j}, \quad x \in [0,1], \quad j=0, \ldots,n.
+$$
+````
+
+Bernstein polynomials exhibit interesting properties highlighted in the following proposition,
+
+```python
+# needed imports
+import numpy as np
+import matplotlib.pyplot as plt
+```
+
+We first consider the evaluation of the Bernstein polynomials. 
+
+## Evaluation of Bernstein polynomials
+
+The following function evaluates all Bernstein polynomials of degree $n$ at $x$
+
+```python
+def all_bernstein(n, x):
+    b = np.zeros(n+1)
+    b[0] = 1.
+    x1 = 1.-x
+    for j in range(1, n+1):
+        saved = 0.
+        for i in range(0, j):
+            tmp = b[i]
+            b[i] = saved + x1*tmp
+            saved = x*tmp
+        b[j] = saved
+    return b
+```
+
+## Bernstein polynomials properties
+
+````{prf:proposition}
+
+We have the following properties of Bernstein Polynomials,
+
+- Positivity: $B_{j,n}(x) \ge 0$, for all $x \in \left[ 0, 1 \right]$ and $0 \leq j \leq n$.
+- Partition of unity: $\sum_{j=0}^n B_{j,n}(x) = 1$, for all $x \in \left[ 0, 1 \right]$.
+- $B_{0,n}(0) = B_{n,n}(1) = 1$. 
+- $B_{j,n}$ has exactly one maximum on the interval $\left[ 0, 1 \right]$, at $\frac{j}{n}$. 
+- Symmetry: $B_{j,n}$ is symmetric with respect to $x = \frac{1}{2}$ for $j=0,\ldots,n$.
+- Bernstein polynomials can be defined recursively using the formula 
+
+$$
+\begin{align}
+  B_{j,n}(x) = (1-x) B_{j,n-1}(x) + x B_{k-1,n-1}(x),
+\end{align}
+$$
+
+  where we assume $B_{j,n}(x) = 0$ if $j < 0$ or $j > n$.
+
+- Bernstein derivatives can be computed using the formulae  
+
+$$
+\begin{align}
+  {B_{j,n}}^\prime(x) = n \left(B_{j-1,n-1}(x) - B_{j,n-1}(x) \right).
+\end{align}
+$$
+using the same assumption as before.
+````
diff --git a/_sources/chapter0/bsplines-curves.md b/_sources/chapter0/bsplines-curves.md
new file mode 100644
index 0000000..07b808a
--- /dev/null
+++ b/_sources/chapter0/bsplines-curves.md
@@ -0,0 +1,593 @@
+# B-Splines curves
+
+Let $(\mathbf{P}_i)_{ 0 \leqslant i \leqslant n}\in \mathbb{R}^d$  be a sequence of control points. Following the same approach as for Bézier curves, we define B-Splines curves as
+
+````{prf:definition} B-Spline curve
+The B-spline curve in $\mathbb{R}^d$ associated to $T=(t_i)_{0\leqslant i \leqslant n + p + 1}$ and $(\mathbf{P}_i)_{ 0 \leqslant i \leqslant n}$ is defined by :
+
+$$
+\mathcal{C}(t) = \sum_{i=0}^n N_i^p(t) \textbf{P}_i
+$$
+
+````
+
+```{prf:remark}
+  The use of open knot vector leads to an **interpolating curve** at the endpoints.
+```
+
+## Properties of B-Splines curves
+
+We have the following properties for a B-spline curve:
+
+
+- If $n=p$, then $\mathcal{C}$ is a Bézier-curve,
+- $\mathcal{C}$ is a piecewise polynomial curve,
+- The curve interpolates its extremas if the associated multiplicity of the first and the last knot are maximum (*i.e.* equal to $p+1$),
+- Invariance with respect to affine transformations,
+- B-Spline curves are affinely invariant; *i.e.* the image curve $\Phi(\sum_{i=0}^n \mathbf{P}_i N_i^p)$ of a B-Spline curve, by an affine mapping $\Phi$, is the B-Spline curve having $\left( \Phi(\mathbf{P}_i) \right)_{0 \le i \le n}$ as control points and the same knot vector,
+- strong convex-hull property:
+  if $t_j \leq t \leq t_{j+1}$, then $\mathcal{C}(t)$ is inside the convex-hull associated to the control points $\mathbf{P}_{j-p},\cdots,\mathbf{P}_{j}$,
+- local modification : moving $\mathbf{P}_{j}$ affects $\mathcal{C}(t)$, only in the interval $[t_j,t_{j+p+1}]$,
+- the control polygon is a linear approximation of the curve. We will see later that the control polygon converges to the curve under knot insertion and degree elevation (with different *speeds*).
+- *Variation diminishing*: no plane intersects the curve more than the control polygon.
+
+
+## Derivative of a B-spline curve
+Using the derivative formula for *B-spline*, we can compute the derivative of a B-Spline curve:
+
+$$
+\begin{align}
+\mathcal{C}^{\prime}(t) &= \sum_{i=0}^{n} \left( {N_{i}^{p}}^{\prime}(t) \mathbf{P}_i \right) 
+\\
+                        &= \sum_{i=0}^{n} \left( \frac{p}{t_{i+p}-t_{i}}N_{i}^{p-1}(t) \mathbf{P}_i - \frac{p}{t_{i+1+p}-t_{i+1}}N_{i+1}^{p-1}(t) \mathbf{P}_i \right)
+\\
+                        &= \sum_{i=0}^{n} \frac{p}{t_{i+p}-t_{i}}N_{i}^{p-1}(t) \mathbf{P}_i - \sum_{i=1}^{n+1} \frac{p}{t_{i+p}-t_{i}}N_{i}^{p-1}(t) \mathbf{P}_{i-1}
+\\
+                        &= \sum\limits_{i} {N_{i}^{p-1}}(t)\frac{p}{t_{i+p}-t_{i}} \left( \mathbf{P}_i - \mathbf{P}_{i-1} \right) 
+\\
+                        &= \sum\limits_{i} {N_{i}^{p-1}}(t) \frac{p}{t_{i+p}-t_{i}} \nabla \mathbf{P}_i
+\end{align}
+$$
+
+where we introduced the **first backward difference** operator $\nabla \mathbf{P}_i := \mathbf{P}_i - \mathbf{P}_{i-1}$.
+
+````{prf:example}
+:label: bsplines-curves-ex-1
+
+we consider a quadratic B-Spline curve with the knot vector $T=\{000~\frac{2}{5}~\frac{3}{5}~111 \}$. 
+
+We have $\mathcal{C}(t) = \sum_{i=0}^{4} {N_{i}^{2}}^{\prime}(t) \mathbf{P}_i $, then
+
+$$
+\mathcal{C}^{\prime}(t) = \sum\limits_{i} {N_{i}^{1}}(t) \mathbf{Q}_i
+$$
+
+where 
+
+$$
+\mathbf{Q}_0 = 5 \{\mathbf{P}_{2} - \mathbf{P}_1\} ~~~~\mathbf{Q}_1 = \frac{10}{3} \{ \mathbf{P}_{3} - \mathbf{P}_2\}
+$$
+
+$$
+\mathbf{Q}_2 = \frac{10}{3} \{ \mathbf{P}_{4} - \mathbf{P}_3\}~~~~\mathbf{Q}_3 = 5 \{\mathbf{P}_{5} - \mathbf{P}_4\}
+$$
+
+the *B-splines* $\{ N_{i}^{1},~0 \leq i \leq 3\}$ are associated to the knot vector $T^{\ast}=\{00~\frac{2}{5}~\frac{3}{5}~11 \}$.
+````
+
+```python
+# needed imports
+import numpy as np
+import matplotlib.pyplot as plt
+```
+
+```python
+# importing bsplines utilities
+from bsplines_utilities import find_span, all_bsplines, point_on_bspline_curve
+```
+
+## Examples
+
+````{prf:example}
+:label: bsplines-curves-ex-2
+
+```python
+knots  = [0., 0., 0., 1., 1., 1.]
+degree = 2
+n      = len(knots) - degree - 1
+
+P = np.zeros((n, 2))
+P[:, 0] = [1., 1., 0.]
+P[:, 1] = [0., 1., 1.]    
+
+nx = 101
+xs = np.linspace(0., 1., nx)
+
+Q = np.zeros((nx, 2))
+for i,x in enumerate(xs):
+    Q[i,:] = point_on_bspline_curve(knots, P, x)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, n):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-0.2, 1.2, -0.2, 1.2])
+```
+    
+![png](images/bsplines-curves/output_5_0.png)
+````    
+
+````{prf:example}
+:label: bsplines-curves-ex-3
+
+```python
+knots  = [0., 0., 0., 0., 1., 1., 1., 1.]
+degree = 3
+n      = len(knots) - degree - 1
+
+P = np.zeros((n, 2))
+P[:, 0] = [1., 1., 0., -1.]
+P[:, 1] = [0., 1., 1.,  0.]  
+
+nx = 101
+xs = np.linspace(0., 1., nx)
+
+Q = np.zeros((nx, 2))
+for i,x in enumerate(xs):
+    Q[i,:] = point_on_bspline_curve(knots, P, x)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, n):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-1.2, 1.2, -0.2, 1.2])
+```
+    
+![png](images/bsplines-curves/output_6_0.png)
+````    
+
+````{prf:example}
+:label: bsplines-curves-ex-4
+
+```python
+knots  = [0., 0., 0., 0.25, 0.5, 0.75, 1., 1., 1.]
+degree = 2
+n      = len(knots) - degree - 1
+
+P = np.zeros((n, 2))
+P[:, 0] = [0.,  1., 1., -2., -1., -2.]
+P[:, 1] = [.5, -1., 1.,  1.,  0., -1.]   
+
+nx = 101
+xs = np.linspace(0., 1., nx)
+
+Q = np.zeros((nx, 2))
+for i,x in enumerate(xs):
+    Q[i,:] = point_on_bspline_curve(knots, P, x)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, n):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-2.2, 1.3, -1.2, 1.4])
+```
+    
+![png](images/bsplines-curves/output_7_0.png)
+````    
+
+````{prf:example}
+:label: bsplines-curves-ex-5
+
+```python
+knots  = [0., 0., 0., 0.25, 0.5, 0.5, 0.75, 1., 1., 1.]
+degree = 2
+n      = len(knots) - degree - 1
+
+P = np.zeros((n, 2))
+P[:, 0] = [0.,  1., 1., 0., -2., -1., -2.]
+P[:, 1] = [.5, -1., 1., 2.,  1.,  0., -1.]   
+
+nx = 101
+xs = np.linspace(0., 1., nx)
+
+Q = np.zeros((nx, 2))
+for i,x in enumerate(xs):
+    Q[i,:] = point_on_bspline_curve(knots, P, x)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, n):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-2.2, 1.3, -1.2, 2.2])
+```
+    
+![png](images/bsplines-curves/output_8_0.png)
+````    
+
+````{prf:example}
+:label: bsplines-curves-ex-6
+
+```python
+knots  = [0., 0., 0., 0.25, 0.5, 0.5, 0.75, 1., 1., 1.]
+degree = 2
+n      = len(knots) - degree - 1
+
+P = np.zeros((n, 2))
+P[:, 0] = [0.,  1., 1., 0., -2., -1., -2.]
+P[:, 1] = [.5, -1., 1., 1.,  1.,  0., -1.]   
+
+nx = 101
+xs = np.linspace(0., 1., nx)
+
+Q = np.zeros((nx, 2))
+for i,x in enumerate(xs):
+    Q[i,:] = point_on_bspline_curve(knots, P, x)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, n):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-2.2, 1.3, -1.2, 1.2])
+```
+    
+![png](images/bsplines-curves/output_9_0.png)
+````    
+
+````{prf:example}
+:label: bsplines-curves-ex-7
+
+```python
+knots  = [0., 0., 0., 0., 0., 0., 0., 1., 1., 1., 1., 1., 1., 1.]
+degree = 6
+n      = len(knots) - degree - 1
+
+P = np.zeros((n, 2))
+P[:, 0] = [7., 6., 5.,  4., 3., 2., 1.]
+P[:, 1] = [0., 1., 1., -1., .5, -.5, 0.]   
+
+nx = 101
+xs = np.linspace(0., 1., nx)
+
+Q = np.zeros((nx, 2))
+for i,x in enumerate(xs):
+    Q[i,:] = point_on_bspline_curve(knots, P, x)
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, n):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([0.6, 7.4, -1.4, 1.4])
+```
+    
+![png](images/bsplines-curves/output_10_0.png)
+````    
+
+````{prf:example}
+:label: bsplines-curves-ex-8
+
+```python
+    knots  = [0., 0., 0., .2, .4, .6, .8, 1., 1., 1.]
+    degree = 2
+    n      = len(knots) - degree - 1
+    
+    P = np.zeros((n, 2))
+    P[:, 0] = [7., 6., 5.,  4., 3., 2., 1.]
+    P[:, 1] = [0., 1., 1., -1., .5, -.5, 0.]   
+    
+    nx = 101
+    xs = np.linspace(0., 1., nx)
+    
+    Q = np.zeros((nx, 2))
+    for i,x in enumerate(xs):
+        Q[i,:] = point_on_bspline_curve(knots, P, x)
+
+    plt.plot(Q[:,0], Q[:,1], '-b')
+    plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+    
+    for i in range(0, n):
+        x,y = P[i,:]
+        plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+    plt.axis([0.6, 7.4, -1.4, 1.4])
+```
+    
+![png](images/bsplines-curves/output_11_0.png)
+````    
+
+## Rational B-Splines (NURBS) curves
+
+We recall that NURBS are defined as
+
+$$
+  R_i^p(t) := \frac{w_i N_i^p(t)}{\sum_{j=0}^n w_j N_j^p(t)}
+$$
+
+where $w_i > 0, \forall i \in \left[0, n \right]$ are non-negative real numbers, called the weights.
+
+NURBS curves are then defined as
+
+$$
+  \mathcal{C}(t) = \sum_{i=0}^n \mathbf{P}_i R_i^p(t) 
+$$
+
+Notice that a NURBS curve in $\mathbb{R}^d$ can be described as a NURBS curve in $\mathbb{R}^{d+1}$ using the control points:
+
+$\textbf{P}^{\omega}_i = ( \omega_i \textbf{P}_i, \omega_i )$
+
+This remark is used for the evaluation and also to extend most of the B-Splines fundamental geometric operations to NURBS curves.
+
+```{prf:remark}
+Although there is a function **point_on_nurbs_curve** to evaluate NURBS curves, we make use of an explicit call to the spline version in this notebook.
+```
+
+### Examples
+
+````{prf:example} Quarter circle 
+:label: nurbs-curves-ex-1
+
+```python
+knots  = [0., 0., 0., 1., 1., 1.]
+degree = 2
+n      = len(knots) - degree - 1
+
+P = np.zeros((n, 2))
+P[:, 0] = [1., 1., 0.]
+P[:, 1] = [0., 1., 1.]
+
+# weights
+s2 = 1./np.sqrt(2)
+W = np.zeros(n)
+W[:] = [1., s2, 1.]
+
+# weithed control points in 3D
+Pw = np.zeros((n,3))
+for i in range(0, n):
+    Pw[i,:2] = W[i]*P[i,:]
+    Pw[i,2]  = W[i]
+
+nx = 200
+xs = np.linspace(0., 1., nx)    
+
+Qw = np.zeros((nx, 3))
+for i,x in enumerate(xs):
+    Qw[i,:] = point_on_bspline_curve(knots, Pw, x)
+
+Q = np.zeros((nx, 2))
+Q[:,0] = Qw[:,0]/Qw[:,2]
+Q[:,1] = Qw[:,1]/Qw[:,2]
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, n):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-0.2, 1.2, -0.2, 1.2])    
+```
+    
+![png](images/bsplines-curves/output_15_0.png)
+````    
+
+````{prf:example} Circular arc of $120°$ 
+:label: nurbs-curves-ex-2
+
+```python
+knots  = [0., 0., 0., 1., 1., 1.]
+degree = 2
+n      = len(knots) - degree - 1
+
+P = np.zeros((n, 2))
+a = np.cos(np.pi/6.)
+P[:, 0] = [ a, 0., -a]
+P[:, 1] = [.5, 2., .5]
+
+# weights
+W = np.zeros(n)
+W[:] = [1., 1./2., 1.]
+
+# weithed control points in 3D
+Pw = np.zeros((n,3))
+for i in range(0, n):
+    Pw[i,:2] = W[i]*P[i,:]
+    Pw[i,2]  = W[i]
+
+nx = 200
+xs = np.linspace(0., 1., nx)    
+
+Qw = np.zeros((nx, 3))
+for i,x in enumerate(xs):
+    Qw[i,:] = point_on_bspline_curve(knots, Pw, x)
+
+Q = np.zeros((nx, 2))
+Q[:,0] = Qw[:,0]/Qw[:,2]
+Q[:,1] = Qw[:,1]/Qw[:,2]
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, n):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-1.2, 1.2, .0, 2.2])    
+```
+    
+![png](images/bsplines-curves/output_17_0.png)
+````    
+
+````{prf:example} half circle 
+:label: nurbs-curves-ex-3
+
+```python
+knots  = [0., 0., 0., 0., 1., 1., 1., 1.]
+degree = 3
+n      = len(knots) - degree - 1
+
+P = np.zeros((n, 2))
+P[:, 0] = [1., 1., -1., -1.]
+P[:, 1] = [0., 2.,  2.,  0.]
+
+# weights
+W = np.zeros(n)
+W[:] = [1., 1./3., 1./3., 1.]
+
+# weithed control points in 3D
+Pw = np.zeros((n,3))
+for i in range(0, n):
+    Pw[i,:2] = W[i]*P[i,:]
+    Pw[i,2]  = W[i]
+
+nx = 200
+xs = np.linspace(0., 1., nx)    
+
+Qw = np.zeros((nx, 3))
+for i,x in enumerate(xs):
+    Qw[i,:] = point_on_bspline_curve(knots, Pw, x)
+
+Q = np.zeros((nx, 2))
+Q[:,0] = Qw[:,0]/Qw[:,2]
+Q[:,1] = Qw[:,1]/Qw[:,2]
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(0, n):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+
+plt.axis([-1.2, 1.2, -.2, 2.2])    
+```
+    
+![png](images/bsplines-curves/output_19_0.png)
+````    
+
+````{prf:example} Circle as 4 arcs 
+:label: nurbs-curves-ex-4
+
+We already saw how to construct a circular arc (with angle = $\frac{\pi}{2}$, using Bézier curves.
+
+Constructing a circular arc can be done in different ways. In this example, we consider a B-Spline curve as a composite of 4 Bézier curves.
+
+
+```python
+knots  = [0., 0., 0., .25, .25, .5, .5, .75, .75, 1., 1., 1.]
+degree = 2
+n      = len(knots) - degree - 1
+
+P = np.zeros((n, 2))
+P[:, 0] = [1., 1., 0., -1., -1., -1.,  0.,  1., 1.]
+P[:, 1] = [0., 1., 1.,  1.,  0., -1., -1., -1., 0.]
+
+# weights
+s2 = 1./np.sqrt(2)
+W = np.zeros(n)
+W[:] = [1., s2, 1., s2, 1., s2, 1., s2, 1.]
+
+# weithed control points in 3D
+Pw = np.zeros((n,3))
+for i in range(0, n):
+    Pw[i,:2] = W[i]*P[i,:]
+    Pw[i,2]  = W[i]
+
+nx = 200
+xs = np.linspace(0., 1., nx)    
+
+Qw = np.zeros((nx, 3))
+for i,x in enumerate(xs):
+    Qw[i,:] = point_on_bspline_curve(knots, Pw, x)
+
+Q = np.zeros((nx, 2))
+Q[:,0] = Qw[:,0]/Qw[:,2]
+Q[:,1] = Qw[:,1]/Qw[:,2]
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(1, n-1):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+    
+# plot P_0 = P_8
+x,y = P[0,:]
+plt.text(x+0.05,y+0.05,'$\mathbf{P}_0 = \mathbf{P}_8$')
+
+plt.axis([-1.2, 1.2, -1.2, 1.2])    
+```
+    
+![png](images/bsplines-curves/output_21_0.png)
+````    
+
+````{prf:example} Circle as 3 arcs 
+:label: nurbs-curves-ex-5
+
+We already saw how to construct a circular arc (with angle = $120°$, using Bézier curves.
+
+In this example, we consider a B-Spline curve as a composite of 3 Bézier curves.
+
+
+```python
+knots  = [0., 0., 0., 1./3., 1./3., 2./3., 2./3., 1., 1., 1.]
+degree = 2
+n      = len(knots) - degree - 1
+
+P = np.zeros((n, 2))
+a = np.cos(np.pi/6.)
+P[:, 0] = [ a, 0., -a, -2*a,  0., 2*a,  a]
+P[:, 1] = [.5, 2., .5,  -1., -1., -1., .5]
+
+# weights
+W = np.zeros(n)
+W[:] = [1., 1./2., 1., 1./2., 1., 1./2., 1.]
+
+# weithed control points in 3D
+Pw = np.zeros((n,3))
+for i in range(0, n):
+    Pw[i,:2] = W[i]*P[i,:]
+    Pw[i,2]  = W[i]
+
+nx = 200
+xs = np.linspace(0., 1., nx)    
+
+Qw = np.zeros((nx, 3))
+for i,x in enumerate(xs):
+    Qw[i,:] = point_on_bspline_curve(knots, Pw, x)
+
+Q = np.zeros((nx, 2))
+Q[:,0] = Qw[:,0]/Qw[:,2]
+Q[:,1] = Qw[:,1]/Qw[:,2]
+
+plt.plot(Q[:,0], Q[:,1], '-b')
+plt.plot(P[:,0], P[:,1], '--ok', linewidth=0.7)
+
+for i in range(1, n-1):
+    x,y = P[i,:]
+    plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')
+    
+# plot P_0 = P_6
+x,y = P[0,:]
+plt.text(x+0.05,y+0.05,'$\mathbf{P}_0 = \mathbf{P}_8$')
+
+plt.axis([-2.2, 2.2, -1.2, 2.2])    
+```
+    
+![png](images/bsplines-curves/output_23_0.png)
+````
diff --git a/_sources/chapter0/bsplines-operations.md b/_sources/chapter0/bsplines-operations.md
new file mode 100644
index 0000000..73751af
--- /dev/null
+++ b/_sources/chapter0/bsplines-operations.md
@@ -0,0 +1,489 @@
+# Fundamental geometric operations for B-Splines
+
+
+Having more control on a curve, adding new control points, can be done in two different ways:
+
+1. insert new knots
+
+2. elevate the polynomial degree
+
+In general, one way need to use both approaches.
+For each strategy, we shall consider one algorithm.
+
+**Remark**: 
+In this notebook we only use B-Splines curves, but similar functions are also available for NURBS.
+
+```python
+# needed imports
+import numpy as np
+import matplotlib.pyplot as plt
+```
+
+## Knot insertion
+
+Assuming an initial B-Spline curve defined by:
+
+* its degree $p$
+* knot vector $T=(t_i)_{0\leqslant i \leqslant n + p + 1}$
+* control points $(\mathbf{P}_i)_{ 0 \leqslant i \leqslant n}$
+
+We are interested in the new B-Spline curve as the result of the insertion of the knot $t$, $m$ times (with a span $j$, i.e. $t_j \leqslant t < t_{j+1}$).
+
+After such operation, the degree is remain unchanged, while the knot vector is enriched by $t$, $m$ times. The aim is then to compute the new control points $(\mathbf{Q}_i)_{ 0 \leqslant i \leqslant n + m}$
+
+For this purpose we use the DeBoor algorithm:
+
+$$
+\begin{align}
+   n &:= n+m
+   \\
+   p &:= p
+   \\
+   T &:= \{ t_0,.., t_j, \underbrace{t, ..., t}_{m}, t_{j+1},.., t_{n+k}\}
+   \\
+   \textbf{Q}_i &:= \textbf{Q}_i^{m}
+\end{align}
+$$
+
+where
+
+$$
+\begin{align}
+   \textbf{Q}_i^0 &= \textbf{P}_i
+   \\
+   \textbf{Q}_i^r &= \alpha_i^r \textbf{Q}_i^{r-1} + (1-\alpha_i^r) \textbf{Q}_{i-1}^{r-1}
+\end{align}
+$$
+
+with
+
+$$
+\begin{align}
+   \alpha_i^r &= 
+      \left\{\begin{array}{cc}
+         1 &  i \leqslant j-p+r-1 \\
+         \frac{t-t_i}{t_{i+p-r+1}-t_i} & j-p+r \leqslant i \leqslant j-m \\
+         0 & j-m+1 \leqslant i 
+      \end{array}\right.   
+\end{align}
+$$
+
+The following python function implements the previous algorithm. 
+
+
+```python
+# import the knot insertion function
+from bsplines_utilities import insert_knot_bspline_curve
+```
+
+From now on, we shall also use the following function to plot curves and their control points:
+
+
+```python
+from bsplines_utilities import plot_curve
+```
+
+**Examples**
+
+
+```python
+def example_1(times=1):
+    knots  = [0., 0., 0., 0., 1., 1., 1., 1.]
+    degree = 3
+    n      = len(knots) - degree - 1
+
+    P = np.zeros((n, 2))
+    P[:, 0] = [1., 1., 0., -1.]
+    P[:, 1] = [0., 1., 1.,  0.]
+    plot_curve(knots, degree, P, with_ctrl_pts=False)
+    
+    # we don't plot the first and last points
+    # they will be plotted later
+    for i in range(1, n-1):
+        x,y = P[i,:]
+        plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')    
+
+    t = 0.5
+    knots, degree, P = insert_knot_bspline_curve(knots, degree, P, t, times=times)
+    plot_curve(knots, degree, P, color='r', label='Q', with_ctrl_pts=False)
+    
+    # we don't plot the first and last points
+    # they will be plotted later
+    n = len(knots) - degree - 1
+    for i in range(1, n-1):
+        x,y = P[i,:]
+        plt.text(x+0.05,y+0.05,'$\mathbf{Q}_{' + str(i) + '}$')    
+     
+    # plot endpoints
+    for i in [0, n-1]:
+        x = P[i,0] ; y = P[i,1]
+        if i == 0:
+            Pi = '\mathbf{P}_{' + str(i) + '}'
+        elif i == n-1:
+            Pi = '\mathbf{P}_{' + str(i-1) + '}'
+            
+        Qi = '\mathbf{Q}_{' + str(i) + '}'
+        text = '${Pi} = {Qi}$'.format(Pi=Pi, Qi=Qi)
+        plt.text(x+0.05,y+0.05,text)     
+```
+
+
+```python
+example_1(times=1)
+```
+
+
+    
+![png](images/bsplines-operations/output_9_0.png)
+    
+
+
+
+```python
+example_1(times=2)
+```
+
+
+    
+![png](images/bsplines-operations/output_10_0.png)
+    
+
+
+
+```python
+example_1(times=3)
+```
+
+
+    
+![png](images/bsplines-operations/output_11_0.png)
+    
+
+
+
+```python
+def example_2a(nt = 3):
+    knots  = [0., 0., 0., 0., 1., 1., 1., 1.]
+    degree = 3
+    n      = len(knots) - degree - 1
+
+    P = np.zeros((n, 2))
+    P[:, 0] = [1., 1., 0., -1.]
+    P[:, 1] = [0., 1., 1.,  0.]
+    plot_curve(knots, degree, P, with_ctrl_pts=False)
+    
+    # we don't plot the first and last points
+    # they will be plotted later
+    for i in range(1, n-1):
+        x,y = P[i,:]
+        plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')    
+
+    ts = np.linspace(0., 1., nt+2)[1:-1]
+    for t in ts:
+        knots, degree, P = insert_knot_bspline_curve(knots, degree, P, t, times=1)
+        
+    plot_curve(knots, degree, P, color='r', label='Q', with_ctrl_pts=False)
+    
+    # we don't plot the first and last points
+    # they will be plotted later
+    n = len(knots) - degree - 1
+    for i in range(1, n-1):
+        x,y = P[i,:]
+        plt.text(x+0.05,y+0.05,'$\mathbf{Q}_{' + str(i) + '}$')    
+     
+    # plot endpoints
+    for i in [0, n-1]:
+        x = P[i,0] ; y = P[i,1]
+        if i == 0:
+            Pi = '\mathbf{P}_{' + str(i) + '}'
+        elif i == n-1:
+            Pi = '\mathbf{P}_{' + str(i-1) + '}'
+            
+        Qi = '\mathbf{Q}_{' + str(i) + '}'
+        text = '${Pi} = {Qi}$'.format(Pi=Pi, Qi=Qi)
+        plt.text(x+0.05,y+0.05,text)     
+```
+
+
+```python
+# curve after inserting [0.25 0.5  0.75]
+example_2a(nt=3) 
+```
+
+
+    
+![png](images/bsplines-operations/output_13_0.png)
+    
+
+
+
+```python
+# curve after inserting [0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9]
+example_2a(nt=9) 
+```
+
+
+    
+![png](images/bsplines-operations/output_14_0.png)
+    
+
+
+## Order elevation
+
+There are also many algorithms for degree elevation of a B-Spline curve. In the sequel, we will be using the one developped by Huand, Hu and Martin (2005).
+
+The previous algorithm is implemented by the following function
+
+
+```python
+# import the degree elevation function
+from bsplines_utilities import elevate_degree_bspline_curve
+```
+
+**Example 1.**
+
+
+```python
+def example_1(m=1):
+    knots  = [0., 0., 0., 0., 0.5, 1., 1., 1., 1.]
+    degree = 3
+    n      = len(knots) - degree - 1
+
+    P = np.zeros((n, 2))
+    P[:, 0] = [1., 1., .5, -.5, -1.]
+    P[:, 1] = [0., .5, 1.,  .5,  0.]
+
+    plot_curve(knots, degree, P, with_ctrl_pts=False)
+    
+    # we don't plot the first and last points
+    # they will be plotted later
+    for i in range(1, n-1):
+        x,y = P[i,:]
+        plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')    
+    
+    knots, degree, P = elevate_degree_bspline_curve(knots, degree, P, m=1)
+    plot_curve(knots, degree, P, color='r', label='Q', with_ctrl_pts=False)
+        
+    # we don't plot the first and last points
+    # they will be plotted later
+    n = len(knots) - degree - 1
+    for i in range(1, n-1):
+        x,y = P[i,:]
+        plt.text(x+0.05,y+0.05,'$\mathbf{Q}_{' + str(i) + '}$')    
+
+    # plot endpoints
+    for i in [0, n-1]:
+        x = P[i,0] ; y = P[i,1]
+        if i == 0:
+            Pi = '\mathbf{P}_{' + str(i) + '}'
+        elif i == n-1:
+            Pi = '\mathbf{P}_{' + str(i-1) + '}'
+            
+        Qi = '\mathbf{Q}_{' + str(i) + '}'
+        text = '${Pi} = {Qi}$'.format(Pi=Pi, Qi=Qi)
+        plt.text(x+0.05,y+0.05,text)     
+    
+example_1()
+```
+
+
+    
+![png](images/bsplines-operations/output_18_0.png)
+    
+
+
+**Example 2.**
+
+
+```python
+def example_2(m=1):
+    knots  = [0., 0., 0., 0., 0.5, 0.5, 1., 1., 1., 1.]
+    degree = 3
+    n      = len(knots) - degree - 1
+    n_initial = n
+
+    P = np.zeros((n, 2))
+    P[:, 0] = [260., 100., 260., 420., 580., 420.]
+    P[:, 1] = [100., 260., 420., 420., 260., 100.]
+    plot_curve(knots, degree, P, with_ctrl_pts=False)
+    
+    # we don't plot the first and last points
+    # they will be plotted later
+    for i in range(1, n-1):
+        x,y = P[i,:]
+        plt.text(x+0.05,y+0.05,'$\mathbf{P}_{' + str(i) + '}$')        
+
+    knots, degree, P = elevate_degree_bspline_curve(knots, degree, P, m=m)
+    plot_curve(knots, degree, P, color='r', label='Q', with_ctrl_pts=False)    
+        
+    # we don't plot the first and last points
+    # they will be plotted later
+    n = len(knots) - degree - 1
+    for i in range(1, n-1):
+        x,y = P[i,:]
+        plt.text(x+0.05,y+0.05,'$\mathbf{Q}_{' + str(i) + '}$')    
+
+    # plot endpoints
+    for i in [0, n-1]:
+        x = P[i,0] ; y = P[i,1]
+        if i == 0:
+            Pi = '\mathbf{P}_{' + str(i) + '}'
+        elif i == n-1:
+            Pi = '\mathbf{P}_{' + str(n_initial-1) + '}'
+            
+        Qi = '\mathbf{Q}_{' + str(i) + '}'
+        text = '${Pi} = {Qi}$'.format(Pi=Pi, Qi=Qi)
+        plt.text(x+0.05,y+0.05,text)         
+```
+
+
+```python
+example_2(m=1)
+```
+
+
+    
+![png](images/bsplines-operations/output_21_0.png)
+    
+
+
+
+```python
+example_2(m=2)
+```
+
+
+    
+![png](images/bsplines-operations/output_22_0.png)
+    
+
+
+
+```python
+example_2(m=4)
+```
+
+
+    
+![png](images/bsplines-operations/output_23_0.png)
+    
+
+
+
+```python
+example_2(m=8)
+```
+
+
+    
+![png](images/bsplines-operations/output_24_0.png)
+    
+
+
+## Translation
+
+Since B-Spline curves are invariant under affine transformations, whan translating a curve with a given displacement, the new curve has the same knot vector while the control points are obtained by applying the same translation to the initial ones.
+
+The following function implements this utility
+
+
+```python
+# import the translation of a B-Spline curve
+from bsplines_utilities import translate_bspline_curve
+```
+
+
+```python
+def example_1():
+    knots  = [0., 0., 0., 0., 1., 1., 1., 1.]
+    degree = 3
+    n      = len(knots) - degree - 1
+
+    P = np.zeros((n, 2))
+    P[:, 0] = [1., 1., 0., -1.]
+    P[:, 1] = [0., 1., 1.,  0.]
+    plot_curve(knots, degree, P)
+    
+    knots, P = translate_bspline_curve(knots, P, [1., 1.5])
+    plot_curve(knots, degree, P, color='r', label='Q')
+    
+example_1()    
+```
+
+
+    
+![png](images/bsplines-operations/output_27_0.png)
+    
+
+
+## Rotation
+
+Rotating a B-Spline curve with an angle $\theta$ is done by applying it on the control points, as for the translation algorithm.
+
+The following function implements this utility.
+
+
+```python
+# import the rotation of a B-Spline curve
+from bsplines_utilities import rotate_bspline_curve
+```
+
+
+```python
+def example_1():
+    knots  = [0., 0., 0., 0., 1., 1., 1., 1.]
+    degree = 3
+    n      = len(knots) - degree - 1
+
+    P = np.zeros((n, 2))
+    P[:, 0] = [1., 1., 0., -1.]
+    P[:, 1] = [0., 1., 1.,  0.]
+    plot_curve(knots, degree, P)
+    
+    knots, P = rotate_bspline_curve(knots, P, np.pi/3., center=[-1.,0.])
+    plot_curve(knots, degree, P, color='r', label='Q')
+    
+example_1() 
+```
+
+
+    
+![png](images/bsplines-operations/output_30_0.png)
+    
+
+
+## Homothetic transformation
+
+The following function implements a hmothetic transformation for B-Spline curves.
+
+
+```python
+# import the homothetic transformation of a B-Spline curve
+from bsplines_utilities import homothetic_bspline_curve
+```
+
+
+```python
+def example_1():
+    knots  = [0., 0., 0., 0., 1., 1., 1., 1.]
+    degree = 3
+    n      = len(knots) - degree - 1
+
+    P = np.zeros((n, 2))
+    P[:, 0] = [1., 1., 0., -1.]
+    P[:, 1] = [0., 1., 1.,  0.]
+    plot_curve(knots, degree, P)
+    
+    knots, P = homothetic_bspline_curve(knots, P, 0.5, center=[0.,0.])
+    plot_curve(knots, degree, P, color='r', label='Q')
+    
+example_1() 
+```
+
+
+    
+![png](images/bsplines-operations/output_33_0.png)
+    
+
+
+
diff --git a/_sources/chapter0/bsplines-surfaces.md b/_sources/chapter0/bsplines-surfaces.md
new file mode 100644
index 0000000..c945976
--- /dev/null
+++ b/_sources/chapter0/bsplines-surfaces.md
@@ -0,0 +1,255 @@
+# B-Splines surfaces
+
+The B-spline surface in $\mathbb{R}^d$ associated to knots $(T_u, T_v)$ where $T_u=(u_i)_{0\leqslant i \leqslant n_u + p_u + 1}$ and $T_v=(v_i)_{0\leqslant i \leqslant n_v + p_v + 1}$, and control points $(\mathbf{P}_{ij})_{ 0 \leqslant i \leqslant n_u,  0 \leqslant j \leqslant n_v}$ is defined by :
+
+$$
+\mathcal{C}(u,v) = \sum_{i=0}^{n_u} \sum_{j=0}^{n_v} N_i^{p_u}(u) N_j^{p_v}(v) \textbf{P}_{i,j}
+$$
+
+```python
+# needed imports
+import numpy as np
+import matplotlib.pyplot as plt
+
+from mpl_toolkits.mplot3d import Axes3D
+from matplotlib import cm
+```
+
+
+```python
+# importing bsplines utilities
+from bsplines_utilities import find_span, all_bsplines, point_on_nurbs_surface, point_on_bspline_surface, insert_knot_bspline_surface, insert_knot_nurbs_surface
+```
+
+**Example 1**:
+
+In this first example, we consider a surface that approximate the function $(u,v) \mapsto \sin(\pi u) \sin(\pi v)$
+
+
+```python
+def example_1():    
+    Tu = [0., 0., 0., 1., 1., 1.]
+    pu = 2
+    nu = len(Tu) - pu - 1
+
+    Tv = [0., 0., 0., 1., 1., 1.]
+    pv = 2
+    nv = len(Tv) - pv - 1
+
+    P = np.zeros((nu, nv,1))
+    
+    gridu = np.linspace(0., 1., nu)
+    gridv = np.linspace(0., 1., nv)
+    
+    for i,u in enumerate(gridu):
+        for j,v in enumerate(gridv):
+            P[i, j, 0] = np.sin(np.pi*u)*np.sin(np.pi*v)
+    
+    nx = 101
+    xs = np.linspace(0., 1., nx)
+    
+    ny = 101
+    ys = np.linspace(0., 1., ny)
+    
+    Q = np.zeros((nx, ny, 1))
+    for i,x in enumerate(xs):
+        for j,y in enumerate(ys):
+            Q[i,j,:] = point_on_bspline_surface(Tu, Tv, P, x, y)
+            
+    fig = plt.figure()
+    ax = fig.gca(projection='3d')
+  
+    X, Y = np.meshgrid(xs, ys)
+    Z = Q[:,:,0]
+
+    # Plot the surface.
+    surf = ax.plot_surface(X, Y, Z, cmap=cm.coolwarm,
+                           linewidth=0, antialiased=False)
+    
+    # Add a color bar which maps values to colors.
+    fig.colorbar(surf, shrink=0.5, aspect=5)
+
+example_1()
+```
+
+
+    
+![png](images/bsplines-surfaces/output_5_0.png)
+    
+
+
+**Example 2**:
+
+This example shows the construction of a square in 2D.
+The B-Spline surface is created using the function **make_square**
+
+
+```python
+def make_square(origin=(0,0), length=1.):
+    Tu  = [0., 0., 1., 1.]
+    Tv  = [0., 0., 1., 1.]
+    pu = 1
+    pv = 1
+    nu = len(Tu) - pu - 1
+    nv = len(Tv) - pv - 1
+    gridu = np.unique(Tu)
+    gridv = np.unique(Tv)
+
+    origin = np.asarray(origin)
+
+    P = np.asarray([[[0.,0.],[0.,1.]],[[1.,0.],[1.,1.]]])
+    for i in range(0, 2):
+        for j in range(0, 2):
+            P[i,j,:] = origin + P[i,j,:]*length
+
+    return (Tu, Tv), (pu, pv), P
+```
+
+
+```python
+def plot_surface(knots, degrees, P, weights=None, Nu=101, Nv=101, color='b'):
+    Tu, Tv = knots
+    pu, pv = degrees
+
+    nu = len(Tu) - pu - 1
+    nv = len(Tv) - pv - 1
+    gridu = np.unique(Tu)
+    gridv = np.unique(Tv)
+
+    us = np.linspace(0., 1., Nu)
+    vs = np.linspace(0., 1., Nv)
+
+    # ...
+    Q = np.zeros((Nv, 2))
+    if weights is None:
+        for i,u in enumerate(gridu):
+            for j,v in enumerate(vs):
+                Q[j,:] = point_on_bspline_surface(Tu, Tv, P, u, v)
+                
+            plt.plot(Q[:,0], Q[:,1], '-'+color)
+
+    else:
+        for i,u in enumerate(gridu):
+            for j,v in enumerate(vs):
+                Q[j,:] = point_on_nurbs_surface(Tu, Tv, P, weights, u, v)
+                
+            plt.plot(Q[:,0], Q[:,1], '-'+color)
+    # ...
+
+    # ...
+    Q = np.zeros((Nu, 2))
+    if weights is None:
+        for j,v in enumerate(gridv):
+            for i,u in enumerate(us):
+                Q[i,:] = point_on_bspline_surface(Tu, Tv, P, u, v)
+
+            plt.plot(Q[:,0], Q[:,1], '-'+color)                
+
+    else:
+        for j,v in enumerate(gridv):
+            for i,u in enumerate(us):
+                Q[i,:] = point_on_nurbs_surface(Tu, Tv, P, weights, u, v)
+                
+            plt.plot(Q[:,0], Q[:,1], '-'+color)                
+    # ...
+# ...
+
+```
+
+
+```python
+def example_2():    
+    knots, degrees, P = make_square(origin=(0,0), 
+                                    length=1.)
+    
+    t = 0.5
+    Tu, Tv, pu, pv, P = insert_knot_bspline_surface(*knots, *degrees, P, t, 
+                                times=1, 
+                                axis=None)
+    
+    plot_surface((Tu, Tv), (pu, pv), P, 
+                 weights=None, 
+                 Nu=101, Nv=101, 
+                 color='b')
+    
+example_2()
+```
+
+
+    
+![png](images/bsplines-surfaces/output_9_0.png)
+    
+
+
+**Example 3**:
+
+The following example shows how to create and plots a circle using NURBS.
+
+We shall need the following function:
+
+
+```python
+def make_circle(center=(0.,0.), radius=1.):
+    Tu  = [0., 0., 0., 1, 1., 1.]
+    Tv  = [0., 0., 0., 1, 1., 1.]
+    pu = 2
+    pv = 2
+    nu = len(Tu) - pu - 1
+    nv = len(Tv) - pv - 1
+    gridu = np.unique(Tu)
+    gridv = np.unique(Tv)
+
+
+    s = 1./np.sqrt(2)
+    P          = np.zeros((nu,nv,2))
+    P[0,0,:]   = np.asarray([-s   , -s   ])
+    P[1,0,:]   = np.asarray([-2*s , 0.   ])
+    P[2,0,:]   = np.asarray([-s   , s    ])
+    P[0,1,:]   = np.asarray([0.   , -2*s ])
+    P[1,1,:]   = np.asarray([0.   , 0.0  ])
+    P[2,1,:]   = np.asarray([0.   , 2*s  ])
+    P[0,2,:]   = np.asarray([s    , -s   ])
+    P[1,2,:]   = np.asarray([2*s  , 0.   ])
+    P[2,2,:]   = np.asarray([s    , s    ])
+
+    P *= radius
+    P[:,:,0] += center[0]
+    P[:,:,1] += center[1]
+
+    W       = np.zeros((3,3))
+    W[0,0]  = 1.
+    W[1,0]  = s
+    W[2,0]  = 1.
+    W[0,1]  = s
+    W[1,1]  = 1.
+    W[2,1]  = s
+    W[0,2]  = 1.
+    W[1,2]  = s
+    W[2,2]  = 1.
+
+    return (Tu, Tv), (pu, pv), P, W
+
+```
+
+
+```python
+def example_3():    
+    knots, degrees, P, W = make_circle(center=(0.,0.), radius=1.)
+    
+    t = 0.5
+    Tu, Tv, pu, pv, P, W = insert_knot_nurbs_surface(*knots, *degrees, P, W, t, 
+                                times=1, 
+                                axis=None)
+    
+    plot_surface((Tu, Tv), (pu, pv), P, 
+                 weights=W, 
+                 Nu=101, Nv=101, 
+                 color='b')
+    
+example_3()
+```
+
+
+    
+![png](images/bsplines-surfaces/output_12_0.png)
+    
diff --git a/_sources/chapter0/bsplines.md b/_sources/chapter0/bsplines.md
new file mode 100644
index 0000000..f5d01fe
--- /dev/null
+++ b/_sources/chapter0/bsplines.md
@@ -0,0 +1,529 @@
+# B-Splines
+
+Given a subdivision $\{x_0 <  x_1 < \cdots < x_r\}$ of the interval $I = [x_0, x_r]$, the \textbf{Schoenberg space} is the space of piecewise polynomials of degree $p$, on the interval $I$ and given regularities $\{k_1, k_2, \cdots, k_{r-1}\}$ at the internal points $\{x_1, x_2, \cdots, x_{r-1}\}$. 
+
+Given $m$ and $p$ as natural numbers, let us consider a sequence of non decreasing real numbers $T=\{t_i\}_{0\leqslant i \leqslant m}$. $T$ is called **knots sequence**. From a knots sequence, we can generate a B-splines family using the reccurence formula \ref{eq:bspline-reccurence}.
+
+````{prf:definition} B-Splines using Cox-DeBoor Formula
+:label: def-bspline 
+The j-th B-spline of degree $p$ is defined by the recurrence relation:
+
+$$
+\begin{align}
+  N_j^p = \frac{t-t_j}{t_{j+p}-t_{j}} N_j^{p-1} + \frac{t_{j+p+1}-t}{t_{j+p+1}-t_{j+1}} N_{j+1}^{p-1},
+  \label{eq:bspline-reccurence}
+\end{align}
+$$
+
+where
+
+$$
+N_j^0(t) = \chi_{ \left[ t_j, t_{j+1} \right[ }(t)
+$$
+
+for $0 \leq j \leq m-p-1$.
+````
+
+```{prf:remark}
+  When working with Bernstein polynomials, we introduced the convention $B_i^n=0$ for all $i<0$ or $i>n$. For B-Splines, we have a similar convetion $N_j^p=0$ if $j<0$ or $j>n$. In addition, we also assume $\frac{0}{0} = 0$, when using the formula \ref{eq:bspline-reccurence} and $N_j^0 = 0$ if $t_j = t_{j+1}$.
+```
+
+```python
+# needed imports
+import numpy as np
+from numpy import empty
+import matplotlib.pyplot as plt
+```
+
+````{prf:example}
+:label: bsplines-ex-1
+
+We consider a linear B-Spline with the knot vector $T = [0, 1, 2]$
+
+
+```python
+def N1_0(t):
+    if t >= 0 and t< 1: return t
+    if t >= 1 and t< 2: return 2-t
+    return 0.
+
+xs = np.linspace(0., 2., 200)
+plt.plot(xs, [N1_0(x) for x in xs], label='$N_0^1$')
+plt.legend()
+```
+![png](images/bsplines/output_5_1.png)
+````
+
+````{prf:example}
+:label: bsplines-ex-2
+
+We consider a linear B-Spline with the knot vector $T = [0, 0, 1]$
+
+```python
+def N1_0(t):
+    if t >= 0 and t< 1: return 1-t
+    return 0.
+
+xs = np.linspace(0., 1., 200)
+plt.plot(xs, [N1_0(x) for x in xs], label='$N_0^1$')
+plt.legend()
+```
+![png](images/bsplines/output_7_1.png)
+````
+
+````{prf:example}
+:label: bsplines-ex-3
+
+We consider a linear B-Spline with the knot vector $T = [0, 1, 1]$
+
+```python
+def N1_0(t):
+    if t >= 0 and t< 1: return t
+    return 0.
+
+xs = np.linspace(0., 1., 201)[:-1]
+plt.plot(xs, [N1_0(x) for x in xs], label='$N_0^1$')
+plt.legend()
+```
+    
+![png](images/bsplines/output_9_1.png)
+````
+
+````{prf:example}
+:label: bsplines-ex-4
+
+We consider linear B-Splines with the knot vector $T = [0, 0, 1, 1]$
+
+```python
+def N1_0(t):
+    if t >= 0 and t< 1: return 1-t
+    return 0.
+
+def N1_1(t):
+    if t >= 0 and t< 1: return t
+    return 0.
+
+xs = np.linspace(0., 1., 201)[:-1]
+plt.plot(xs, [N1_0(x) for x in xs], label='$N_0^1$')
+plt.plot(xs, [N1_1(x) for x in xs], label='$N_1^1$')
+plt.legend()
+```
+    
+![png](images/bsplines/output_11_1.png)
+````
+
+````{prf:example}
+:label: bsplines-ex-5
+
+We consider linear B-Splines with the knot vector $T = [0, 0, 1, 2]$
+
+```python
+def N1_0(t):
+    if t >= 0 and t< 1: return 1-t
+    return 0.
+
+def N1_1(t):
+    if t >= 0 and t< 1: return t
+    if t >= 1 and t< 2: return 2-t
+    return 0.
+
+xs = np.linspace(0., 2., 201)[:-1]
+plt.plot(xs, [N1_0(x) for x in xs], label='$N_0^1$')
+plt.plot(xs, [N1_1(x) for x in xs], label='$N_1^1$')
+plt.legend()
+```
+    
+![png](images/bsplines/output_13_1.png)
+````
+    
+````{prf:example}
+:label: bsplines-ex-6
+
+We consider a quadratic B-Spline with the knot vector $T = [0, 0, 1, 1]$
+
+```python
+def N2_0(t):
+    if t >= 0 and t< 1: return 2*t*(1-t)
+    return 0.
+
+xs = np.linspace(0., 1., 201)[:-1]
+plt.plot(xs, [N2_0(x) for x in xs], label='$N_0^2$')
+plt.legend()
+```
+    
+![png](images/bsplines/output_15_1.png)
+````
+
+````{prf:example}
+:label: bsplines-ex-7
+
+We consider a quadratic B-Spline with the knot vector $T = [0, 0, 1, 2]$
+
+
+```python
+T = [0, 0, 1, 2]
+def N2_0(t):
+    if t >= 0 and t< 1: return 2*t-3./2.*t**2
+    if t >= 1 and t< 2: return 0.5*(2-t)**2
+    return 0.
+
+xs = np.linspace(0., 2., 200)
+plt.plot(xs, [N2_0(x) for x in xs], label='$N_0^2$')
+plt.legend()
+```
+
+![png](images/bsplines/output_17_1.png)
+````
+
+````{prf:example}
+:label: bsplines-ex-8
+
+We consider linear B-Splines with the knot vector $T = [0, 0, 1, 2, 3, 3]$
+
+```python
+def N1_0(t):
+    if t >= 0 and t< 1: return 1-t
+    return 0.
+
+def N1_1(t):
+    if t >= 0 and t< 1: return t
+    if t >= 1 and t< 2: return 2-t
+    return 0.
+
+def N1_2(t):
+    if t >= 1 and t< 2: return t-1
+    if t >= 2 and t< 3: return 3-t
+    return 0.
+
+def N1_3(t):
+    if t >= 2 and t< 3: return t-2
+    return 0.
+
+
+xs = np.linspace(0., 3., 201)[:-1]
+plt.plot(xs, [N1_0(x) for x in xs], label='$N_0^1$')
+plt.plot(xs, [N1_1(x) for x in xs], label='$N_1^1$')
+plt.plot(xs, [N1_2(x) for x in xs], label='$N_2^1$')
+plt.plot(xs, [N1_3(x) for x in xs], label='$N_3^1$')
+plt.legend()
+```
+    
+![png](images/bsplines/output_19_1.png)
+````
+
+````{prf:example}
+:label: bsplines-ex-9
+
+We consider quadratic B-Splines with the knot vector $T = [0, 0, 0, 1, 1, 1]$
+
+```python
+def N2_0(t):
+    if t >= 0 and t< 1: return (1-t)**2
+    return 0.
+
+def N2_1(t):
+    if t >= 0 and t< 1: return 2*t*(1-t)
+    return 0.
+
+def N2_2(t):
+    if t >= 0 and t< 1: return t**2
+    return 0.
+
+xs = np.linspace(0., 1., 201)[:-1]
+plt.plot(xs, [N2_0(x) for x in xs], label='$N_0^2$')
+plt.plot(xs, [N2_1(x) for x in xs], label='$N_1^2$')
+plt.plot(xs, [N2_2(x) for x in xs], label='$N_2^2$')
+plt.legend()
+```
+
+![png](images/bsplines/output_21_1.png)
+````
+
+## B-Splines properties
+
+B-Splines have many interesting properties that are listed below. In general, most of the proofs are done by induction on the B-Spline degree. 
+
+````{prf:proposition}
+  B-splines are piecewise polynomial of degree $p$
+````
+
+```{prf:remark}
+  The B-splines functions associated to open knots sequences without internal knots, \textit{i.e.} the length of the knots sequence is exactly $2p+2$, are the Bernstein polynomials of degree$p$.
+```
+
+````{prf:proposition} compact support
+  $N_j^p(t) = 0$ for all $t \notin [t_j, t_{j+p+1})$.
+````
+
+````{prf:proposition}
+  \label{prop:non-vanishing-bsplines}
+  If $t \in~ [ t_j,t_{j+1} ) $, then only the \textit{B-splines} $\{ N_{j-p}^p,\cdots,N_{j}^p \}$ are non vanishing at $t$.
+````
+
+
+````{prf:proposition} non-negativity
+  $N_j^p(t) \ge 0, \quad \forall t \in [ t_j, t_{j+p+1} ) $
+````
+
+````{prf:proposition} Partition of unity
+  $\sum N_i^{p}(t) = 1, \forall t \in \mathbb{R}$
+````
+
+```{prf:remark}
+The previous sum $\sum N_i^{p}(t)$ has a meaning since for every $t \in \mathbb{R}$, only $p+1$ B-Splines are non-vanishing.
+```
+
+```{prf:remark}
+  For the sake of simplicity, we shall avoid using summation indices on linear expansion of B-Splines.
+```
+
+````{prf:lemma}
+$$
+  \begin{align}
+  \sum a_j N_j^p = \sum \left(\frac{t_{j+p}-t}{t_{j+p}-t_{j}}a_{j-1} + \frac{t-t_j}{t_{j+p}-t_{j}} a_j\right) N_j^{p-1}  
+  \end{align}
+$$
+````
+
+````{prf:proposition} Marsden's idenity
+$$
+  \begin{align}
+    \left( x-y \right)^p = \sum\limits_{j} \rho_j^p(y) N_j^p(x)
+    \label{eq:marsden-spline}
+  \end{align}
+$$
+
+where 
+
+$$
+  \begin{align}
+    \rho_j^p(y) = \prod\limits_{i=j+1}^{j+p} (t_{i} -y)   
+  \end{align}
+$$
+````
+
+Thanks to Marsden's identity and the proposition \ref{prop:non-vanishing-bsplines}, we now are able to prove the local linear independence of the B-Splines basis functions.
+
+````{prf:proposition} Local linear independence
+  On each interval $[t_j, t_{j+1})$, the B-Splines are lineary independent.
+````
+
+```{prf:remark}
+  We will see in the Approximation theory part, that the B-Splines family has also a global linear independence property.
+```
+
+````{prf:proposition} Derivatives of B-Splines
+The derivative of B-Splines can be computed reccursivly by deriving the formula \ref{eq:bspline-reccurence}, which gives
+
+$$
+\begin{align}
+  {N_j^p}^\prime = \frac{p}{t_{j+p}-t_{j}} N_j^{p-1} - \frac{p}{t_{j+p+1}-t_{j+1}} N_{j+1}^{p-1}
+  \label{eq:bspline-ders-reccurence}
+\end{align}
+$$
+
+````
+    
+## Evaluation of B-Splines
+
+Given a knot sequence $T=\{t_i\}_{0\leqslant i \leqslant n + p}$, we are interested in the algorithmic evaluation of B-Splines of degree $p$. 
+
+```{warning}
+Add evaluation diagram
+```
+
+For a given real point $x$, it is done in two steps:
+
+1. find the knot span index $j$, such that $x \in~ ] t_j,t_{j+1} [ $  
+2. evaluate all non-vanishing B-Splines $N_{j-p}^p, \cdots, N_j^p$ 
+
+The first point is achieved by the function implemented by the following function:
+
+```python
+def find_span( knots, degree, x ):
+    # Knot index at left/right boundary
+    low  = degree
+    high = 0
+    high = len(knots)-1-degree
+
+    # Check if point is exactly on left/right boundary, or outside domain
+    if x <= knots[low ]: returnVal = low
+    elif x >= knots[high]: returnVal = high-1
+    else:
+        # Perform binary search
+        span = (low+high)//2
+        while x < knots[span] or x >= knots[span+1]:
+            if x < knots[span]:
+                high = span
+            else:
+                low  = span
+            span = (low+high)//2
+        returnVal = span
+
+    return returnVal
+```
+
+The second point is implemented by the following function, that returns all non-vanishing B-Splines at $x$
+
+
+```python
+def all_bsplines( knots, degree, x, span ):
+    left   = empty( degree  , dtype=float )
+    right  = empty( degree  , dtype=float )
+    values = empty( degree+1, dtype=float )
+
+    values[0] = 1.0
+    for j in range(0,degree):
+        left [j] = x - knots[span-j]
+        right[j] = knots[span+1+j] - x
+        saved    = 0.0
+        for r in range(0,j+1):
+            temp      = values[r] / (right[r] + left[j-r])
+            values[r] = saved + right[r] * temp
+            saved     = left[j-r] * temp
+        values[j+1] = saved
+
+    return values
+```
+
+The following function plots all B-Splines given a knot vector and a polynomial degree.
+
+
+```python
+def plot_splines(knots, degree, nx=100):
+    xmin = knots[degree]
+    xmax = knots[-degree-1]
+    
+    # grid points for evaluation
+    xs = np.linspace(xmin,xmax,nx)
+
+    # this is the number of the BSplines in the Schoenberg space
+    N = len(knots) - degree - 1
+
+    ys = np.zeros((N,nx), dtype=np.double)
+    for ix,x in enumerate(xs):
+        span = find_span( knots, degree, x )    
+        b    = all_bsplines( knots, degree, x, span )  
+        ys[span-degree:span+1, ix] = b[:]
+        
+    for i in range(0,N):
+        plt.plot(xs,ys[i,:], label='$N_{}$'.format(i+1))
+    plt.legend(loc=9, ncol=4)
+```
+
+## Knots vector families
+
+There are two kind of **knots vectors**, called **clamped** and **unclamped**. Both families contains **uniform** and **non-uniform** sequences. 
+
+The following are examples of such knots vectors
+
+### Clamped knots (open knots vector)
+
+#### uniform
+
+````{prf:example}
+:label: bsplines-ex-10
+
+```python
+T = np.array([0, 0, 0, 1, 2, 3, 4, 5, 5, 5])
+
+plot_splines(T, degree=2, nx=100)
+```
+    
+![png](images/bsplines/output_30_0.png)
+````
+
+````{prf:example}
+:label: bsplines-ex-11
+
+```python
+T = [-0.2, -0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 0.8]
+
+plot_splines(T, degree=2, nx=100)
+```
+    
+![png](images/bsplines/output_31_0.png)
+````    
+
+#### non-uniform 
+
+````{prf:example}
+:label: bsplines-ex-12
+
+```python
+T = [0, 0, 0, 1, 3, 4, 5, 5, 5]
+
+plot_splines(T, degree=2, nx=100)
+```
+    
+![png](images/bsplines/output_33_0.png)
+````
+
+
+````{prf:example}
+:label: bsplines-ex-13
+
+```python
+T = [-0.2, -0.2, 0.4, 0.6, 0.8, 0.8]
+
+plot_splines(T, degree=2, nx=100)
+```
+    
+![png](images/bsplines/output_34_0.png)
+````
+
+### Unclamped knots
+
+#### uniform
+
+````{prf:example}
+:label: bsplines-ex-14
+
+```python
+T = [0, 1, 2, 3, 4, 5, 6, 7, 8]
+
+plot_splines(T, degree=2, nx=100)
+```
+    
+![png](images/bsplines/output_36_0.png)
+````
+    
+
+````{prf:example}
+:label: bsplines-ex-15
+
+```python
+T = [-0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0]
+
+plot_splines(T, degree=2, nx=100)
+```
+    
+![png](images/bsplines/output_37_0.png)
+````
+
+#### non-uniform 
+
+
+````{prf:example}
+:label: bsplines-ex-16
+
+```python
+T = [0, 0, 3, 4, 7, 8, 9]
+
+plot_splines(T, degree=2, nx=100)
+```
+    
+![png](images/bsplines/output_39_0.png)
+````    
+
+````{prf:example}
+:label: bsplines-ex-17
+
+```python
+T = [-0.2, 0.2, 0.4, 0.6, 1.0, 2.0, 2.5]
+
+plot_splines(T, degree=2, nx=100)
+```
+    
+![png](images/bsplines/output_40_0.png)
+```` 
diff --git a/_sources/chapter0/cad.md b/_sources/chapter0/cad.md
new file mode 100644
index 0000000..843df56
--- /dev/null
+++ b/_sources/chapter0/cad.md
@@ -0,0 +1,3 @@
+# Computer Aided Design
+
+TODO
diff --git a/_sources/chapter0/data-structure.md b/_sources/chapter0/data-structure.md
new file mode 100644
index 0000000..6a15a16
--- /dev/null
+++ b/_sources/chapter0/data-structure.md
@@ -0,0 +1,146 @@
+# Data Structure
+
+In the sequel, we shall use **StencilMatrix** and **StencilVector** from the **psydac** library.
+
+For the moment, we are only interested about having an appropriate data structure to store our computations.
+Once the assembly is done, we will convert the matrix into a scipy sparse matrix, while the vector will be converted to numpy array.
+
+We will see later how one can impose boundary conditions.
+
+# 1D Case
+
+
+```python
+import numpy as np
+
+from psydac.fem.splines import SplineSpace
+from psydac.fem.tensor  import TensorFemSpace
+from psydac.linalg.stencil import StencilMatrix
+from psydac.linalg.stencil import StencilVector
+from psydac.ddm.cart import DomainDecomposition
+```
+
+
+```python
+from gallery_section_03 import assemble_stiffness_1d
+from gallery_section_03 import assemble_vector_1d
+```
+
+
+```python
+xmin = 0. ; xmax = 1.
+nelements = 8
+degree = 3
+grid = np.linspace( xmin, xmax, num=nelements+1 )
+V = SplineSpace(degree=degree, grid=grid)
+dd = DomainDecomposition(ncells=[V.ncells], periods=[False])
+V = TensorFemSpace(dd, V)
+```
+
+
+```python
+stiffness = StencilMatrix(V.vector_space, V.vector_space)
+```
+
+
+```python
+stiffness = assemble_stiffness_1d( V, matrix=stiffness )
+```
+
+
+```python
+stiffness = stiffness.tosparse()
+```
+
+
+```python
+f = lambda x: 2.    
+```
+
+
+```python
+rhs = StencilVector(V.vector_space)
+```
+
+
+```python
+rhs = assemble_vector_1d( f, V, rhs=rhs )
+```
+
+
+```python
+rhs = rhs.toarray()
+```
+
+## 2D Case
+
+
+```python
+from gallery_section_03 import assemble_stiffness_2d
+from gallery_section_03 import assemble_vector_2d
+```
+
+
+```python
+x1min = 0. ; x1max = 1.
+x2min = 0. ; x2max = 1.
+nelements1 = 8
+nelements2 = 8
+degree1 = 3
+degree2 = 3
+grid1 = np.linspace( x1min, x1max, num=nelements1+1 )
+grid2 = np.linspace( x2min, x2max, num=nelements2+1 )
+
+V1 = SplineSpace(degree=degree1, grid=grid1)
+V2 = SplineSpace(degree=degree2, grid=grid2)
+
+dd = DomainDecomposition(ncells=[V1.ncells, V2.ncells], periods=[False, False])
+V = TensorFemSpace(dd, V1, V2)
+```
+
+
+```python
+stiffness = StencilMatrix(V.vector_space, V.vector_space)
+```
+
+
+```python
+stiffness = assemble_stiffness_2d( V, matrix=stiffness )
+```
+
+
+```python
+stiffness = stiffness.tosparse()
+```
+
+
+```python
+# convert the sparse matrix to dense
+stiffness = stiffness.toarray()
+```
+
+
+```python
+f = lambda x,y: 2.    
+```
+
+
+```python
+rhs = StencilVector(V.vector_space)
+```
+
+
+```python
+rhs = assemble_vector_2d( f, V, rhs=rhs )
+```
+
+### Matrix profile
+
+
+```python
+import matplotlib.pyplot as plt 
+
+plt.spy(stiffness)
+```
+    
+![png](images/data-structure/output_24_1.png)
diff --git a/_sources/chapter0/fem.md b/_sources/chapter0/fem.md
new file mode 100644
index 0000000..b0e99ed
--- /dev/null
+++ b/_sources/chapter0/fem.md
@@ -0,0 +1,209 @@
+# Introduction to B-Splines FEM
+
+Let $\Omega \subset \mathbb{R}^d$ be a computational domain that is the image of a logical domain $\mathcal{P}$, *i.e.* a unit line (in *1d*), square (in *2d*) or a cube (in *3d*) with a **mapping** function 
+
+$$
+\Omega = \mathcal{G} (\mathcal{P}) 
+$$
+
+We consider the Poisson problem with Homogeneous Dirichlet boundary conditions:
+
+$$
+  \mbox{Find} ~  u \in H^1_0(\Omega), ~ \mbox{such that}
+$$
+$$
+  - \nabla^2 u = f, ~~ \Omega
+$$
+
+Using a Galerkin-Rietz method, we introduce a discrete finite elements space $\mathcal{V}_h = \mathbf{span}\{ \phi_j, 1 \leq j \leq n_V \}$ for trial and test functions. Here the index $j$ is a multi-index. For example, in $2D$, we have $j = (j_1, j_2)$
+
+Let $u_h \in \mathcal{V}_h$ such that $u_h = \sum_{j=1}^{n_V} u_j \phi_j$. Then the weak formulation for the Poisson equation writes
+
+$$
+  \sum_{j=1}^{n} u_j \int_{\Omega} \nabla \phi_j \cdot \nabla \phi_i = \int_{\Omega} f \phi_i, \quad \forall  1 \leq i \leq n 
+$$
+
+which can be written in a matrix form
+
+$$
+  M U = F
+$$
+
+where $U$ denotes the coefficients $(u_j, ~ 1 \leq j \leq n)$ and $F$ is a vector consisting of the terms $\int_{\Omega} f \phi_i$ for $1 \leq i \leq n$. Finally, the matrix $M$ is called the **stiffness** matrix and its entries are defined as
+
+$$
+  M_{ij} = \int_{\Omega} \nabla \phi_j \cdot \nabla \phi_i
+$$
+
+We will denote our basis function by $b_i$ and $b_j$ rather than $\phi_i$ and $\phi_j$. In this case, in order to solve the Poisson problem, one needs to
+
+1. Assemble the matrix $M$ of entries
+$$
+  M_{ij} = \int_{\Omega} \nabla b_j \cdot \nabla b_i
+$$
+2. Assemble the right hand side $F$ of entries
+$$
+  F_{i} = \int_{\Omega} f b_i
+$$
+3. Solve the linear system 
+$$
+  M U = F
+$$
+
+Since at the end, we need to compute the $L^2$-norm of the error $u-u_h$, we need also to evaluate (discretize) the following (scalar) expression:
+
+$$
+\| e_h \|_{L^2}^2 := \| u-u_h \|_{L^2}^2 = \int_{\Omega} (u-u_h)^2
+$$
+
+
+## Discretization of the Poisson problem
+
+Now let's go back to a matrix entry $M_{ij}$, we have
+$$
+M_{ij} = \sum_e \int_{Q_e} \nabla b_i \cdot \nabla b_j
+$$
+We know that every basis function $b_i$ is a polynomial over the element $Q_i$. We can then use one of the Gauss quadrature formulae. 
+
+**Note**:
+In the case of *B-splines* we will avoid having to evaluate on the boundary of our element (B-Splines are only defined on the right, if we don't do so, evaluating derivatives may lead to wrong results). Hence we will use the Gauss-Legendre formula.
+
+**Note**:
+In general, the quadrature formulae are given on the interval $[-1, 1]$. We therefor need to remap these points for every element, in out partition.
+
+Let's assume now that our quadrature points have been remapped, and let's call them $\xi_k$, $\omega_k$ will denote the associated weight to $\xi_k$.
+In this case $M_{ij}$ can be written as
+
+$$
+M_{ij} = \sum_e \sum_k \omega_k \nabla b_i(\xi_k) \cdot \nabla b_j( \xi_k)
+$$
+
+Now let's go back to a rhs entry $F_{i}$, we have
+$$
+F_{i} = \sum_e \int_{Q_e} f b_i 
+$$
+We know that every basis function $b_i$ is a polynomial over the element $Q_i$. We can then use one of the Gauss quadrature formulae. 
+
+Since our quadrature points have been remapped ( points : $\xi_k$, weights : $\omega_k$ ).
+
+In this case $F_{i}$ can be written as
+
+$$
+F_{i} = \sum_e \sum_k \omega_k b_i(\xi_k) f( \xi_k)
+$$
+
+Now let's go back to a norm $e_h$, we have
+$$
+\| e_h \|_{L^2}^2 = \sum_e \int_{Q_e} (u-u_h)^2 
+$$
+
+which leads to
+
+$$
+\| e_h \|_{L^2}^2 = \sum_e \sum_k \omega_k (u(\xi_k) - u_h( \xi_k))^2
+$$
+
+The missing part here, is how to evaluate the discrete field $u_h$. This will be covered in the next part.
+
+
+## Expressions
+
+When using a Finite Elements method, one needs to compute the following kind of expressions:
+
+* **bilinear form**. Example : $\int_{\Omega} \phi_i \phi_j ~ d\Omega$
+* **linear form**. Example : $\int_{\Omega} f \phi_i ~ d\Omega$ 
+* **functional**, i.e. integral of a scalar (vector) expression. Example : $\int_{\Omega} (u-u_h)^2$  
+
+A general formal definition of a variational form $a$ is given by Eq. \ref{eq:var-form-def}, as real-valued \textit{multi-linear} map on the product of a sequence of function spaces  $\{V_j\}_{j=1}^\rho$ :
+
+\begin{align}
+  a = 
+  \begin{cases}
+  V_\rho \times \ldots \times V_2 \times V_1 \times \mathcal{P} \rightarrow \mathbb{R}
+  \\
+  a(v_\rho, \dots, v_1; w_1, \dots, w_\nu, c_1, \dots, c_\eta)
+  \end{cases}
+  \label{eq:var-form-def}
+\end{align}
+
+where $\rho$ is the arity of the variational form, and the special cases $\rho=0,1,2$ yield to the notions of \emph{functional}, \emph{linear form} and \emph{bilinear form}, respectively.
+Each of the spaces $V_j$ may be a scalar/vector function space, or a product of other function spaces.
+$\mathcal{P}$ is the space of all parameters (which can be empty), containing either functions or constants (with respect to differential operators).
+A function parameter is referred to as a \emph{field} and is an element of a function space, therefore it may be represented by its coefficients over a given basis.
+The general definition of the space of parameters is given as $\mathcal{P} := \left( W_1 \times \ldots \times W_\nu \right) \times \mathbb{R}^\eta$ where $\left( W_i \right)_{1 \le i \le \nu}$ can be scalar or vector function spaces and $\eta$ is the number of (real) constants.
+We shall refer to the parameters as \textit{free variables}, as they will not appear in the signature of the variational form, and are defined as global variables (or local to the current scope).
+
+In the sequel, we shall see how the different expressions can be evaluated. We will developp the different needed algorithms, that we will use in the next chapters.
+
+Since, we will end up by computing an integral over the domain, we will need the following procedures
+
+* Assembly over the domain: this will loop over the elements of the domain. In each element, we shall call a kernel procedure that performs the computation over one single element. 
+* depending on the multi-linear map (expression) to evaluate, we shall have three kinds of kernels:
+  - a kernel for computing expressions with arity of 2
+  - a kernel for computing expressions with arity of 1 
+  - a kernel for computing expressions with arity of 0
+* we shall not discuss for the moment the data structure used to store the results of the kernels
+
+
+## Notations
+
+In the sequel, we shall use the following arguments
+
+* n_elements_d (d=1,2,3) number of elements of the grid in each direction
+* an element will always refer to a grid element, which considered as the a tuple
+* k_d (d=1,2,3) will denote the number of quadrature points in each direction, with respect to current element
+* The operator $\texttt{coeff}$ will denote the local coefficients of a field over the current element 
+* The operator $\texttt{value}$ will denote the values of a field over the current element 
+
+
+## Assembly procedures
+
+The generic form of the assembly procedure is described next. In general, it is splitted in 3 parts:
+
+* Pre-process: select the needed data, related to the current element. 
+* Process: call the kernel for computations on the given element
+* Post-process: 
+
+### Pre-process
+
+For each function space (both $V_i$ and $W_i$) involved in the expression, we need to get the basis functions over the current element, as well as their spans.  
+
+The following examples are not general, since we consider only one function space.
+
+
+## Evaluation of a discrete field on a given element
+
+When a free-variable is used in the expression, one needs to evaluate it on the quadrature points. In what follows, we describe the evaluation of a discrete field, with respect to a given element
+
+![title](images/algorithms/eval_field_1d.png)
+![title](images/algorithms/eval_field_2d.png)
+
+## Kernel for an expression with arity = 1
+
+![title](images/algorithms/kernel_arity1_1d.png)
+![title](images/algorithms/kernel_arity1_2d.png)
+
+### Computing contributions of test basis functions (1D)
+
+![title](images/algorithms/contribution_arity1_1d_ex1.png)
+![title](images/algorithms/contribution_arity1_1d_ex2.png)
+
+### Computing contributions of test basis functions (2D)
+
+![title](images/algorithms/contribution_arity1_2d_ex1.png)
+![title](images/algorithms/contribution_arity1_2d_ex2.png)
+
+## Kernel for an expression with arity = 2
+
+![title](images/algorithms/kernel_arity2_1d.png)
+![title](images/algorithms/kernel_arity2_2d.png)
+
+### Computing contributions between test and trial basis functions (1D)
+
+![title](images/algorithms/contribution_arity2_1d_ex1.png)
+![title](images/algorithms/contribution_arity2_1d_ex2.png)
+
+### Computing contributions between test and trial basis functions (2D)
+
+![title](images/algorithms/contribution_arity2_2d_ex1.png)
+![title](images/algorithms/contribution_arity2_2d_ex2.png)
diff --git a/_sources/chapter0/howto.md b/_sources/chapter0/howto.md
new file mode 100644
index 0000000..62df7f4
--- /dev/null
+++ b/_sources/chapter0/howto.md
@@ -0,0 +1,2 @@
+# What to expect from IGA-Python
+
diff --git a/_sources/chapter0/iga.md b/_sources/chapter0/iga.md
new file mode 100644
index 0000000..1ffca4b
--- /dev/null
+++ b/_sources/chapter0/iga.md
@@ -0,0 +1,3 @@
+# Isogeometric Analysis 
+
+TODO
diff --git a/_sources/chapter0/performance-acceleration.md b/_sources/chapter0/performance-acceleration.md
new file mode 100644
index 0000000..0b03c46
--- /dev/null
+++ b/_sources/chapter0/performance-acceleration.md
@@ -0,0 +1,316 @@
+# Performance and Acceleration
+
+In this section, we shall see how to accelerate our Python code assembly and get native speed.
+We will be using [Numba](https://numba.pydata.org/) and [Pyccel](https://github.com/pyccel/pyccel).
+
+The **StencilMatrix** format are based on a negative indexing, which we provide through a syntactic-sugar approach by overiding the **__getitem__** and **__setitem__** methods. The entries are in fact stored in the the private attribute **\_data** which is a *numpy.NdArray*. We therefor use a shift by adding **p** as described in the **StencilMatrix** method
+
+```python
+class StencilMatrix( object ):
+    ...
+    def _shift_index( index, shift ):
+        if isinstance( index, slice ):
+            start = None if index.start is None else index.start + shift
+            stop  = None if index.stop  is None else index.stop  + shift
+            return slice(start, stop, index.step)
+        else:
+            return index + shift
+```
+
+In order to use a Python accelerator, we will first need to pass the *NdArray* as argument to the assembly function and not the **StencilMatrix** object.
+
+Since this will be *ugly* it is highly recommanded that one creates an *interface* function that will be calling the *core* assembly function. It is the later one that we will accelerate using Numba or Pyccel.
+
+
+```python
+from psydac.fem.splines import SplineSpace
+from psydac.fem.tensor  import TensorFemSpace
+from psydac.linalg.stencil import StencilMatrix
+from psydac.linalg.stencil import StencilVector
+from psydac.ddm.cart import DomainDecomposition
+```
+
+## 1D Case
+
+### Pure Python
+
+The novelties here are
+
+* we add Python Typing Syntax to our function
+* we use the matrix as **inout** argument, in the spirit of Fortran. This means that we shall have a procedure and not a function.
+* we shift the access to memory when storing the Matrix entries
+
+
+```python
+def assemble_stiffness_1d_pure(nk1: 'int', 
+                               p1: 'int',
+                               nq1: 'int', 
+                               spans_1: 'int[:]', 
+                               basis_1: 'double[:,:,:,:]', 
+                               weights_1: 'double[:,:]', 
+                               points_1: 'double[:,:]', 
+                               matrix: 'double[:,:]'):
+    """
+    assembling the stiffness matrix using stencil forms
+    """
+
+    # ... build matrices
+    for ie1 in range(0, nk1):
+        i_span_1 = spans_1[ie1]
+        for il_1 in range(0, p1+1):
+            for jl_1 in range(0, p1+1):
+                i1 = i_span_1 - p1 + il_1
+                j1 = i_span_1 - p1 + jl_1
+
+                v = 0.0
+                for g1 in range(0, nq1):
+                    bi_0 = basis_1[ie1, il_1, 0, g1]
+                    bi_x = basis_1[ie1, il_1, 1, g1]
+
+                    bj_0 = basis_1[ie1, jl_1, 0, g1]
+                    bj_x = basis_1[ie1, jl_1, 1, g1]
+
+                    wvol = weights_1[ie1, g1]
+
+                    v += (bi_x * bj_x) * wvol
+
+                # we shift the test index by p1
+                matrix[i1, p1+j1-i1]  += v
+    # ...
+
+    # NOTE: we will not return the matrix. 
+    #       explainations will come later
+    #return matrix
+```
+
+#### Timing using pure Python
+
+
+```python
+import numpy as np
+
+xmin = 0. ; xmax = 1.
+nelements = 400
+degree = 3
+grid = np.linspace( xmin, xmax, num=nelements+1 )
+V = SplineSpace(degree=degree, grid=grid)
+dd = DomainDecomposition(ncells=[V.ncells], periods=[False])
+V = TensorFemSpace(dd, V)
+
+M = StencilMatrix(V.vector_space, V.vector_space)
+
+# Sizes
+[s1] = V.vector_space.starts
+[e1] = V.vector_space.ends
+[p1] = V.vector_space.pads
+
+# Quadrature data
+nk1       = V.quad_grids()[0].num_elements
+nq1       = V.quad_grids()[0].num_quad_pts
+spans_1   = V.quad_grids()[0].spans
+basis_1   = V.quad_grids()[0].basis
+points_1  = V.quad_grids()[0].points
+weights_1 = V.quad_grids()[0].weights
+
+%timeit assemble_stiffness_1d_pure( nk1, p1, nq1, spans_1, basis_1, weights_1, points_1, M._data )
+```
+
+    13.2 ms ± 126 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
+
+
+### Using Pyccel
+
+
+```python
+from pyccel.epyccel import epyccel
+```
+
+
+```python
+assemble_stiffness_1d_pyccel = epyccel(assemble_stiffness_1d_pure)
+```
+
+#### Timing
+
+
+```python
+import numpy as np
+
+xmin = 0. ; xmax = 1.
+nelements = 400
+degree = 3
+grid = np.linspace( xmin, xmax, num=nelements+1 )
+V = SplineSpace(degree=degree, grid=grid)
+dd = DomainDecomposition(ncells=[V.ncells], periods=[False])
+V = TensorFemSpace(dd, V)
+
+M = StencilMatrix(V.vector_space, V.vector_space)
+
+# Sizes
+[s1] = V.vector_space.starts
+[e1] = V.vector_space.ends
+[p1] = V.vector_space.pads
+
+# Quadrature data
+nk1       = V.quad_grids()[0].num_elements
+nq1       = V.quad_grids()[0].num_quad_pts
+spans_1   = V.quad_grids()[0].spans
+basis_1   = V.quad_grids()[0].basis
+points_1  = V.quad_grids()[0].points
+weights_1 = V.quad_grids()[0].weights
+
+%timeit assemble_stiffness_1d_pyccel( nk1, p1, nq1, spans_1, basis_1, weights_1, points_1, M._data )
+```
+
+    12.2 µs ± 139 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
+
+
+## 2D case
+
+### Pure Python
+
+
+```python
+def assemble_stiffness_2d_pure(nk1: 'int', nk2: 'int', 
+                               p1:  'int', p2:  'int', 
+                               nq1:  'int', nq2:  'int', 
+                               spans_1:          'int[:]', spans_2:          'int[:]', 
+                               basis_1: 'double[:,:,:,:]', basis_2: 'double[:,:,:,:]', 
+                               weights_1:   'double[:,:]', weights_2:   'double[:,:]', 
+                               points_1:    'double[:,:]', points_2:    'double[:,:]', 
+                               matrix: 'double[:,:,:,:]'):
+    """
+    assembling the stiffness matrix using stencil forms
+    """
+
+    # ... build matrices
+    for ie1 in range(0, nk1):
+        i_span_1 = spans_1[ie1]
+        for ie2 in range(0, nk2):
+            i_span_2 = spans_2[ie2]
+            # evaluation dependant uniquement de l'element
+
+            for il_1 in range(0, p1+1):
+                for il_2 in range(0, p2+1):
+                    for jl_1 in range(0, p1+1):
+                        for jl_2 in range(0, p2+1):
+                            i1 = i_span_1 - p1 + il_1
+                            j1 = i_span_1 - p1 + jl_1
+
+                            i2 = i_span_2 - p2 + il_2
+                            j2 = i_span_2 - p2 + jl_2
+
+                            v = 0.0
+                            for g1 in range(0, nq1):
+                                for g2 in range(0, nq2):
+                                    bi_0 = basis_1[ie1, il_1, 0, g1] * basis_2[ie2, il_2, 0, g2]
+                                    bi_x = basis_1[ie1, il_1, 1, g1] * basis_2[ie2, il_2, 0, g2]
+                                    bi_y = basis_1[ie1, il_1, 0, g1] * basis_2[ie2, il_2, 1, g2]
+
+                                    bj_0 = basis_1[ie1, jl_1, 0, g1] * basis_2[ie2, jl_2, 0, g2]
+                                    bj_x = basis_1[ie1, jl_1, 1, g1] * basis_2[ie2, jl_2, 0, g2]
+                                    bj_y = basis_1[ie1, jl_1, 0, g1] * basis_2[ie2, jl_2, 1, g2]
+
+                                    wvol = weights_1[ie1, g1] * weights_2[ie2, g2]
+
+                                    v += (bi_x * bj_x + bi_y * bj_y) * wvol
+
+                            matrix[i1, i2, p1+j1-i1, p2+j2-i2]  += v
+    # ...
+```
+
+#### Timing
+
+
+```python
+# create the spline space for each direction
+x1min = 0. ; x1max = 1.
+nelements1 = 32
+degree1 = 3
+grid1 = np.linspace( x1min, x1max, num=nelements1+1 )
+V1 = SplineSpace(degree=degree1, grid=grid1)
+
+x2min = 0. ; x2max = 1.
+nelements2 = 32
+degree2 = 3
+grid2 = np.linspace( x2min, x2max, num=nelements2+1 )
+V2 = SplineSpace(degree=degree2, grid=grid2)
+
+# create the tensor space
+dd = DomainDecomposition(ncells=[V1.ncells, V2.ncells], periods=[False, False])
+V = TensorFemSpace(dd, V1, V2)
+
+M = StencilMatrix(V.vector_space, V.vector_space)
+
+# Sizes
+[s1, s2] = V.vector_space.starts
+[e1, e2] = V.vector_space.ends
+[p1, p2] = V.vector_space.pads
+
+# Quadrature data
+[      nk1,       nk2] = [g.num_elements for g in V.quad_grids()]
+[      nq1,       nq2] = [g.num_quad_pts for g in V.quad_grids()]
+[  spans_1,   spans_2] = [g.spans        for g in V.quad_grids()]
+[  basis_1,   basis_2] = [g.basis        for g in V.quad_grids()]
+[ points_1,  points_2] = [g.points       for g in V.quad_grids()]
+[weights_1, weights_2] = [g.weights      for g in V.quad_grids()]
+
+%timeit assemble_stiffness_2d_pure( nk1, nk2, p1, p2, nq1, nq2, spans_1, spans_2, basis_1, basis_2, weights_1, weights_2, points_1, points_2, M._data )
+```
+
+    5.97 s ± 491 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
+
+
+### Using Pyccel
+
+
+```python
+assemble_stiffness_2d_pyccel = epyccel(assemble_stiffness_2d_pure)
+```
+
+#### Timing
+
+
+```python
+# create the spline space for each direction
+x1min = 0. ; x1max = 1.
+nelements1 = 32
+degree1 = 3
+grid1 = np.linspace( x1min, x1max, num=nelements1+1 )
+V1 = SplineSpace(degree=degree1, grid=grid1)
+
+x2min = 0. ; x2max = 1.
+nelements2 = 32
+degree2 = 3
+grid2 = np.linspace( x2min, x2max, num=nelements2+1 )
+V2 = SplineSpace(degree=degree2, grid=grid2)
+
+# create the tensor space
+dd = DomainDecomposition(ncells=[V1.ncells, V2.ncells], periods=[False, False])
+V = TensorFemSpace(dd, V1, V2)
+
+M = StencilMatrix(V.vector_space, V.vector_space)
+
+# Sizes
+[s1, s2] = V.vector_space.starts
+[e1, e2] = V.vector_space.ends
+[p1, p2] = V.vector_space.pads
+
+# Quadrature data
+[      nk1,       nk2] = [g.num_elements for g in V.quad_grids()]
+[      nq1,       nq2] = [g.num_quad_pts for g in V.quad_grids()]
+[  spans_1,   spans_2] = [g.spans        for g in V.quad_grids()]
+[  basis_1,   basis_2] = [g.basis        for g in V.quad_grids()]
+[ points_1,  points_2] = [g.points       for g in V.quad_grids()]
+[weights_1, weights_2] = [g.weights      for g in V.quad_grids()]
+
+%timeit assemble_stiffness_2d_pyccel( nk1, nk2, p1, p2, nq1, nq2, spans_1, spans_2, basis_1, basis_2, weights_1, weights_2, points_1, points_2, M._data )
+```
+
+    3.56 ms ± 127 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
+
+
+## Exercises
+
+1. Write the accelerated version of the rhs assembly produce using Pyccel and Numba and compute their timing.
+2. Perform a benchmark between Pyccel and Numba while varying the B-Spline degree.
diff --git a/_sources/chapter0/poisson-1d.md b/_sources/chapter0/poisson-1d.md
new file mode 100644
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--- /dev/null
+++ b/_sources/chapter0/poisson-1d.md
@@ -0,0 +1,303 @@
+# B-splines FEM solver for Poisson equation (1D)
+
+Following the previous [section](http://nbviewer.jupyter.org/github/ratnania/IGA-Python/blob/main/lessons/Chapter1/01_introduction_fem.ipynb), we implement here a B-Splines FEM for the Poisson problem in 1D, with homogeneous boundary conditions.
+
+Appropriate data-structures for matrices and vectors, involved when assembling a bilinear and linear form, will be discussed in the next section.
+
+We consider the following Poisson problem in 1D:
+$$
+-u^{\prime \prime} = f, \quad (0,1)
+\\
+u(0) = 0  ~~ \mbox{and} ~~ u(1) = 0
+$$
+
+Using the algorithms developped in section 1., for assembly a bilinear form and a linear form, we see that we need to assemble
+
+* The Stiffness matrix
+* The right hand side (rhs)
+
+
+```python
+# needed imports
+from numpy import zeros, ones, linspace, zeros_like, asarray
+from matplotlib.pyplot import plot, show
+%matplotlib inline
+```
+
+
+```python
+# ... assembling the stiffness matrix using stencil forms
+def assemble_stiffness(nelements, degree, spans, basis, weights, points, matrix):
+
+    # ... sizes
+    ne1       = nelements
+    p1        = degree
+    spans_1   = spans
+    basis_1   = basis
+    weights_1 = weights
+    points_1  = points
+    
+    k1 = weights.shape[1]
+    # ...
+
+    # ... build matrices
+    for ie1 in range(0, ne1):
+        i_span_1 = spans_1[ie1]
+        for il_1 in range(0, p1+1):
+            for jl_1 in range(0, p1+1):
+                i1 = i_span_1 - p1 + il_1
+                j1 = i_span_1 - p1 + jl_1
+
+                v = 0.0
+                for g1 in range(0, k1):
+                    bi_0 = basis_1[ie1, il_1, 0, g1]
+                    bi_x = basis_1[ie1, il_1, 1, g1]                    
+
+                    bj_0 = basis_1[ie1, jl_1, 0, g1]
+                    bj_x = basis_1[ie1, jl_1, 1, g1]                    
+
+                    wvol = weights_1[ie1, g1]
+
+                    v += (bi_x * bj_x) * wvol
+
+                matrix[i1, j1]  += v
+    # ...
+
+    return matrix    
+# ...
+```
+
+
+```python
+# ... Assembly procedure for the rhs
+def assemble_vector(f, nelements, degree, spans, basis, weights, points, rhs):
+
+    # ... sizes
+    ne1       = nelements
+    p1        = degree
+    spans_1   = spans
+    basis_1   = basis
+    weights_1 = weights
+    points_1  = points
+    
+    k1 = weights.shape[1]
+    # ...
+
+    # ... build rhs
+    for ie1 in range(0, ne1):
+        i_span_1 = spans_1[ie1]
+        for il_1 in range(0, p1+1):
+            i1 = i_span_1 - p1 + il_1
+
+            v = 0.0
+            for g1 in range(0, k1):
+                bi_0 = basis_1[ie1, il_1, 0, g1]
+                bi_x = basis_1[ie1, il_1, 1, g1]  
+
+                x1    = points_1[ie1, g1]
+                wvol  = weights_1[ie1, g1]
+
+                v += bi_0 * f(x1) * wvol
+
+            rhs[i1] += v
+    # ...
+
+    # ...
+    return rhs
+    # ...
+# ...
+```
+
+---
+**Note**: 
+We added the function **f** as an argument of the *assemble_rhs* function. In general, we will avoid calling a function over the quadrature points for performance reasons. 
+
+---
+
+We will need the following imports from the **psydac** library.
+
+
+```python
+# nedded imports
+from psydac.core.bsplines import elements_spans  # computes the span for each element
+from psydac.core.bsplines import make_knots      # create a knot sequence from a grid
+from psydac.core.bsplines import quadrature_grid # create a quadrature rule over the whole 1d grid
+from psydac.core.bsplines import basis_ders_on_quad_grid # evaluates all bsplines and their derivatives on the quad grid
+from psydac.utilities.quadratures import gauss_legendre # computes gauss legendre quadrature points & weights
+from utilities.plot import plot_field_1d # plot a solution for 1d problems
+```
+
+Now, let's create a B-spline Finite Element Space $V$ over a grid of $32$ elements, and using **cubic** *B-splines*.
+
+A B-Spline space is defined here through a grid, we then call the function **make_knots** to create an open knot sequence. We finally, compute the spans for every element, to avoid doing it for each quadrature point on the current element, in the assembly procedure.
+
+
+```python
+p  = 3    # spline degree
+ne = 32   # number of elements
+
+grid  = linspace(0., 1., ne+1)
+knots = make_knots(grid, p, periodic=False)
+spans = elements_spans(knots, p)  
+```
+
+
+```python
+nelements = len(grid) - 1
+nbasis    = len(knots) - p - 1
+
+# we need the value a B-Spline and its first derivative
+nderiv = 1
+
+# create the gauss-legendre rule, on [-1, 1]
+u, w = gauss_legendre( p )
+
+# for each element on the grid, we create a local quadrature grid
+points, weights = quadrature_grid( grid, u, w )
+
+# for each element and a quadrature points, 
+# we compute the non-vanishing B-Splines
+basis = basis_ders_on_quad_grid( knots, p, points, nderiv, normalization=False )
+```
+
+Assembling the **Stiffness** matrix is then done using
+
+
+```python
+stiffness = zeros((nbasis, nbasis))
+stiffness = assemble_stiffness(nelements, p, spans, basis, weights, points, matrix=stiffness)
+```
+
+### Analytical solution
+
+In finite elements, one of the first tests we do is to check polynomial solutions of with a degree that smaller or equal to the one we chose for our discretization.
+Here, we take the analytical solution $u(x) = x(1-x)$ which leads to the rhs $f(x) = - \partial_{xx} u(x) = 2$.
+
+
+```python
+f = lambda x: 2.    
+rhs = zeros(nbasis)
+rhs = assemble_vector(f, nelements, p, spans, basis, weights, points, rhs=rhs)
+```
+
+In order to apply the homogeneous Dirichlet boundary conditions, we only have to remove the first and the last splines, since their values at the boundary is $1$. Due to the partition of unity, the remaining splines are $0$ on the boundary.
+
+
+```python
+# apply homogeneous dirichlet boundary conditions
+rhs = rhs[1:-1]
+stiffness = stiffness[1:-1, 1:-1]
+```
+
+We use a **cg** solver from **scipy** to solve the linear system.
+
+
+```python
+from scipy.sparse.linalg import cg
+u, info = cg( stiffness, rhs, rtol=1e-6, maxiter=5000 )
+```
+
+Now, we need to add the first and last control points, which are equal to **0**, then proceed to the plot.
+
+
+```python
+u = [0.] + list(u) + [0.]
+u = asarray(u)
+plot_field_1d(knots, p, u, nx=401)
+```
+
+
+    
+![png](images/poisson-1d/output_20_0.png)
+    
+
+
+## The Spline Space class
+
+Since we will be calling over and over the block-lines [5] and [6], we created a class called **SplineSpace** that you can import from **psydac** library.
+
+### Matrix profile
+
+Let's check the profile of our matrix.
+
+```python
+import matplotlib.pyplot as plt 
+
+plt.spy(stiffness)
+```
+    
+![png](images/poisson-1d/output_23_1.png)
+
+
+As expected, it is a sparse matrix and it has a typical profile (band one).
+
+### Next step
+
+In fact, we are not done yet!
+The problem now, is that the data structure we are using for the matrix (here as a dense matrix) will consume too much memory, and is not taken into account the locality!
+
+Since on each element, there are exactly $p+1$ consecutive non-vanishing *B-splines*, we know that on each element there are at most $2p+1$ non zeros entries (in every direction). This means that we will need to set entries as the following, in $1D$:
+
+```python
+M[i1, j1 - i1]  += v_s
+```
+
+in $2D$:
+
+```python
+M[i1, i2, j1 - i1, j2 - i2]  += v_s
+```
+
+and in $3D$:
+
+
+```python
+M[i1, i2, i3, j1 - i1, j2 - i2, j3 - i3]  += v_s
+```
+
+Unfortunatly, this can not be represented as a **numpy.ndarray**, because of the negative indexing!
+Moreover, if our aim is to write a parallel code, other consideration must be taken into account.
+
+More details will be found in the next [section](http://nbviewer.jupyter.org/github/ratnania/IGA-Python/blob/main/lessons/Chapter1/03_data_structure.ipynb).
+
+
+## Exercises
+
+### Exercise 1.
+
+1. Implement a 1D Poisson solver using non homogeneous Dirichlet boundary conditions.
+
+2. Implement a 2D Poisson solver using homogeneous and non homogeneous Dirichlet boundary conditions.
+
+### Exercise 2.
+
+We consider the Heat equation in 1D
+
+$$
+\partial_t u = \nabla^2 u
+$$
+
+with homogeneous boundary conditions.
+
+1. Write an explicit FEM solver. What do you notice about the dependency of the CFL with respect to the degree? what about the regularity?
+
+2. Write an implicit FEM solver.
+
+### Exercise 3.
+
+We consider a unidirectional flow in the velocity/pressure formulation
+
+$$
+\partial u_x = - \frac{1}{\rho} \partial_x p + \nu \partial_{yy}^2 u_x + g_x
+$$
+
+where the unknown $u_x$ is a function of $y$ only. 
+
+We will consider $\partial_x p = -2$ and $g_x = 0.4$. The boundary conditions are given by $u_x(y=0) = u_x(y=h) = 0$.
+
+1. Write an explicit and implicit time schemes and solvers.
+
+2. Find numerical CFL in the case of explicit time scheme.
+
+3. Find the steady state.
+
diff --git a/_sources/chapter0/poisson-2d.md b/_sources/chapter0/poisson-2d.md
new file mode 100644
index 0000000..c5f3bca
--- /dev/null
+++ b/_sources/chapter0/poisson-2d.md
@@ -0,0 +1,185 @@
+# B-splines FEM solver for Poisson equation (2D)
+
+In this section, we show hoa to use **simplines** to solve a 2D Poisson problem with homogeneous boundary conditions
+$$
+- \nabla^2 u = f, \Omega,
+\\
+u = 0, \partial \Omega
+$$
+where the computation domain $\Omega$ is the unit square.
+
+
+```python
+# needed imports
+from numpy import zeros, ones, linspace, zeros_like, asarray
+import numpy as np
+from matplotlib.pyplot import plot, show
+import matplotlib.pyplot as plt
+
+%matplotlib inline
+```
+
+
+```python
+from psydac.fem.splines import SplineSpace
+from psydac.fem.tensor  import TensorFemSpace
+from psydac.linalg.stencil import StencilMatrix
+from psydac.linalg.stencil import StencilVector
+from psydac.ddm.cart import DomainDecomposition
+```
+
+
+```python
+from gallery_section_04 import assemble_stiffness_2d
+from gallery_section_04 import assemble_vector_2d
+```
+
+## Create the Finite Elements Space
+
+In 2D, our Spline function space is defined as
+
+$$
+\mathcal{V}_h := \texttt{span}\{ B_{i_1}^{p_1} B_{i_2}^{p_1}, ~ 1 \le i_1 \le n_1, ~ 1 \le i_2 \le n_2\}
+$$
+
+which is basicaly 
+$$
+\mathcal{V}_h = \mathcal{V}_h^1 \otimes \mathcal{V}_h^2 
+$$
+where
+$$
+\mathcal{V}_h^1 := \texttt{span}\{ B_{i_1}^{p_1}, ~ 1 \le i_1 \le n_1\}
+$$
+and
+$$
+\mathcal{V}_h^2 := \texttt{span}\{ B_{i_2}^{p_2}, ~ 1 \le i_2 \le n_2\}
+$$
+
+```python
+# create the spline space for each direction
+x1min = 0. ; x1max = 1.
+nelements1 = 1
+degree1 = 2
+grid1 = np.linspace( x1min, x1max, num=nelements1+1 )
+V1 = SplineSpace(degree=degree1, grid=grid1)
+
+x2min = 0. ; x2max = 1.
+nelements2 = 1
+degree2 = 2
+grid2 = np.linspace( x2min, x2max, num=nelements2+1 )
+V2 = SplineSpace(degree=degree2, grid=grid2)
+```
+
+```python
+dd = DomainDecomposition(ncells=[V1.ncells, V2.ncells], periods=[False, False])
+# create the tensor space
+V = TensorFemSpace(dd, V1, V2)
+```
+
+## Assemble the Stiffness Matrix
+
+The stiffness matrix entries are defined as
+
+$$
+M_{\textbf{i}, \textbf{j}} := \int_{\Omega} \nabla B_{\textbf{i}} \cdot \nabla B_{\textbf{j}}
+$$
+
+where 
+$$
+B_{\textbf{i}}(x_1,x_2) := B_{i_1}(x_1)B_{i_2}(x_2), \quad \textbf{i} := (i_1,i_2) 
+$$
+and
+$$
+B_{\textbf{j}}(x_1,x_2) := B_{j_1}(x_1)B_{j_2}(x_2), \quad \textbf{j} := (j_1,j_2)
+$$
+
+
+```python
+stiffness = StencilMatrix(V.vector_space, V.vector_space)
+stiffness = assemble_stiffness_2d( V, matrix=stiffness )
+```
+
+## Assemble the rhs
+
+The right hand side entries are defined as
+
+$$
+F_{\textbf{i}} := \int_{\Omega} f B_{\textbf{i}} ~d\Omega
+$$
+
+
+```python
+rhs = StencilVector(V.vector_space)
+
+f = lambda x,y: 2*x*(1 - x) + 2*y*(1 - y) 
+rhs = assemble_vector_2d( f, V, rhs=rhs )
+```
+
+## Imposing boundary conditions
+
+
+```python
+s1, s2 = V.vector_space.starts
+e1, e2 = V.vector_space.ends
+
+# ... needed for iterative solvers
+# left  bc at x=0.
+stiffness[s1,:,:,:] = 0.
+rhs[s1,:]           = 0.
+# right bc at x=1.
+stiffness[e1,:,:,:] = 0.
+rhs[e1,:]           = 0.
+# lower bc at y=0.
+stiffness[:,s2,:,:] = 0.
+rhs[:,s2]           = 0.
+# upper bc at y=1.
+stiffness[:,e2,:,:] = 0.
+rhs[:,e2]           = 0.
+# ...
+
+# ... needed for direct solvers
+# boundary x = 0
+#stiffness[s1,:,0,:] = 1.
+# boundary x = 1
+#stiffness[e1,:,0,:] = 1.
+# boundary y = 0
+#stiffness[:,s2,:,0] = 1.
+# boundary y = 1
+#stiffness[:,e2,:,0] = 1.    
+# ...
+```
+
+From now on, you can use the function **apply_dirichlet** to set the dirichlet boundary conditions for both the matrix and rhs.
+
+```python
+# convert the stencil matrix to scipy sparse
+stiffness = stiffness.tosparse() 
+
+
+```python
+# convert the stencil vector to a nd_array
+rhs = rhs.toarray()
+```
+
+```python
+#from scipy.sparse import csc_matrix, linalg as sla
+
+#lu = sla.splu(csc_matrix(stiffness))
+#x = lu.solve(rhs)
+```
+
+```python
+from scipy.sparse.linalg import cg
+x, info = cg( stiffness, rhs, rtol=1e-7, maxiter=100 )
+```
+
+```python
+from utilities.plot import plot_field_2d
+nbasis  = [W.nbasis for W in V.spaces]
+knots   = [W.knots for W in V.spaces]
+degrees = [W.degree for W in V.spaces]
+u = x.reshape(nbasis)
+plot_field_2d(knots, degrees, u) ; plt.colorbar()
+```
+
+![png](images/poisson-2d/output_20_1.png)
diff --git a/_sources/chapter1/install.md b/_sources/chapter1/install.md
new file mode 100644
index 0000000..12fa121
--- /dev/null
+++ b/_sources/chapter1/install.md
@@ -0,0 +1,31 @@
+# Installation
+
+It is recommanded to use a virtual environement.
+
+
+```shell
+# create a virtual environement
+python3 -m venv .iga-python
+
+# activate this environement using 
+source .iga-python/bin/activate
+```
+
+Then install dependencies using
+
+```shell
+pip3 install wheel
+pip3 install numpy
+pip3 install pyccel 
+pip3 install ipykernel
+pip3 install git+https://github.com/pyccel/sympde.git
+pip3 install git+https://github.com/pyccel/gelato.git
+pip3 install git+https://github.com/pyccel/psydac.git
+pip3 install git+https://github.com/girving/igakit.git
+```
+
+Adding the virtual environement to Jupyter notebook, can be done using
+
+```shell
+python3 -m ipykernel install --user --name=.iga-python
+```
diff --git a/_sources/chapter1/poisson.ipynb b/_sources/chapter1/poisson.ipynb
new file mode 100644
index 0000000..92fc8fc
--- /dev/null
+++ b/_sources/chapter1/poisson.ipynb
@@ -0,0 +1,330 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9f28f9af",
+   "metadata": {},
+   "source": [
+    "# Your first code using SymPDE & PsyDAC\n",
+    "\n",
+    "We first start by writing our first example using SymPDE.\n",
+    "Let $\\Omega := (0,1)^2$. We consider the Poisson problem with homogeneous Dirichlet boundary conditions.  \n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  - \\nabla^2 u = f \\quad \\text{in $\\Omega$}, \\quad \\quad \n",
+    "  u = 0            \\quad \\text{on $\\partial \\Omega$}. \n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "## Variational Formulation\n",
+    "\n",
+    "An $H^1$-conforming variational formulation of the previous problem reads\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\text{find $u \\in V$ such that} \\quad a(u,v) = l(v) \\quad \\forall v \\in V,\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where \n",
+    "\n",
+    "- $V \\subset H^1_0(\\Omega)$, \n",
+    "- $a(u,v) := \\int_{\\Omega} \\nabla u \\cdot \\nabla v ~ d\\Omega$, \n",
+    "- $l(v) := \\int_{\\Omega} f v ~ d\\Omega$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a958607",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "id": "d742586c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import ScalarFunctionSpace, Square, element_of\n",
+    "from sympde.calculus import grad, dot\n",
+    "\n",
+    "from sympy import pi, sin\n",
+    "\n",
+    "domain = Square()\n",
+    "\n",
+    "V = ScalarFunctionSpace('V', domain)\n",
+    "\n",
+    "x,y = domain.coordinates\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
+    "\n",
+    "# bilinear form\n",
+    "a = BilinearForm((u,v), integral(domain , dot(grad(v), grad(u))))\n",
+    "\n",
+    "# linear form\n",
+    "f = 2*pi**2*sin(pi*x)*sin(pi*y)\n",
+    "l = LinearForm(v, integral(domain, f*v))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(u, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "62ac1fd4",
+   "metadata": {},
+   "source": [
+    "\n",
+    "This very simple Python code reflects well the abstract mathematical framework needed for variational formulations.\n",
+    "The structure of the code is as follows,\n",
+    "\n",
+    "1. Create a domain.\n",
+    "2. Create a space of *scalar* functions over the domain.\n",
+    "3. Create elements from this function space. These elements will denote the test and trial functions.\n",
+    "4. Create the Bilinear and Linear forms, $a$ and $l$ respectively.\n",
+    "5. Create Essential Boundary Conditions.\n",
+    "6. Create the variational problem.\n",
+    "\n",
+    "Most of the time, you will need to follow the same steps, with some minor variants depending on the problem you're considering."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f983ece",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51095918",
+   "metadata": {},
+   "source": [
+    "We shall need the **discretize** function from **PsyDAC**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "id": "a2a0a2a1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from psydac.api.discretization import discretize"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "id": "00e54163",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [3,3]\n",
+    "ncells = [16,16]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "id": "5999c62b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b29fbcf",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "id": "541192ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "uh = equation_h.solve()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5174c4b5",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm\n",
+    "\n",
+    "When the analytical solution is available, you might be interested in computing the $L^2$ norm or $H^1_0$ semi-norm.\n",
+    "SymPDE allows you to do so, by creating the **Norm** object.\n",
+    "In this example, the analytical solution is given by\n",
+    "\n",
+    "$$\n",
+    "u_e = \\sin(\\pi x) \\sin(\\pi y)\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a31c46f",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "id": "5925c6cd",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.49756716724664e-07\n"
+     ]
+    }
+   ],
+   "source": [
+    "ue = sin(pi*x)*sin(pi*y)\n",
+    "\n",
+    "u = element_of(V, name='u')\n",
+    "\n",
+    "# create the formal Norm object\n",
+    "l2norm = Norm(u - ue, domain, kind='l2')\n",
+    "\n",
+    "# discretize the norm\n",
+    "l2norm_h = discretize(l2norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "l2_error = l2norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(l2_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6cbfeae",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ semi-norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "id": "e5c1a8b8",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.768702410721585e-05\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "h1norm = SemiNorm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3c09131c",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "id": "d829e410",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.769164089397408e-05\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "h1norm = Norm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9250897b",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter1/rules.md b/_sources/chapter1/rules.md
new file mode 100644
index 0000000..76b5aed
--- /dev/null
+++ b/_sources/chapter1/rules.md
@@ -0,0 +1,60 @@
+# Algebraic and differential operators evalution rules
+
+## Evaluation of the $\mathrm{grad}$ operator 
+
+Rule ID | symbolic expression | evaluation 
+--- | --- | ---
+`[OG1]` | $\mathrm{grad}(f+g)$ | $\mathrm{grad}(f) + \mathrm{grad}(g)$ 
+`[OG2]` | $\mathrm{grad}(\alpha ~ f)$ | $\alpha~\mathrm{grad}(f)$ 
+`[OG3]` | $\mathrm{grad}(f ~ g)$ | $f~\mathrm{grad}(g) + g~\mathrm{grad}(f)$ 
+`[OG4]` | $\mathrm{grad}(\frac{f}{g})$ | $-\frac{f}{g^2}\mathrm{grad}(g) + \frac{1}{g}\mathrm{grad}(f)$ 
+`[OG5]` | $\mathrm{grad}(F+G)$ | $\mathrm{grad}(F) + \mathrm{grad}(G)$ 
+`[OG6]` | $\mathrm{grad}(\alpha ~ F)$ | $\alpha~\mathrm{grad}(F)$ 
+`[OG7]` | $\mathrm{grad}(\mathrm{dot}(F, G))$ | $\mathrm{convect}(F, G) + \mathrm{convect}(G, F) + \mathrm{cross}(F, \mathrm{curl}(G)) - \mathrm{cross}(\mathrm{curl}(F), G)$ 
+
+## Evaluation of $\mathrm{curl}$ and $\mathrm{rot}$ operators in 2D 
+
+Rule ID | symbolic expression | evaluation 
+--- | --- | ---
+`[OC1a]` | $\mathrm{curl}(f+g)$ | $\mathrm{curl}(f) + \mathrm{curl}(g)$ 
+`[OC2a]` | $\mathrm{curl}(\alpha ~ f)$ | $\alpha~\mathrm{curl}(f)$ 
+`[OC1b]` | $\mathrm{rot}(F+G)$ | $\mathrm{rot}(F) + \mathrm{rot}(G)$ 
+`[OC2b]` | $\mathrm{rot}(\alpha ~ F)$ | $\alpha~\mathrm{rot}(F)$ 
+
+
+## Evaluation of $\mathrm{curl}$ operator in 3D
+
+Rule ID | symbolic expression | evaluation 
+--- | --- | ---
+`[OC1]` | $\mathrm{curl}(F+G)$ | $\mathrm{curl}(F) + \mathrm{curl}(G)$ 
+`[OC2]` | $\mathrm{curl}(\alpha ~ F)$ | $\alpha~\mathrm{curl}(F)$ 
+`[OC3]` | $\mathrm{curl}(f  ~ F)$ | $f~\mathrm{curl}(F) + \mathrm{cross}(\mathrm{grad}(f), F)$ 
+`[OC4]` | $\mathrm{curl}(\mathrm{cross}(F, G))$ | $\mathrm{div}(G)~F - \mathrm{div}(F)~G - \mathrm{convect}(F, G) +\mathrm{convect}(G, F)$ 
+
+## Evaluation of $\mathrm{div}$ operator 
+
+Rule ID | symbolic expression | evaluation 
+--- | --- | ---
+`[OD1]` | $\mathrm{div}(F+G)$ | $\mathrm{div}(F) + \mathrm{div}(G)$ 
+`[OD2]` | $\mathrm{div}(\alpha ~ F)$ | $\alpha~\mathrm{div}(F)$ 
+`[OD3]` | $\mathrm{div}(f ~ G)$ | $f~\mathrm{div}(G) + \mathrm{dot}(G, \mathrm{grad}(f))$ 
+`[OD4]` | $\mathrm{div}(\mathrm{cross}(F, G))$ | $-\mathrm{dot}(F, \mathrm{curl}(G)) + \mathrm{dot}(G, \mathrm{curl}(F))$ 
+
+## Evaluation of $\mathrm{laplace}$ operator 
+
+Rule ID | symbolic expression | evaluation 
+--- | --- | ---
+`[OL1]` | $\mathrm{laplace}(f+g)$ | $\mathrm{laplace}(f) + \mathrm{laplace}(g)$ 
+`[OL2]` | $\mathrm{laplace}(\alpha ~ f)$ | $\alpha~\mathrm{laplace}(f)$ 
+`[OL3]` | $\mathrm{laplace}(f~g)$ | $f ~ \mathrm{laplace}(g) + g ~ \mathrm{laplace}(f) + 2 \mathrm{dot}(\mathrm{grad}(f), \mathrm{grad}(g))$ 
+
+## Evaluation of specific combination of operators 
+
+Rule ID | symbolic expression | evaluation 
+--- | --- | ---
+`[OS1]` | $\mathrm{curl}(\mathrm{grad}(f))$ | $0$ 
+`[OS2]` | $\mathrm{div}(\mathrm{curl}(F))$ | $0$ 
+`[OS3]` | $\mathrm{div}(\mathrm{cross}(\mathrm{grad}(F), \mathrm{grad}(G)))$ | $0$ 
+`[OS4]` | $\mathrm{curl}(\mathrm{curl}(F))$ | $\mathrm{grad}(\mathrm{div}(F)) - \mathrm{laplace}(F)$ 
+`[OS5]` | $\mathrm{curl}(f~\mathrm{grad}(g))$ | $\mathrm{cross}(\mathrm{grad}(f), \mathrm{grad}(g))$ 
+
diff --git a/_sources/chapter1/space.md b/_sources/chapter1/space.md
new file mode 100644
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--- /dev/null
+++ b/_sources/chapter1/space.md
@@ -0,0 +1,76 @@
+# Function Space concepts
+
+SymPDE provides two Python classes to describe scalar and vector function spaces, respectively. There is no notion of a discrete representation; for example, we do not need to mention that a space is a Brezzi-Douglas-Marini (BDM) space. In fact, these kind of function spaces can be seen as parametric types having as a basic type **ScalarFunctionSpace** or **VectorFunctionSpace**. SymPDE only needs to know if an element of the space can be indexed or not. A BDM space would then be identified by an annotation added by a third party library to uniquely define a function space at the discrete level.
+
+## BasicFunctionSpace 
+
+This represents the base class of all our function spaces. It does not reflect a continuous space, but it is more a **type** at the formal/abstract level.
+
+## ScalarFunctionSpace 
+
+```python
+>>> domain = Domain('Omega', dim=2)
+>>> V1 = ScalarFunctionSpace('V1', domain)
+```
+
+## VectorFunctionSpace 
+
+```python
+>>> domain = Domain('Omega', dim=2)
+>>> W1 = VectorFunctionSpace('W1', domain)
+```
+
+## Typed Spaces
+```python
+>>> domain = Domain('Omega', dim=2)
+>>> H1    = ScalarFunctionSpace('V0', domain, kind='H1')
+>>> Hcurl = VectorFunctionSpace('V1', domain, kind='Hcurl')
+>>> Hdiv  = VectorFunctionSpace('V2', domain, kind='Hdiv')
+>>> L2    = ScalarFunctionSpace('V3', domain, kind='L2')
+>>> assert(H1.kind    == H1Space)                
+>>> assert(Hcurl.kind == HcurlSpace)
+>>> assert(Hdiv.kind  == HdivSpace)
+>>> assert(L2.kind    == L2Space)
+>>> assert(V.kind     == UndefinedSpace)
+>>> assert(W.kind     == UndefinedSpace)
+>>> assert(H1.regularity    > L2.regularity)
+>>> assert(H1.regularity    > Hcurl.regularity)
+>>> assert(Hcurl.regularity > L2.regularity)
+```
+
+## ProductSpace 
+
+```python
+>>> domain = Domain('Omega', dim=2)
+>>> V1 = ScalarFunctionSpace('V1', domain)
+>>> V2 = ScalarFunctionSpace('V2', domain)
+>>> W1 = VectorFunctionSpace('W1', domain)
+>>> X = V1 * V2 * W1
+>>> # or 
+>>> X = ProductSpace(V1, V2, W1)
+```
+
+## Derham 
+
+## Functions
+
+### ScalarFunction
+A scalar function is necessary an element of the **ScalarFunctionSpace**.
+
+### IndexedVectorFunction
+### VectorFunction
+A vector function is necessary an element of the **VectorFunctionSpace**. It is an extension of the **IndexedVectorFunction**, which means that one can access to its components.
+
+## Trace 
+
+### trace_0
+### trace_1
+
+## Utilities
+### element_of and elements_of
+
+## Projection (Experimental)
+
+## Projector (Experimental) 
+
+ 
diff --git a/_sources/chapter1/sympde.md b/_sources/chapter1/sympde.md
new file mode 100644
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--- /dev/null
+++ b/_sources/chapter1/sympde.md
@@ -0,0 +1,128 @@
+# SymPDE concepts and their mathematical meaning
+
+## Domain
+
+sympde notation | Mathematical notion 
+--- | ---
+`Domain('Omega', dim=n)`  | an abstract domain $\Omega \subset \mathbb{R}^n$ 
+`Domain('Omega', interiors, boundaries, connectivity)`  | an abstract domain $\Omega \subset \mathbb{R}^n$, represented by its interiors, boundaries and connectivity
+`Interval('I', coordinate=x)`  | an abstract (interior) interval $I \subset \mathbb{R}$, with the coordinate $x$
+`Union(A, B, C)`  | union of domains $A \cup B \cup C$
+`ProductDomain(A, B)`  | product of two domains $A \times B$
+`Indicator(A)`  | Indicator/characteristic function on $A$, *i.e.* $\mathbf{1}\_A$
+`Line('Omega')`    | an abstract line segment, $\Omega \subset \mathbb{R}$, having one interval as interior domain
+`Square('Omega')`  | an abstract square, $\Omega \subset \mathbb{R}^2$, having one interior domain as product of two intervals 
+`Cube('Omega')`    | an abstract cube, $\Omega \subset \mathbb{R}^3$, having one interior domain as product of three intervals 
+`NormalVector('n')` | Normal vector $\mathbf{n}$ 
+`TangentVector('t')` | Tangent vector $\mathbf{t}$ 
+`e = ElementDomain(Omega)` | an element of an abstract domain $e \in \mathcal{T}(\Omega)$ or $e = d\Omega$, where $\mathcal{T}(\Omega)$ is a given tessellation of $\Omega$ 
+`DomainArea(Omega)` | Area of an abstract domain $\Omega$ 
+`ElementArea(Omega)` | Area of an abstract element of a domain $\Omega$ 
+`Area(A)` | Area of an expression of topological domain notions 
+
+## Mapping
+
+sympde notation | Mathematical notion 
+--- | ---
+`Mapping('F', n)` | a mapping functor $\mathbf{F}\_{\Omega} := \left( \mathbf{F}, \Omega \right): \Omega \rightarrow \mathbb{R}^n$
+`F[i]`            | $i^{th}$ physical coordinate from $\{x,y,z\}$
+`F.jacobian`      | the jacobian matrix $\mathcal{D}\_\mathbf{F}$
+`F.det_jacobian`  | determinant of the jacobian matrix, $J_\mathbf{F}:= \mathrm{det} ~\mathcal{D}\_\mathbf{F}$
+`F.covariant`     | the covariant matrix $\left( \mathcal{D}\_\mathbf{F} \right)^{-T}$
+`F.contravariant` | the contravariant matrix $\frac{1}{J_\mathbf{F}} \mathcal{D}\_\mathbf{F}$ 
+`F.hessian`       | the hessian matrix $\mathcal{H}\_\mathbf{F}$
+`Covariant(F, v)` | action of the covariant matrix of $\mathbf{F}$ on $\mathbf{v}$, *i.e.* $\left( \mathcal{D}\_\mathbf{F} \right)^{-T} \mathbf{v}$
+`Contravariant(F, v)` | action of the contravariant matrix of $\mathbf{F}$ on $\mathbf{v}$, *i.e.* $\frac{1}{J_\mathbf{F}} \mathcal{D}\_\mathbf{F} \mathbf{v}$
+
+## Function spaces
+
+sympde notation | Mathematical notion 
+--- | ---
+`ScalarFunctionSpace('V', Omega)`     | scalar function space $\mathcal{V}$
+`VectorFunctionSpace('W', Omega)`     | vector function space $\mathbf{\mathcal{W}}$
+`ProductSpace(V, W)` or `V*W`        | product of spaces, *i.e.* $\mathcal{V} \times \mathcal{W}$
+`element_of_space(V, 'v')`          | scalar function $v \in \mathcal{V}$ 
+`element_of_space(W, 'w')`          | vector function $\mathbf{w} \in \mathcal{W}$  
+`element_of_space(V*W, ['v', 'w'])` | $\left(v,\mathbf{w}\right) \in \mathcal{V} \times \mathcal{W}$ 
+
+### Function space types
+
+sympde notation | Mathematical notion 
+--- | ---
+`H1SpaceType`    | $H^1$    
+`HcurlSpaceType` | ${H}{\mbox{curl}}$ 
+`HdivSpaceType`  | ${H}{\mbox{div}}$  
+`L2SpaceType`    | $L^2$    
+
+### Typed function spaces
+
+sympde notation | Mathematical notion 
+--- | ---
+`ScalarFunctionSpace('V0', Omega, kind='H1')`    | scalar function space $\mathcal{V}\_0 \subseteq H^1(\Omega)$    
+`VectorFunctionSpace('V1', Omega, kind='Hcurl')` | vector function space $\mathcal{V}\_1 \subseteq H(\mbox{curl}, \Omega)$ 
+`VectorFunctionSpace('V2', Omega, kind='Hdiv') ` | vector function space $\mathcal{V}\_2 \subseteq H(\mbox{div}, \Omega)$  
+`ScalarFunctionSpace('V3', Omega, kind='L2')`    | scalar function space $\mathcal{V}\_3 \subseteq L^2(\Omega)$    
+
+## Projections
+
+sympde notation | Mathematical notion 
+--- | ---
+`P_V := Projector(V)`                 | Projector onto the function space $\mathcal{V}$  
+`P_V(expr)`                           | Projection of the expression `expr` onto the function space $\mathcal{V}$  
+`Pi_V := Projector(V, 'commuting')`   | Commuting projector onto the typed function space $\mathcal{V}$  
+`Pi_V(expr)`                          | Commuting projection of the expression `expr` onto the typed function space $\mathcal{V}$  
+
+## Atomic variables
+
+sympde notation | Mathematical notion 
+--- | ---
+`ScalarFunction(V, 'u')`    |  $u \in \mathcal{V}$
+`VectorFunction(W, 'w')`    |  $\mathbf{w} \in \mathcal{W}$
+`Constant('mu', real=True)` | constant number $\mu \in \mathbb{R}$ 
+`int`                       | integer number 
+`float`                     | floating-point number
+
+## Algebraic operators 
+
+sympde notation | Mathematical notion 
+--- | ---
+`dot(u, v)`    | $\mathbf{u} \cdot \mathbf{v}$   
+`inner(u, v)`  | $\mathbf{u} : \mathbf{v}$       
+`cross(u, v)`  | $\mathbf{u} \times \mathbf{v}$  
+`outer(u, v)`  | $\mathbf{u} \otimes \mathbf{v}$  
+
+## Differential operators 
+
+boldface font is used for vector functions/expressions
+
+sympde notation | Mathematical notion 
+--- | ---
+`dx(u)`, `dy(u)` or `dz(u)`   |  $\partial_x u$, $\partial_y u$ or $\partial_z u$                               
+`grad(u)`                     |  $\nabla u$ or $\nabla \mathbf{u}$                                               
+`div(u)`                      |  $\nabla \cdot \mathbf{u}$                                                         
+`curl(u)`                     |  $\nabla \times \mathbf{u}$                                                        
+`rot(u) `                     |  2D rotational $\nabla \times u$                                                   
+`convect(a, u)`               |  $\left( \mathbf{a} \cdot \nabla \right) \mathbf{u}$                               
+`trace(u)`                    |  trace $\gamma(u)$                                                                 
+`Dn(u)`                       |  normal derivative $\partial_{\mathbf{n}} u$                                       
+`D(u)`                        |  Strain tensor $\frac{1}{2}\left(\nabla \mathbf{u} + \nabla \mathbf{u}^T \right) $ 
+`laplace(u)`                  |  Laplace $\Delta u$                                                                
+`hessian(u)`                  |  Hessian $H(u)$                                                                    
+`bracket(u,v)`                |  Poisson Bracket $[u,v]$                                                           
+              
+## Additional operators
+
+sympde notation | Mathematical notion 
+--- | ---
+`jump(u)`      |  jump of $u$, *i.e.* $[u]$                            
+`avg(u)`       |  average of $u$, *i.e.* $\langle u \rangle$           
+`conv(K,u)`    |  convolution of $u$ with a kernel $K$, *i.e.* $K * u$ 
+
+## Integral operators 
+
+sympde notation | Mathematical notion 
+--- | ---
+`DomainIntegral(f)`   | integral over a domain, *i.e.* $(f, \Omega) \mapsto \int_{\Omega} f ~d\Omega$
+`BoundaryIntegral(f)` | integral over a boundary, *i.e.* $(f, \Gamma) \mapsto \int_{\Gamma} f ~d\partial\Omega$
+`PathIntegral(F)`     | integral over an oriented path, *i.e.* $(\mathbf{F}, C) \mapsto \int_{C} \mathbf{F} \cdot d\mathbf{s}$
+`SurfaceIntegral(F)`  | integral over an oriented surface, *i.e.* $(\mathbf{F}, S) \mapsto \int_{S} \mathbf{F} \cdot d\mathbf{S}$
diff --git a/_sources/chapter1/topology.md b/_sources/chapter1/topology.md
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+++ b/_sources/chapter1/topology.md
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+# Topology concepts
+
+## Domain
+
+Mathematicaly speaking, a domain $\Omega$ is defined as an interior $\a{\Omega}$ and a boundary $\partial \Omega$.
+A domain is defined by its **interior** and its **boundaries**. For instance, a domain without a boundary is periodic, since, **sympde** does not consider infinite domains.
+
+In addition to these two attributes, a **Domain** can also be defined as the image of a *logical domain* using a *mapping*.
+
+Finally, a **Domain** can have subdomains, in which case, a **connectivity** must be provided as well.
+
+The simplest **Domain** is the **NCube** object. For instance, **Line**, **Square** and **Cube** are specific cases of **NCube**.
+
+Let's take a look at the **Line** object.
+
+```python
+>>> from sympde import topology
+>>> domain = topology.Line()
+>>> print(type(domain))
+sympde.topology.domain.Line
+```
+
+The parent class of **Line** is **NCube** as we can see in our example
+
+```python
+>>> print(type(topology.Line.__bases__))
+sympde.topology.domain.NCube
+```
+
+In fact, the objects **Line**, **Square** and **Cube** are subclasses of **NCube** which is a subclasse of **Domain** that is an extension of the **BasicDomain** object.
+
+
+### Properties
+
+- **name**: returns the domain's name
+- **interior**: returns the interior of the domain
+- **boundary**: returns the boundary of the domain
+- **mapping**: returns the mapping, if associated to the domain
+- **logical_domain**: returns the logical domain if a mapping is used
+- **connectivity**: returns information about the interfaces of the subdomains
+- **dim**: dimension of the space
+- **dtype**: a dictionary that contains information about the domain
+- **interfaces**: returns the interfaces of the subdomains 
+- **corners**: returns the corners of the domain
+
+**TODO** add inherited properties
+
+### Examples
+
+#### Example 1
+
+```python
+>>> from sympde.topology import InteriorDomain, Boundary, Domain
+
+>>> Omega_1 = InteriorDomain('Omega_1', dim=2)
+>>> Omega_2 = InteriorDomain('Omega_2', dim=2)
+
+>>> Gamma_1 = Boundary('Gamma_1', Omega_1)
+>>> Gamma_2 = Boundary('Gamma_2', Omega_2)
+>>> Gamma_3 = Boundary('Gamma_3', Omega_2)
+
+>>> Omega = Domain('Omega',
+                   interiors=[Omega_1, Omega_2],
+                   boundaries=[Gamma_1, Gamma_2, Gamma_3])
+
+>>> print( Omega.dim )
+2
+>>> print( len(Omega.interior) )
+2
+>>> print( len(Omega.boundary) )
+3
+```
+
+#### Example 2
+
+```python
+>>> from sympde.topology import InteriorDomain, Boundary, Domain
+
+>>> Omega_1 = InteriorDomain('Omega_1', dim=2)
+
+>>> Gamma_1 = Boundary('Gamma_1', Omega_1)
+>>> Gamma_2 = Boundary('Gamma_2', Omega_1)
+
+>>> Omega = Domain('Omega',
+                   interiors=[Omega_1],
+                   boundaries=[Gamma_1, Gamma_2])
+
+>>> assert(Omega.boundary == Union(Gamma_1, Gamma_2))
+>>> assert(Omega.boundary.complement(Gamma_1) == Gamma_2)
+>>> assert(Omega.boundary - Gamma_1 == Gamma_2)
+```
+
+## PeriodicDomain
+
+Similar to the **Domain** object, a **PeriodicDomain** is defined by giving a **Domain** and the associated **periods**.
+
+### Properties
+
+- **domain**: returns the domain
+- **periods**: returns the flags for each axis
+- **boundary**: return the boundary of the domain
+- **dim** returns the dimension of the domain
+- **coordinates** return the coordinates symbols
+
+## NCube
+
+An **NCube** is a **Domain** that models hypercubes in $\mathbb{R}^n$. It is defined by specifying its **name**, **dim**, **min_coords** and **max_coords**.
+
+The interior of a **NCube** is defined as **NCubeInterior**.
+To make things easier, we provide specific classes in the case where $n \in \{ 1, 2, 3 \}$.
+
+### Line
+This corresponds to the case where $n=1$, meaning it is $(x_{min}, x_{max})$.
+
+#### Properties
+- **bounds**: the bounds for the line, i.e. $x_{min}$ and $x_{max}$.
+
+### Square
+This corresponds to the case where $n=2$, meaning it is $(x_{min}, x_{max}) \times (y_{min}, y_{max})$.
+
+#### Properties
+- **bounds1**: the bounds in the first axis, i.e. $x_{min}$ and $x_{max}$.
+- **bounds2**: the bounds in the first axis, i.e. $y_{min}$ and $y_{max}$.
+
+### Cube
+This corresponds to the case where $n=3$, meaning it is $(x_{min}, x_{max}) \times (y_{min}, y_{max}) \times (z_{min}, z_{max})$.
+
+#### Properties
+- **bounds1**: the bounds in the first axis, i.e. $x_{min}$ and $x_{max}$.
+- **bounds2**: the bounds in the first axis, i.e. $y_{min}$ and $y_{max}$.
+- **bounds3**: the bounds in the first axis, i.e. $z_{min}$ and $z_{max}$.
+
+
+## InteriorDomain
+
+**TODO**
+
+### Examples
+#### Example 1
+
+In this example, we create two interior domains and then we build their union.
+
+```python
+>>> from sympde.topology import InteriorDomain, Union
+
+>>> D1 = InteriorDomain('D1', dim=2)
+>>> D2 = InteriorDomain('D2', dim=2)
+
+>>> # Union is commutative
+>>> assert( Union(D2, D1) == Union(D1, D2) )
+
+>>> D = Union(D1, D2)
+>>> # the union will be of dimension 2, and has a length of 2
+>>> print(D.dim)
+2
+>>> print(len(D))
+2
+>>> # we see it clearly when converting it to a dictionary
+>>> print(D.todict())
+[{'name': 'D1', 'mapping': 'None'}, {'name': 'D2', 'mapping': 'None'}]
+```
+
+
+## NCubeInterior
+
+This class defines the interior of a domain which is **NCube**.
+
+### Properties
+
+- **min_coords**
+- **max_coords**
+- **boundary**
+
+
+## Additional geometry objects
+
+### Vectors
+
+#### BoundaryVector
+
+is an extension of the **IndexedBase** object from **sympy**.
+
+#### NormalVector
+
+defines the normal vector to boundaries. It is an extension of the **BoundaryVector** object.
+
+#### MinusNormalVector
+
+defines the inward normal vector to a boundary. It is an extension of the **NormalVector** object.
+
+#### MinusNormalVector
+
+defines the outward normal vector to a boundary. It is an extension of the **NormalVector** object.
+
+#### TangentVector
+
+defines the tangent vector to boundaries. It is an extension of the **BoundaryVector** object.
+
+### ElementDomain
+
+defines an element of the domain.
+
+### BasicArea
+
+A basic class defining the concept of an area. It is an extension of the **AtomicExpr** object from **sympy**, this way we can use it in arithmetic expressions.
+
+### DomainArea
+
+defines the area of a **Domain**.
+
+### ElementArea
+
+defines the area of an element of a **Domain**.
+
+### Geometry Operations
+
+**TODO**
+
+#### BasicGeometryOperator
+
+**TODO**
+
+#### Area
+
+**TODO**
diff --git a/_sources/chapter2/advection-diffusion-stabilized.ipynb b/_sources/chapter2/advection-diffusion-stabilized.ipynb
new file mode 100644
index 0000000..f70ea54
--- /dev/null
+++ b/_sources/chapter2/advection-diffusion-stabilized.ipynb
@@ -0,0 +1,356 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9f28f9af",
+   "metadata": {},
+   "source": [
+    "# Stabilized advection-diffusion equation\n",
+    "\n",
+    "In the sequel we consider two different stabilization methods, leading to formulations of the form\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\text{find $u \\in V$ such that} \\quad \n",
+    "  a(u,v) + a_s(u,v) = l(v) + l_s(v) \\quad \\forall v \\in V,\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $a_s$ and $l_s$ are additional terms depending on the stabilization method.\n",
+    "\n",
+    "The streamline upwind Petrov-Galerkin (SUPG) method is given by\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "    a_s(u,v) = \\int_{\\Omega} \\tau ~ L(u) ~ \\left( \\mathbf{b} \\cdot \\nabla v \\right) ~d\\Omega \n",
+    "    \\quad \\mbox{and} \\quad\n",
+    "    l_s(v) = \\int_{\\Omega} \\tau ~ f ~ \\left( \\mathbf{b} \\cdot \\nabla v \\right) ~d\\Omega \n",
+    "  && \\text{[SUPG]}\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "while the Galerkin least squares (GLS) method is as follows, \n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "    a_s(u,v) = \\int_{\\Omega} \\tau ~ L(u) ~ L(v) ~d\\Omega \n",
+    "    \\quad \\mbox{and} \\quad\n",
+    "    l_s(v) = \\int_{\\Omega} \\tau ~ f ~ L(v) ~d\\Omega \n",
+    "  && \\text{[GLS]}\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where we introduced the differential operator $L(u) := - \\kappa \\nabla^2 u + \\mathbf{b} \\cdot \\nabla u$.\n",
+    "\n",
+    "Notice that in these formulations $\\tau$ stands for a piecewise function that is constant on each element of the tessellation associated to the computational domain $\\Omega$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a958607",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cccc1cc5",
+   "metadata": {},
+   "source": [
+    "### Build a manifactured solution"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "d742586c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import ScalarFunctionSpace, Square, element_of\n",
+    "from sympde.calculus import grad, dot, laplace\n",
+    "from sympde.core     import Constant\n",
+    "\n",
+    "from sympy import pi, sin, Tuple\n",
+    "\n",
+    "domain = Square()\n",
+    "\n",
+    "x,y = domain.coordinates\n",
+    "\n",
+    "kappa = Constant('kappa', is_real=True)\n",
+    "b1 = 1.\n",
+    "b2 = 0.\n",
+    "b = Tuple(b1, b2)\n",
+    "\n",
+    "L = lambda w: -kappa * laplace(w) + dot(b,grad(w))\n",
+    "\n",
+    "ue = sin(pi*x)*sin(pi*y)\n",
+    "f = L(ue)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "76d3d381",
+   "metadata": {},
+   "source": [
+    "You can have the full expression of **f** by calling the **TerminalExpr** as follows"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "87e62069",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "2*kappa*pi**2*sin(pi*x1)*sin(pi*x2) + 1.0*pi*sin(pi*x2)*cos(pi*x1)\n"
+     ]
+    }
+   ],
+   "source": [
+    "from sympde.expr import TerminalExpr\n",
+    "\n",
+    "print(TerminalExpr(f, domain))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fcddd2ca",
+   "metadata": {},
+   "source": [
+    "Let's go back now to our formal model."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "e4bbd1a8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "V = ScalarFunctionSpace('V', domain)\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
+    "\n",
+    "# bilinear form\n",
+    "expr = kappa * dot(grad(v), grad(u)) + dot(b, grad(u)) * v\n",
+    "a = BilinearForm((u,v), integral(domain, expr))\n",
+    "\n",
+    "# linear form\n",
+    "l = LinearForm(v, integral(domain, f*v))\n",
+    "\n",
+    "tau   = Constant('tau', is_real=True)\n",
+    "\n",
+    "# ...\n",
+    "s_supg = BilinearForm((v,u), integral(domain, tau * L(u) * dot(b, grad(v))))\n",
+    "\n",
+    "l_supg = LinearForm(v, integral(domain, tau * f * dot(b, grad(v))))\n",
+    "# ...\n",
+    "\n",
+    "# ...\n",
+    "s_gls = BilinearForm((v,u), integral(domain, tau * L(u) * L(v)))\n",
+    "\n",
+    "l_gls = LinearForm(v, integral(domain, tau * f * L(v)))\n",
+    "# ...\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(u, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation_supg = find(u, forall=v, lhs=a(u, v) + s_supg(u,v), rhs=l(v) + l_supg(v), bc=bc)\n",
+    "equation_gls  = find(u, forall=v, lhs=a(u, v) + s_gls(u,v), rhs=l(v) + l_gls(v), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f983ece",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51095918",
+   "metadata": {},
+   "source": [
+    "We shall need the **discretize** function from **PsyDAC**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "a2a0a2a1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from psydac.api.discretization import discretize"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "00e54163",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2]\n",
+    "ncells = [8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "5999c62b",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/home/ranania/PYCCEL/IGA-Python/.iga-python/lib/python3.10/site-packages/sympy/matrices/repmatrix.py:90: SymPyDeprecationWarning: \n",
+      "\n",
+      "non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use\n",
+      "list of lists, TableForm or some other data structure instead. See\n",
+      "https://github.com/sympy/sympy/issues/21497 for more info.\n",
+      "\n",
+      "  ).warn()\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation SUPG\n",
+    "equation_supg_h = discretize(equation_supg, domain_h, [Vh, Vh])\n",
+    "\n",
+    "# Discretize equation GLS\n",
+    "equation_gls_h = discretize(equation_gls, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b29fbcf",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE\n",
+    "\n",
+    "Since the problem is not symmetric, we shall use **gmres** for the linear solver."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "67b32148",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "equation_supg_h.set_solver('gmres', info=False, tol=1e-8)\n",
+    "equation_gls_h.set_solver('gmres', info=False, tol=1e-8)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "541192ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "uh_supg = equation_supg_h.solve(kappa=1.e-1, tau=1.e-3)\n",
+    "uh_gls  = equation_gls_h.solve(kappa=1.e-1, tau=1.e-3)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5174c4b5",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a31c46f",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "5925c6cd",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      ">>> SUPG =  0.002139370600719921\n",
+      ">>> GLS =  0.00021873265193415864\n"
+     ]
+    }
+   ],
+   "source": [
+    "u = element_of(V, name='u')\n",
+    "\n",
+    "# create the formal Norm object\n",
+    "l2norm = Norm(u - ue, domain, kind='l2')\n",
+    "\n",
+    "# discretize the norm\n",
+    "l2norm_h = discretize(l2norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "l2_error = l2norm_h.assemble(u=uh_supg)\n",
+    "\n",
+    "# print the result\n",
+    "print('>>> SUPG = ', l2_error)\n",
+    "\n",
+    "# assemble the norm\n",
+    "l2_error = l2norm_h.assemble(u=uh_gls)\n",
+    "\n",
+    "# print the result\n",
+    "print('>>> GLS = ', l2_error)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0a972a4f",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/advection-diffusion.ipynb b/_sources/chapter2/advection-diffusion.ipynb
new file mode 100644
index 0000000..ebe73e1
--- /dev/null
+++ b/_sources/chapter2/advection-diffusion.ipynb
@@ -0,0 +1,377 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9f28f9af",
+   "metadata": {},
+   "source": [
+    "# Advection-diffusion equation\n",
+    "\n",
+    "We consider the advection-diffusion problem consisting of finding a scalar-valued function $u$ such that\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\begin{cases}\n",
+    "    - \\kappa \\nabla^2 u + \\mathbf{b} \\cdot \\nabla u = f &\\text{in $\\Omega$}, \\\\\n",
+    "    u = 0                                               &\\text{on $\\partial \\Omega$}.\n",
+    "  \\end{cases}\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "## The Variational Formulation\n",
+    "\n",
+    "The variational formulation reads\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\text{find $u \\in V$ such that} \\quad a(u,v) = l(v) \\quad \\forall v \\in V,\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where \n",
+    "\n",
+    "- $V \\subset H_0^1(\\Omega)$, \n",
+    "- $a(u,v) := \\int_{\\Omega} \\left( \\kappa \\nabla u \\cdot \\nabla v + \\left( \\mathbf{b} \\cdot \\nabla u \\right) v \\right) d\\Omega$,\n",
+    "- $l(v) := \\int_{\\Omega} f v~d\\Omega$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a958607",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "cccc1cc5",
+   "metadata": {},
+   "source": [
+    "### Build a manifactured solution"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "d742586c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import ScalarFunctionSpace, Square, element_of\n",
+    "from sympde.calculus import grad, dot, laplace\n",
+    "from sympde.core     import Constant\n",
+    "\n",
+    "from sympy import pi, sin, Tuple\n",
+    "\n",
+    "domain = Square()\n",
+    "\n",
+    "x,y = domain.coordinates\n",
+    "\n",
+    "kappa = Constant('kappa', is_real=True)\n",
+    "b1 = 1.\n",
+    "b2 = 0.\n",
+    "b = Tuple(b1, b2)\n",
+    "\n",
+    "L = lambda w: -kappa * laplace(w) + dot(b,grad(w))\n",
+    "\n",
+    "ue = sin(pi*x)*sin(pi*y)\n",
+    "f = L(ue)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "76d3d381",
+   "metadata": {},
+   "source": [
+    "You can have the full expression of **f** by calling the **TerminalExpr** as follows"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "87e62069",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "2*kappa*pi**2*sin(pi*x1)*sin(pi*x2) + 1.0*pi*sin(pi*x2)*cos(pi*x1)\n"
+     ]
+    }
+   ],
+   "source": [
+    "from sympde.expr import TerminalExpr\n",
+    "\n",
+    "print(TerminalExpr(f, domain))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fcddd2ca",
+   "metadata": {},
+   "source": [
+    "Let's go back now to our formal model."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "e4bbd1a8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "V = ScalarFunctionSpace('V', domain)\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
+    "\n",
+    "# bilinear form\n",
+    "expr = kappa * dot(grad(v), grad(u)) + dot(b, grad(u)) * v\n",
+    "a = BilinearForm((u,v), integral(domain, expr))\n",
+    "\n",
+    "# linear form\n",
+    "l = LinearForm(v, integral(domain, f*v))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(u, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f983ece",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51095918",
+   "metadata": {},
+   "source": [
+    "We shall need the **discretize** function from **PsyDAC**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "a2a0a2a1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from psydac.api.discretization import discretize"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "00e54163",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2]\n",
+    "ncells = [8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "5999c62b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b29fbcf",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE\n",
+    "\n",
+    "Since the problem is not symmetric, we shall use **gmres** for the linear solver."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "67b32148",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "equation_h.set_solver('gmres', info=False, tol=1e-8)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "id": "541192ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "uh = equation_h.solve(kappa=1.e-1)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5174c4b5",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a31c46f",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "id": "5925c6cd",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.00022156219679932395\n"
+     ]
+    }
+   ],
+   "source": [
+    "u = element_of(V, name='u')\n",
+    "\n",
+    "# create the formal Norm object\n",
+    "l2norm = Norm(u - ue, domain, kind='l2')\n",
+    "\n",
+    "# discretize the norm\n",
+    "l2norm_h = discretize(l2norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "l2_error = l2norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(l2_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6cbfeae",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ semi-norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "id": "e5c1a8b8",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.013033907531543579\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "h1norm = SemiNorm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3c09131c",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "id": "d829e410",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.013035790553238334\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "h1norm = Norm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "0a972a4f",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/biharmonic.ipynb b/_sources/chapter2/biharmonic.ipynb
new file mode 100644
index 0000000..ea848e1
--- /dev/null
+++ b/_sources/chapter2/biharmonic.ipynb
@@ -0,0 +1,374 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9f28f9af",
+   "metadata": {},
+   "source": [
+    "# The Biharmonic problem\n",
+    "\n",
+    "We consider the (inhomogeneous) biharmonic equation with homogeneous essential boundary conditions,\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\nabla^2 \\nabla^2  u = f       \\quad \\text{in $\\Omega$}         , \\quad \\quad \n",
+    "  u = 0                          \\quad \\text{on $\\partial \\Omega$}, \\quad \\quad\n",
+    "  \\nabla u \\cdot \\mathbf{n} = 0  \\quad \\text{on $\\partial \\Omega$}.\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "## The Variational Formulation\n",
+    "\n",
+    "An $H^2$-conforming variational formulation reads \n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\text{find $u \\in V$ such that} \\quad a(u,v) = l(v) \\quad \\forall v \\in V,\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where \n",
+    "\n",
+    "- $V \\subset H^2(\\Omega)$, \n",
+    "- $a(u,v) := \\int_{\\Omega} \\nabla^2 u ~ \\nabla^2 v ~ d\\Omega$,\n",
+    "- $l(v) := \\int_{\\Omega} f v ~ d\\Omega$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a958607",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f223516b",
+   "metadata": {},
+   "source": [
+    "In this example, the analytical solution is given by\n",
+    "\n",
+    "$$\n",
+    "u_e = \\sin(\\pi x)^2 \\sin(\\pi y)^2\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "811c0b9b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import ScalarFunctionSpace, Square, element_of\n",
+    "from sympde.calculus import laplace, dot\n",
+    "\n",
+    "from sympy import pi, sin"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "d742586c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "domain = Square()\n",
+    "\n",
+    "V = ScalarFunctionSpace('V', domain)\n",
+    "\n",
+    "x,y = domain.coordinates\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
+    "\n",
+    "# the analytical solution and its rhs\n",
+    "ue = sin(pi * x)**2 * sin(pi * y)**2\n",
+    "f  = laplace(laplace(ue))\n",
+    "\n",
+    "# bilinear form\n",
+    "a = BilinearForm((u,v), integral(domain , laplace(u)*laplace(v)))\n",
+    "\n",
+    "# linear form\n",
+    "l = LinearForm(v, integral(domain, f*v))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(u, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f983ece",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51095918",
+   "metadata": {},
+   "source": [
+    "We shall need the **discretize** function from **PsyDAC**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "a2a0a2a1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from psydac.api.discretization import discretize"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "00e54163",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2]\n",
+    "ncells = [8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "5999c62b",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/home/ranania/PYCCEL/IGA-Python/.iga-python/lib/python3.10/site-packages/sympy/matrices/repmatrix.py:90: SymPyDeprecationWarning: \n",
+      "\n",
+      "non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use\n",
+      "list of lists, TableForm or some other data structure instead. See\n",
+      "https://github.com/sympy/sympy/issues/21497 for more info.\n",
+      "\n",
+      "  ).warn()\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b29fbcf",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "541192ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "uh = equation_h.solve()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5174c4b5",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a31c46f",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "5925c6cd",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.5367871671842632\n"
+     ]
+    }
+   ],
+   "source": [
+    "u = element_of(V, name='u')\n",
+    "\n",
+    "# create the formal Norm object\n",
+    "l2norm = Norm(u - ue, domain, kind='l2')\n",
+    "\n",
+    "# discretize the norm\n",
+    "l2norm_h = discretize(l2norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "l2_error = l2norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(l2_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6cbfeae",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ semi-norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "e5c1a8b8",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "2.4343747827631277\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "h1norm = SemiNorm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3c09131c",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "id": "d829e410",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "2.492853996086961\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "h1norm = Norm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fe684489",
+   "metadata": {},
+   "source": [
+    "### Computing the H^2 semi-norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "8a4adf8d",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.115861322114647\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "h2norm = SemiNorm(u - ue, domain, kind='h2')\n",
+    "\n",
+    "# discretize the norm\n",
+    "h2norm_h = discretize(h2norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "h2_error = h2norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(h2_error)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "531105ea",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/elliptic-curl.ipynb b/_sources/chapter2/elliptic-curl.ipynb
new file mode 100644
index 0000000..2e4c928
--- /dev/null
+++ b/_sources/chapter2/elliptic-curl.ipynb
@@ -0,0 +1,341 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9f28f9af",
+   "metadata": {},
+   "source": [
+    "# Elliptic-curl Problem\n",
+    "\n",
+    "Let $\\Omega \\subset \\mathbb{R}^d$ be an open Liptschitz bounded set, and we look for the solution of the following problem\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\left\\{ \n",
+    "  \\begin{array}{rl}\n",
+    "    \\nabla \\times \\nabla \\times \\mathbf{u} + \\mu \\mathbf{u} &= \\mathbf{f}, \\quad \\Omega \n",
+    "    \\\\\n",
+    "    \\mathbf{u} \\times \\mathbf{n} &= 0, \\quad \\partial\\Omega\n",
+    "  \\end{array} \\right.\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\mathbf{f} \\in \\mathbf{L}^2(\\Omega)$,  $\\mu \\in L^\\infty(\\Omega)$ and there exists $\\mu_0 > 0$ such that $\\mu \\geq \\mu_0$ almost everywhere.\n",
+    "\n",
+    "## The Variational Formulation\n",
+    "\n",
+    "We take the Hilbert space $V := \\mathbf{H}\\_0(\\mbox{curl}, \\Omega)$, in which case the variational formulation writes \n",
+    "\n",
+    "Find $\\mathbf{u} \\in V$ such that\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  a(\\mathbf{u},\\mathbf{v}) = l(\\mathbf{v}) \\quad \\forall \\mathbf{v} \\in V \n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where \n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\left\\{ \n",
+    "\\begin{array}{rll}\n",
+    "a(\\mathbf{u}, \\mathbf{v}) &:= \\int_{\\Omega} \\nabla \\times \\mathbf{u} \\cdot \\nabla \\times \\mathbf{v} + \\int_{\\Omega} \\mu \\mathbf{u} \\cdot \\mathbf{v}, & \\forall \\mathbf{u}, \\mathbf{v} \\in V  \\\\\n",
+    "l(\\mathbf{v}) &:= \\int_{\\Omega} \\mathbf{v} \\cdot \\mathbf{f}, & \\forall \\mathbf{v} \\in V  \n",
+    "\\end{array} \\right.\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "We recall that in $\\mathbf{H}_0(\\mbox{curl}, \\Omega)$, the bilinear form $a$ is equivalent to the inner product and is therefor continuous and coercive. Hence, our abstract theory applies and there exists a unique solution to the problem."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8d905be9",
+   "metadata": {},
+   "source": [
+    "```{warning}\n",
+    "This NB needs to be fixed\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a958607",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "id": "d742586c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC\n",
+    "from sympde.topology import VectorFunctionSpace, Cube, element_of\n",
+    "from sympde.calculus import curl, dot, grad, div\n",
+    "from sympde.core     import Constant\n",
+    "\n",
+    "from sympy import pi, sin, cos, Tuple\n",
+    "\n",
+    "mu = Constant('mu', is_real=True)\n",
+    "\n",
+    "domain = Cube()\n",
+    "\n",
+    "V = VectorFunctionSpace('V', domain, kind='Hcurl')\n",
+    "\n",
+    "x,y,z = domain.coordinates\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
+    "\n",
+    "# bilinear form\n",
+    "a = BilinearForm((u,v), integral(domain, dot(curl(v), curl(u)) + mu * dot(u,v)))\n",
+    "\n",
+    "# linear form\n",
+    "f1 = sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "f2 = sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "f3 = sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "f = Tuple(f1, f2, f3)\n",
+    "\n",
+    "l = LinearForm(v, integral(domain, dot(f,v)))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(u, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f983ece",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51095918",
+   "metadata": {},
+   "source": [
+    "We shall need the **discretize** function from **PsyDAC**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "id": "a2a0a2a1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from psydac.api.discretization import discretize"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "id": "00e54163",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2,2]\n",
+    "ncells = [8,8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "id": "5999c62b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b29fbcf",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "id": "541192ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#uh = equation_h.solve()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5174c4b5",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm\n",
+    "\n",
+    "When the analytical solution is available, you might be interested in computing the $L^2$ norm or $H^1_0$ semi-norm.\n",
+    "SymPDE allows you to do so, by creating the **Norm** object.\n",
+    "In this example, the analytical solution is given by\n",
+    "\n",
+    "$$\n",
+    "u_e = \\sin(\\pi x) \\sin(\\pi y)\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a31c46f",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "id": "5925c6cd",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.49756716724664e-07\n"
+     ]
+    }
+   ],
+   "source": [
+    "#u = element_of(V, name='u')\n",
+    "\n",
+    "# create the formal Norm object\n",
+    "#l2norm = Norm(u - ue, domain, kind='l2')\n",
+    "\n",
+    "# discretize the norm\n",
+    "#l2norm_h = discretize(l2norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "#l2_error = l2norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "#print(l2_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6cbfeae",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ semi-norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "id": "e5c1a8b8",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.768702410721585e-05\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "#h1norm = SemiNorm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "#h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "#h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "#print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3c09131c",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "id": "d829e410",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.769164089397408e-05\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "#h1norm = Norm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "#h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "#h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "#print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9250897b",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/elliptic-div.ipynb b/_sources/chapter2/elliptic-div.ipynb
new file mode 100644
index 0000000..f4b6601
--- /dev/null
+++ b/_sources/chapter2/elliptic-div.ipynb
@@ -0,0 +1,350 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9f28f9af",
+   "metadata": {},
+   "source": [
+    "# Elliptic-div Problem\n",
+    "\n",
+    "Let $\\Omega \\subset \\mathbb{R}^d$ be an open Liptschitz bounded set, and we look for the solution of the following problem\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\left\\{ \n",
+    "  \\begin{array}{rl}\n",
+    "    - \\nabla \\nabla \\cdot \\mathbf{u} + \\mu \\mathbf{u} &= \\mathbf{f}, \\quad \\Omega \n",
+    "    \\\\\n",
+    "    \\mathbf{u} \\times \\mathbf{n} &= 0, \\quad \\partial\\Omega\n",
+    "  \\end{array} \\right.\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\mathbf{f} \\in \\mathbf{L}^2(\\Omega)$,  $\\mu \\in L^\\infty(\\Omega)$ and there exists $\\mu_0 > 0$ such that $\\mu \\geq \\mu_0$ almost everywhere.\n",
+    "\n",
+    "## The Variational Formulation\n",
+    "\n",
+    "We take the Hilbert space $V := \\mathbf{H}\\_0(\\mbox{div}, \\Omega)$, in which case the variational formulation writes \n",
+    "\n",
+    "\n",
+    "Find $\\mathbf{u} \\in V$ such that\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  a(\\mathbf{u},\\mathbf{v}) = l(\\mathbf{v}) \\quad \\forall \\mathbf{v} \\in V \n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where \n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\left\\{ \n",
+    "\\begin{array}{rll}\n",
+    "a(\\mathbf{u}, \\mathbf{v}) &:= \\int_{\\Omega} \\nabla \\cdot \\mathbf{u} ~ \\nabla \\cdot \\mathbf{v} + \\int_{\\Omega} \\mu \\mathbf{u} \\cdot \\mathbf{v}, & \\forall \\mathbf{u}, \\mathbf{v} \\in V  \\\\\n",
+    "l(\\mathbf{v}) &:= \\int_{\\Omega} \\mathbf{v} \\cdot \\mathbf{f}, & \\forall \\mathbf{v} \\in V  \n",
+    "\\end{array} \\right.\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "We recall that in $\\mathbf{H}\\_0(\\mbox{div}, \\Omega)$, the bilinear form $a$ is equivalent to the inner product and is therefor continuous and coercive. Hence, our abstract theory applies and there exists a unique solution."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2092b000",
+   "metadata": {},
+   "source": [
+    "```{warning}\n",
+    "This NB needs to be fixed\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a958607",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "d742586c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC\n",
+    "from sympde.topology import VectorFunctionSpace, Cube, element_of\n",
+    "from sympde.calculus import div, dot\n",
+    "from sympde.core     import Constant\n",
+    "\n",
+    "from sympy import pi, sin, Tuple\n",
+    "\n",
+    "mu = Constant('mu', is_real=True)\n",
+    "\n",
+    "domain = Cube()\n",
+    "\n",
+    "V = VectorFunctionSpace('V', domain, kind='Hdiv')\n",
+    "\n",
+    "x,y,z = domain.coordinates\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
+    "\n",
+    "# bilinear form\n",
+    "a = BilinearForm((u,v), integral(domain, div(v)*div(u) + mu * dot(u,v)))\n",
+    "\n",
+    "# linear form\n",
+    "f1 = sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "f2 = sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "f3 = sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "f = Tuple(f1, f2, f3)\n",
+    "\n",
+    "l = LinearForm(v, integral(domain, dot(f,v)))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(u, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f983ece",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51095918",
+   "metadata": {},
+   "source": [
+    "We shall need the **discretize** function from **PsyDAC**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "a2a0a2a1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from psydac.api.discretization import discretize"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "00e54163",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2,2]\n",
+    "ncells = [8,8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "5999c62b",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/home/ranania/PYCCEL/IGA-Python/.iga-python/lib/python3.10/site-packages/sympy/matrices/repmatrix.py:90: SymPyDeprecationWarning: \n",
+      "\n",
+      "non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use\n",
+      "list of lists, TableForm or some other data structure instead. See\n",
+      "https://github.com/sympy/sympy/issues/21497 for more info.\n",
+      "\n",
+      "  ).warn()\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b29fbcf",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "541192ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#uh = equation_h.solve()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5174c4b5",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a31c46f",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "id": "5925c6cd",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.49756716724664e-07\n"
+     ]
+    }
+   ],
+   "source": [
+    "#ue = sin(pi*x)*sin(pi*y)\n",
+    "\n",
+    "#u = element_of(V, name='u')\n",
+    "\n",
+    "# create the formal Norm object\n",
+    "#l2norm = Norm(u - ue, domain, kind='l2')\n",
+    "\n",
+    "# discretize the norm\n",
+    "#l2norm_h = discretize(l2norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "#l2_error = l2norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "#print(l2_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6cbfeae",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ semi-norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "id": "e5c1a8b8",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.768702410721585e-05\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "#h1norm = SemiNorm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "#h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "#h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "#print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3c09131c",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "id": "d829e410",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.769164089397408e-05\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "#h1norm = Norm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "#h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "#h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "#print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9250897b",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/elliptic-general-form.ipynb b/_sources/chapter2/elliptic-general-form.ipynb
new file mode 100644
index 0000000..34a98c7
--- /dev/null
+++ b/_sources/chapter2/elliptic-general-form.ipynb
@@ -0,0 +1,278 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "b5e0474c",
+   "metadata": {},
+   "source": [
+    "# Elliptic equation in the general form\n",
+    "\n",
+    "We consider here, the following general form of an elliptic partial differential equation,\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "- \\nabla \\cdot \\left( A \\nabla u \\right) + \\mathbf{b} \\cdot \\nabla u + c u &= f, \\quad \\Omega \\\\\n",
+    "u &= 0, \\quad \\partial \\Omega\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "## Variational Formulation\n",
+    "\n",
+    "An $H^1$-conforming variational formulation of reads\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\text{find $u \\in V$ such that} \\quad a(u,v) = l(v) \\quad \\forall v \\in V,\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where \n",
+    "\n",
+    "- $V \\subset H^1(\\Omega)$, \n",
+    "- $a(u,v) := \\int_{\\Omega} \\left( \\left( A \\nabla u \\right) \\cdot \\nabla v + \\left( \\mathbf{b} \\cdot \\nabla u \\right) v + cuv \\right) ~ d\\Omega$,\n",
+    "- $l(v) := \\int_{\\Omega} f v ~ d\\Omega$.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c5c944a1",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "c2cee2d9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import ScalarFunctionSpace, Square, element_of\n",
+    "from sympde.calculus import grad, dot, div\n",
+    "from sympde.core import Vector, Matrix\n",
+    "\n",
+    "from sympy import pi, sin\n",
+    "\n",
+    "from psydac.api.discretization import discretize\n",
+    "\n",
+    "domain = Square()\n",
+    "\n",
+    "V = ScalarFunctionSpace('V', domain)\n",
+    "\n",
+    "x,y = domain.coordinates\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
+    "\n",
+    "c = x*y\n",
+    "b = Vector([1e-2, 1e-1], name='b')\n",
+    "A = Matrix([[1,1], [0,1]], name='A')\n",
+    "\n",
+    "# bilinear form\n",
+    "expr = dot(grad(v), A * grad(u)) + dot(b, grad(u))*v + c*u*v\n",
+    "a = BilinearForm((u,v), integral(domain, expr))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "366c2d67",
+   "metadata": {},
+   "source": [
+    "### Manifactured solution"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "2b05c79f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# the analytical solution and its rhs\n",
+    "ue = sin(pi * x) * sin(pi * y)\n",
+    "\n",
+    "L = lambda u: - div(A*grad(u)) + dot(b,grad(u)) + c*u\n",
+    "f  = L(ue)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "66b50430",
+   "metadata": {},
+   "source": [
+    "### Formal Equation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "d0d386a2",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "l = LinearForm(v, integral(domain, f*v))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(u, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0614c740",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "96c29a27",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2]\n",
+    "ncells = [8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "ef6a9709",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/home/ranania/PYCCEL/IGA-Python/.iga-python/lib/python3.10/site-packages/sympy/matrices/repmatrix.py:90: SymPyDeprecationWarning: \n",
+      "\n",
+      "non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use\n",
+      "list of lists, TableForm or some other data structure instead. See\n",
+      "https://github.com/sympy/sympy/issues/21497 for more info.\n",
+      "\n",
+      "  ).warn()\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c286e572",
+   "metadata": {},
+   "source": [
+    "### Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "3bafe9f5",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "equation_h.set_solver('gmres', info=False, tol=1e-8)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "b88daf50",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "uh = equation_h.solve()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d865a17f",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "be88e26c",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "id": "3440e74a",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.00021948770944141364\n"
+     ]
+    }
+   ],
+   "source": [
+    "u = element_of(V, name='u')\n",
+    "\n",
+    "# create the formal Norm object\n",
+    "l2norm = Norm(u - ue, domain, kind='l2')\n",
+    "\n",
+    "# discretize the norm\n",
+    "l2norm_h = discretize(l2norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "l2_error = l2norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(l2_error)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5d7d9180",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/linear-elasticity.ipynb b/_sources/chapter2/linear-elasticity.ipynb
new file mode 100644
index 0000000..82b9f1a
--- /dev/null
+++ b/_sources/chapter2/linear-elasticity.ipynb
@@ -0,0 +1,267 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9f28f9af",
+   "metadata": {},
+   "source": [
+    "# Linear Elasticity Problem\n",
+    "\n",
+    "Analysis of deformable structures is essential in engineering, with the equations of linear elasticity being fundamental in this field. In this section, we present the variational formulation of linear elasticity equations using the principle of virtual work.\n",
+    "\n",
+    "## The PDE problem\n",
+    "The governing equations for small elastic deformations of a body $ \\Omega $ can be expressed as:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "    -\\nabla \\cdot \\sigma(u) &= f & \\text{in } \\Omega \\\\\n",
+    "    \\sigma(u) &= \\kappa \\text{tr}(\\epsilon(u))I + 2 \\mu \\epsilon(u)\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $ \\sigma $ is the stress tensor, $ f $ represents the body force per unit volume, $ \\kappa $ and $ \\mu $ are Lamé's elasticity parameters for the material, $ I $ denotes the identity tensor, and $ \\epsilon $ is the symmetric strain tensor. The displacement vector field is denoted by $ u $.\n",
+    "\n",
+    "By substituting $ \\epsilon(u) $ into $ \\sigma $, we obtain:\n",
+    "\n",
+    "$$\n",
+    "\\sigma(u) = \\kappa (\\nabla \\cdot u)I + \\mu(\\nabla u + (\\nabla u)^T)\n",
+    "$$\n",
+    "\n",
+    "## The Variational Formulation\n",
+    "The variational formulation of the linear elasticity equations involves forming the inner product of the PDE with a vector test function $ v \\in \\hat{V} $ and integrating over the domain $ \\Omega $. This yields:\n",
+    "\n",
+    "$$\n",
+    "\\int_{\\Omega} \\sigma : \\nabla v \\, \\mathrm{d} x = \\int_{\\Omega} f \\cdot v \\, \\mathrm{d} x\n",
+    "$$\n",
+    "\n",
+    "Integrating the term $ \\nabla \\cdot \\sigma \\cdot v $ by parts, considering boundary conditions, we obtain:\n",
+    "\n",
+    "$$\n",
+    "\\int_{\\Omega} \\sigma : \\nabla v \\, \\mathrm{d} x = \\int_{\\Omega} f \\cdot v \\, \\mathrm{d} x + \\int_{\\partial \\Omega_T} T \\cdot v \\, \\mathrm{d} s\n",
+    "$$\n",
+    "\n",
+    "where $ T $ represents the traction vector on the part $ \\partial \\Omega_T $ of the boundary where it's prescribed.\n",
+    "\n",
+    "This leads to the variational formulation: Find $ u \\in V $ such that\n",
+    "\n",
+    "$$\n",
+    "a(u, v) = L(v) \\quad \\forall v \\in \\hat{V}\n",
+    "$$\n",
+    "\n",
+    "where\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "    a(u, v) &= \\int_{\\Omega} \\sigma(u) : \\nabla v \\, \\mathrm{d} x \\\\\n",
+    "    L(v) &= \\int_{\\Omega} f \\cdot v \\, \\mathrm{d} x + \\int_{\\partial \\Omega_T} T \\cdot v \\, \\mathrm{d} s\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "This formulation can be alternatively expressed as:\n",
+    "\n",
+    "$$\n",
+    "a(u, v) = \\int_{\\Omega} \\sigma(u) : \\epsilon(v) \\, \\mathrm{d} x\n",
+    "$$\n",
+    "\n",
+    "where $ \\epsilon(v) = \\frac{1}{2} (\\nabla v + (\\nabla v)^T) $ is the symmetric strain tensor.\n",
+    "\n",
+    "This variational formulation is essential for solving linear elasticity problems numerically using methods like the finite element method (FEM)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a958607",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "d742586c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import VectorFunctionSpace, Cube, element_of\n",
+    "from sympde.calculus import grad, dot, inner, outer, cross, div\n",
+    "from sympde.core import Constant\n",
+    "from sympde.core import Matrix, Vector, Transpose\n",
+    "\n",
+    "domain = Cube()\n",
+    "\n",
+    "I = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]], name='I')\n",
+    "\n",
+    "kappa = Constant('kappa', is_real=True)\n",
+    "mu    = Constant('mu',    is_real=True)\n",
+    "rho   = Constant('rho',   is_real=True)\n",
+    "\n",
+    "epsilon = lambda w: 0.5*(grad(w) + Transpose(grad(w)))\n",
+    "sigma   = lambda w: kappa * div(w) * I + 2 * mu * epsilon(w)\n",
+    "\n",
+    "V = VectorFunctionSpace('V', domain)\n",
+    "\n",
+    "x,y,z = domain.coordinates\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "b16cc745",
+   "metadata": {
+    "scrolled": true
+   },
+   "outputs": [],
+   "source": [
+    "# bilinear form\n",
+    "a = BilinearForm((u,v), integral(domain , inner(sigma(u), epsilon(v))))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "0a0f152c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "L = 1\n",
+    "W = 1 #0.2\n",
+    "delta = W / L\n",
+    "\n",
+    "#mu = 1\n",
+    "#rho = 1\n",
+    "#kappa = 1.25\n",
+    "g = 0.4 * delta**2\n",
+    "\n",
+    "# linear form\n",
+    "f = Vector([0, 0, -rho*g], name='f')\n",
+    "l = LinearForm(v, integral(domain, dot(f,v)))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(u, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f983ece",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51095918",
+   "metadata": {},
+   "source": [
+    "We shall need the **discretize** function from **PsyDAC**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "a2a0a2a1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from psydac.api.discretization import discretize"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "00e54163",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2,2]\n",
+    "ncells = [8,8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "5999c62b",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/home/ranania/PYCCEL/IGA-Python/.iga-python/lib/python3.10/site-packages/sympy/matrices/repmatrix.py:90: SymPyDeprecationWarning: \n",
+      "\n",
+      "non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use\n",
+      "list of lists, TableForm or some other data structure instead. See\n",
+      "https://github.com/sympy/sympy/issues/21497 for more info.\n",
+      "\n",
+      "  ).warn()\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b29fbcf",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "541192ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "uh = equation_h.solve(mu=1, rho=1, kappa=1.25)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9250897b",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/poisson-mixed-v1.ipynb b/_sources/chapter2/poisson-mixed-v1.ipynb
new file mode 100644
index 0000000..5d8106a
--- /dev/null
+++ b/_sources/chapter2/poisson-mixed-v1.ipynb
@@ -0,0 +1,283 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "cce2131b",
+   "metadata": {},
+   "source": [
+    "# First mixed formulation of the Poisson problem\n",
+    "\n",
+    "Instead of having one unknown, we now have two, along with the above two equations.\n",
+    "In order to get a mixed variational formulation, we first take the dot product of the first one by $ \\mathbf{v}$ and integrate by parts\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\int_{\\Omega} \\mathbf{u}\\cdot \\mathbf{v}~\\mathrm{d} \\mathbf{x} -\\int_{\\Omega} p ~ \\nabla\\cdot \\mathbf{v}~\\mathrm{d} \\mathbf{x} + \\int_{\\partial\\Omega} p ~  \\mathbf{v}\\cdot \\mathbf{n}~\\mathrm{d} \\sigma = \\int_{\\Omega} \\mathbf{u}\\cdot \\mathbf{v}~\\mathrm{d} \\mathbf{x} -\\int_{\\Omega} p ~ \\nabla\\cdot \\mathbf{v}~\\mathrm{d} \\mathbf{x}=0,\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "using $p=0$ as a natural boundary condition. Then multiplying the second equation by $q$ and integrating yields\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\int_{\\Omega} \\nabla\\cdot\\mathbf{u} ~ q ~\\mathrm{d} \\mathbf{x} = \\int_{\\Omega} f q ~\\mathrm{d} \\mathbf{x}.\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "No integration by parts is necessary here. And we thus get the following mixed variational formulation:\n",
+    "\n",
+    "Find $(\\mathbf{u},p) \\in H(\\mathrm{div},\\Omega)\\times L^2(\\Omega)$ such that\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\left\\{ \n",
+    "\\begin{array}{llll}\n",
+    "  \\int_{\\Omega} \\mathbf{u}\\cdot \\mathbf{v}~\\mathrm{d} \\mathbf{x} &- \\int_{\\Omega} p ~ \\nabla\\cdot \\mathbf{v}~\\mathrm{d} \\mathbf{x} &=0, & \\forall \\mathbf{v}\\in H(\\mathrm{div},\\Omega) \\\\\n",
+    "  \\int_{\\Omega} \\nabla\\cdot\\mathbf{u} ~ q ~\\mathrm{d} \\mathbf{x} &  &= \\int_{\\Omega} f q ~\\mathrm{d} \\mathbf{x}, & \\forall q\\in L^2(\\Omega)\n",
+    "\\end{array} \\right.\n",
+    "\\end{align}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fc928232",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "bcb6daca",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import (ScalarFunctionSpace, VectorFunctionSpace, Square,\n",
+    "                             element_of)\n",
+    "from sympde.calculus import grad, dot, div, inner\n",
+    "from sympde.core     import Constant\n",
+    "\n",
+    "from psydac.api.discretization import discretize\n",
+    "\n",
+    "from sympy import pi, sin, cos, Tuple\n",
+    "\n",
+    "from psydac.api.discretization import discretize\n",
+    "\n",
+    "domain = Square()\n",
+    "\n",
+    "V1 = VectorFunctionSpace('V1', domain, kind='Hdiv')\n",
+    "V2 = ScalarFunctionSpace('V2', domain, kind='L2')\n",
+    "X  = V1*V2\n",
+    "\n",
+    "x,y = domain.coordinates\n",
+    "\n",
+    "# rhs\n",
+    "f = 2*pi**2*sin(pi*x)*sin(pi*y)\n",
+    "# exact solution\n",
+    "pe = sin(pi*x)*sin(pi*y)\n",
+    "#ue = -grad(pe) # not working when computing the norm\n",
+    "ue = Tuple(-pi*cos(pi*x)*sin(pi*y), -pi*sin(pi*x)*cos(pi*y))\n",
+    "\n",
+    "u,v = [element_of(V1, name=i) for i in ['u', 'v']]\n",
+    "p,q = [element_of(V2, name=i) for i in ['p', 'q']]\n",
+    "\n",
+    "# bilinear form\n",
+    "a  = BilinearForm(((u,p),(v,q)), integral(domain, dot(u,v) - p*div(v) + div(u)*q) )\n",
+    "\n",
+    "# linear form\n",
+    "l  = LinearForm((v,q), integral(domain, f*q))\n",
+    "\n",
+    "# Variational problem\n",
+    "equation = find([u,p], forall=[v,q], lhs=a((u,p),(v,q)), rhs=l(v,q))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3cc341c5",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "e5234456",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2]\n",
+    "ncells = [8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "1d8e59bb",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/home/ranania/PYCCEL/IGA-Python/.iga-python/lib/python3.10/site-packages/sympy/matrices/repmatrix.py:90: SymPyDeprecationWarning: \n",
+      "\n",
+      "non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use\n",
+      "list of lists, TableForm or some other data structure instead. See\n",
+      "https://github.com/sympy/sympy/issues/21497 for more info.\n",
+      "\n",
+      "  ).warn()\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "V1h = discretize(V1, domain_h, degree=degree)\n",
+    "V2h = discretize(V2, domain_h, degree=degree)\n",
+    "Xh  = discretize(X, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Xh, Xh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4c473ba3",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "02abfb7d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "equation_h.set_solver('gmres', info=False, tol=1e-8)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "e2d935e8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#uh, ph = equation_h.solve()\n",
+    "\n",
+    "from psydac.fem.basic          import FemField\n",
+    "\n",
+    "phi_h = equation_h.solve()\n",
+    "\n",
+    "uh = FemField(V1h)\n",
+    "uh.coeffs[0][:] = phi_h.coeffs[0][:]\n",
+    "uh.coeffs[1][:] = phi_h.coeffs[1][:]\n",
+    "\n",
+    "ph = FemField(V2h)\n",
+    "ph.coeffs[:] = phi_h.coeffs[2][:]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9c790f00",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "432b8d59",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "b6c85077",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# L2 error norm of the velocity field\n",
+    "error_u   = [ue[0]-u[0], ue[1]-u[1]]\n",
+    "l2norm_u  = Norm(error_u, domain, kind='l2')\n",
+    "l2norm_uh = discretize(l2norm_u, domain_h, V1h)\n",
+    "\n",
+    "# L2 error norm of the pressure, after removing the average value from the field\n",
+    "l2norm_p  = Norm(pe - p, domain, kind='l2')\n",
+    "l2norm_ph = discretize(l2norm_p, domain_h, V2h)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "52fdeaec",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      ">>> norm-l2 uh =  0.012985071480053038\n",
+      ">>> norm-l2 ph =  0.004127522959699811\n"
+     ]
+    }
+   ],
+   "source": [
+    "l2norm = l2norm_uh.assemble(u=uh)\n",
+    "print('>>> norm-l2 uh = ', l2norm)\n",
+    "\n",
+    "l2norm = l2norm_ph.assemble(p=ph)\n",
+    "print('>>> norm-l2 ph = ', l2norm)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "a82ae495",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "494e2cca",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/poisson-mixed-v2.ipynb b/_sources/chapter2/poisson-mixed-v2.ipynb
new file mode 100644
index 0000000..103f2ab
--- /dev/null
+++ b/_sources/chapter2/poisson-mixed-v2.ipynb
@@ -0,0 +1,284 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "cce2131b",
+   "metadata": {},
+   "source": [
+    "# Second mixed formulation of the Poisson problem\n",
+    "\n",
+    "Here, we get an alternative formulation by not integrating by parts, the mixed term in the first formulation but in the second. The first formulation simply becomes\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\int_{\\Omega} \\mathbf{u}\\cdot \\mathbf{v}~\\mathrm{d} \\mathbf{x} +\\int_{\\Omega} \\nabla p \\cdot \\mathbf{v}~\\mathrm{d} \\mathbf{x}=0,\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "and the second, removing immediately the boundary term due to the essential boundary condition $q=0$\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\int_{\\Omega}\\nabla \\cdot\\mathbf{u}  ~ q ~\\mathrm{d} \\mathbf{x} = -\\int_{\\Omega}  \\mathbf{u} \\cdot \\nabla q  ~\\mathrm{d} \\mathbf{x} = \\int_{\\Omega} f q ~\\mathrm{d} \\mathbf{x},\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "which leads to the variational formulation\n",
+    "\n",
+    "Find $(\\mathbf{u},p) \\in L^2(\\Omega)^3 \\times H^1_0(\\Omega)$ such that\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\left\\{ \n",
+    "\\begin{array}{llll}\n",
+    "  \\int_{\\Omega} \\mathbf{u}\\cdot \\mathbf{v}~\\mathrm{d} \\mathbf{x} &+ \\int_{\\Omega} \\nabla p \\cdot \\mathbf{v}~\\mathrm{d} \\mathbf{x} &=0, & \\forall \\mathbf{v}\\in L^2(\\Omega)^3 \\\\\n",
+    "  \\int_{\\Omega}  \\mathbf{u} \\cdot \\nabla q  ~\\mathrm{d} \\mathbf{x} & & = -\\int_{\\Omega} f q ~\\mathrm{d} \\mathbf{x}, & \\forall q\\in H^1_0(\\Omega)\n",
+    "\\end{array} \\right.\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "\n",
+    "Note that this formulation actually contains the classical variational formulation for the Poisson equation. Indeed for $q\\in H^1_0(\\Omega)$, $\\nabla q \\in L^2(\\Omega)^3$ can be used as a test function in the first equation. And plugging this into the second we get\n",
+    "\n",
+    "$$\n",
+    "\\int_{\\Omega}  \\nabla p \\cdot \\nabla q  ~\\mathrm{d} \\mathbf{x}  = \\int_{\\Omega} f q ~\\mathrm{d} \\mathbf{x}, \\quad \\forall q\\in H^1_0(\\Omega).\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fc928232",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "bcb6daca",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import (ScalarFunctionSpace, VectorFunctionSpace, Square,\n",
+    "                             element_of)\n",
+    "from sympde.calculus import grad, dot, div, inner\n",
+    "from sympde.core     import Constant\n",
+    "\n",
+    "from psydac.api.discretization import discretize\n",
+    "\n",
+    "from sympy import pi, sin, cos, Tuple\n",
+    "\n",
+    "from psydac.api.discretization import discretize\n",
+    "\n",
+    "domain = Square()\n",
+    "\n",
+    "V1 = VectorFunctionSpace('V1', domain, kind='L2')\n",
+    "V2 = ScalarFunctionSpace('V2', domain, kind='H1')\n",
+    "X  = V1*V2\n",
+    "\n",
+    "x,y = domain.coordinates\n",
+    "\n",
+    "# rhs\n",
+    "f = 2*pi**2*sin(pi*x)*sin(pi*y)\n",
+    "# exact solution\n",
+    "pe = sin(pi*x)*sin(pi*y)\n",
+    "#ue = -grad(pe) # not working when computing the norm\n",
+    "ue = Tuple(-pi*cos(pi*x)*sin(pi*y), -pi*sin(pi*x)*cos(pi*y))\n",
+    "\n",
+    "u,v = [element_of(V1, name=i) for i in ['u', 'v']]\n",
+    "p,q = [element_of(V2, name=i) for i in ['p', 'q']]\n",
+    "\n",
+    "# bilinear form\n",
+    "a  = BilinearForm(((u,p),(v,q)), integral(domain, dot(u,v) + dot(grad(p),v) + dot(u,grad(q))) )\n",
+    "\n",
+    "# linear form\n",
+    "l  = LinearForm((v,q), integral(domain, -f*q))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(p, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation = find([u,p], forall=[v,q], lhs=a((u,p),(v,q)), rhs=l(v,q), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3cc341c5",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "e5234456",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2]\n",
+    "ncells = [8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "1d8e59bb",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/home/ranania/PYCCEL/IGA-Python/.iga-python/lib/python3.10/site-packages/sympy/matrices/repmatrix.py:90: SymPyDeprecationWarning: \n",
+      "\n",
+      "non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use\n",
+      "list of lists, TableForm or some other data structure instead. See\n",
+      "https://github.com/sympy/sympy/issues/21497 for more info.\n",
+      "\n",
+      "  ).warn()\n"
+     ]
+    }
+   ],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "V1h = discretize(V1, domain_h, degree=degree)\n",
+    "V2h = discretize(V2, domain_h, degree=degree)\n",
+    "Xh  = discretize(X, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Xh, Xh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4c473ba3",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "02abfb7d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "equation_h.set_solver('gmres', info=False, tol=1e-8)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "e2d935e8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#uh, ph = equation_h.solve()\n",
+    "\n",
+    "from psydac.fem.basic          import FemField\n",
+    "\n",
+    "phi_h = equation_h.solve()\n",
+    "\n",
+    "uh = FemField(V1h)\n",
+    "uh.coeffs[0][:] = phi_h.coeffs[0][:]\n",
+    "uh.coeffs[1][:] = phi_h.coeffs[1][:]\n",
+    "\n",
+    "ph = FemField(V2h)\n",
+    "ph.coeffs[:] = phi_h.coeffs[2][:]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b7f03590",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "62ad7d46",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "31967fb4",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# L2 error norm of the velocity field\n",
+    "error_u   = [ue[0]-u[0], ue[1]-u[1]]\n",
+    "l2norm_u  = Norm(error_u, domain, kind='l2')\n",
+    "l2norm_uh = discretize(l2norm_u, domain_h, V1h)\n",
+    "\n",
+    "# L2 error norm of the pressure, after removing the average value from the field\n",
+    "l2norm_p  = Norm(pe - p, domain, kind='l2')\n",
+    "l2norm_ph = discretize(l2norm_p, domain_h, V2h)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "9c3f7e15",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      ">>> norm-l2 uh =  64.20821457571618\n",
+      ">>> norm-l2 ph =  3254706679445337.5\n"
+     ]
+    }
+   ],
+   "source": [
+    "l2norm = l2norm_uh.assemble(u=uh)\n",
+    "print('>>> norm-l2 uh = ', l2norm)\n",
+    "\n",
+    "l2norm = l2norm_ph.assemble(p=ph)\n",
+    "print('>>> norm-l2 ph = ', l2norm)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d7adf378",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/poisson-mixed.ipynb b/_sources/chapter2/poisson-mixed.ipynb
new file mode 100644
index 0000000..5243e3b
--- /dev/null
+++ b/_sources/chapter2/poisson-mixed.ipynb
@@ -0,0 +1,57 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "1a4e2078",
+   "metadata": {},
+   "source": [
+    "# Mixed FEM for the Poisson problem\n",
+    "\n",
+    "Let $\\Omega \\subset \\mathbb{R}^3$ and consider the Poisson problem\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\left\\{ \n",
+    "  \\begin{array}{clr}\n",
+    "    -\\Delta p & =f & ,~\\Omega    \\\\\n",
+    "    p         & =0 & ,~\\partial \\Omega\n",
+    "  \\end{array} \\right.\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "Using that $\\Delta p = \\nabla\\cdot\\nabla p$, we set $ \\mathbf{u}=\\nabla p$, then the Poisson equation can be written equivalently\n",
+    "\n",
+    "$$ \\mathbf{u}=-\\nabla p, ~~~ \\nabla\\cdot \\mathbf{u}= f.$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "46339ba9",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/poisson-nitsche.ipynb b/_sources/chapter2/poisson-nitsche.ipynb
new file mode 100644
index 0000000..e6c862e
--- /dev/null
+++ b/_sources/chapter2/poisson-nitsche.ipynb
@@ -0,0 +1,210 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "119eb328",
+   "metadata": {},
+   "source": [
+    "# The Poisson equation with weak imposition of Dirichlet conditions"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a5e01a2",
+   "metadata": {},
+   "source": [
+    "In this example, we consider the following Poisson problem\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  - \\nabla^2 u = f \\quad \\text{in~$\\Omega$}, \\quad \\quad \n",
+    "  u = g            \\quad \\text{on~$\\partial \\Omega$}.\n",
+    "\\end{align}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "505861ef",
+   "metadata": {},
+   "source": [
+    "## The variational formulation\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\text{find $u \\in V$ such that} \\quad a(u,v) = l(v) \\quad \\forall v \\in V,\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where \n",
+    "\n",
+    "- $V \\subset H^1(\\Omega)$, \n",
+    "- $a(u,v) := \\int_{\\Omega} \\nabla u \\cdot \\nabla v ~ d\\Omega + \\int_{\\partial \\Omega} \\left( \\kappa u v - u \\partial_n v - v \\partial_n u \\right)~d\\partial\\Omega $,\n",
+    "- $l(v) := \\int_{\\Omega} f v ~ d\\Omega + \\int_{\\partial\\Omega} \\left( \\kappa g v - g \\partial_n v \\right) ~ d\\Gamma$.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d4f55f53",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "ecd1330a",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC\n",
+    "from sympde.topology import ScalarFunctionSpace, Square, element_of\n",
+    "from sympde.calculus import grad, dot, Dn\n",
+    "from sympde.core import Constant\n",
+    "\n",
+    "from psydac.api.discretization import discretize\n",
+    "\n",
+    "from sympy import pi, sin\n",
+    "\n",
+    "kappa = Constant('kappa', is_real=True)\n",
+    "\n",
+    "domain = Square()\n",
+    "\n",
+    "V = ScalarFunctionSpace('V', domain)\n",
+    "\n",
+    "x,y = domain.coordinates\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
+    "\n",
+    "# bilinear form\n",
+    "a = BilinearForm((u,v), integral(domain, dot(grad(v), grad(u)) + kappa*u*v - u*Dn(v) - v*Dn(u)))\n",
+    "\n",
+    "# linear form\n",
+    "g = sin(pi*x)*sin(pi*y)\n",
+    "f = 2*pi**2*sin(pi*x)*sin(pi*y)\n",
+    "\n",
+    "l = LinearForm(v, integral(domain, f*v + kappa*g*v - g*Dn(v)))\n",
+    "\n",
+    "# Variational problem\n",
+    "equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "11ed2bc5",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "3d86674b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2]\n",
+    "ncells = [8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "94c83267",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "#equation_h = discretize(equation, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "85318e35",
+   "metadata": {},
+   "source": [
+    "### Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "8f1b2d68",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#uh = equation_h.solve()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e96f321c",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c7ae0b9e",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9ad386a6",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "068bdb95",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "aa88a918",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/poisson.ipynb b/_sources/chapter2/poisson.ipynb
new file mode 100644
index 0000000..b4d9f01
--- /dev/null
+++ b/_sources/chapter2/poisson.ipynb
@@ -0,0 +1,336 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9f28f9af",
+   "metadata": {},
+   "source": [
+    "# The Poisson equation\n",
+    "\n",
+    "As a first example, we consider the Poisson equation\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  - \\nabla^2 u = f \\quad \\text{in $\\Omega$}, \\quad \\quad \n",
+    "  u = 0            \\quad \\text{on $\\Gamma_D$}, \\quad \\quad \n",
+    "  \\partial_n u = g \\quad \\text{on $\\Gamma_N$}.\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "## Variational Formulation\n",
+    "\n",
+    "An $H^1$-conforming variational formulation of reads\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\text{find $u \\in V$ such that} \\quad a(u,v) = l(v) \\quad \\forall v \\in V,\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where \n",
+    "\n",
+    "- $V \\subset H^1(\\Omega)$, \n",
+    "- $a(u,v) := \\int_{\\Omega} \\nabla u \\cdot \\nabla v ~ d\\Omega$,\n",
+    "- $l(v) := \\int_{\\Omega} f v ~ d\\Omega + \\int_{\\Gamma_N} g v ~ d\\Gamma$.\n",
+    "\n",
+    "## Examples\n",
+    "\n",
+    "- [Poisson on a square domain with Homogeneous Boundary Conditions](https://github.com/pyccel/sympde/blob/devel-documentation/docs/examples/2d_poisson_dir0.py)\n",
+    "- [Poisson on a square domain with Homogeneous and Neumann Boundary Conditions](https://github.com/pyccel/sympde/blob/devel-documentation/docs/examples/2d_poisson_dir0_neu.py)\n",
+    "- [Poisson on a square domain with Homogeneous, Inhomogeneous and Neumann Boundary Conditions](https://github.com/pyccel/sympde/blob/devel-documentation/docs/examples/2d_poisson_dir_neu.py)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a958607",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "id": "d742586c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import ScalarFunctionSpace, Square, element_of\n",
+    "from sympde.calculus import grad, dot\n",
+    "\n",
+    "from sympy import pi, sin\n",
+    "\n",
+    "domain = Square()\n",
+    "\n",
+    "V = ScalarFunctionSpace('V', domain)\n",
+    "\n",
+    "x,y = domain.coordinates\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
+    "\n",
+    "# bilinear form\n",
+    "a = BilinearForm((u,v), integral(domain , dot(grad(v), grad(u))))\n",
+    "\n",
+    "# linear form\n",
+    "f = 2*pi**2*sin(pi*x)*sin(pi*y)\n",
+    "l = LinearForm(v, integral(domain, f*v))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(u, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "62ac1fd4",
+   "metadata": {},
+   "source": [
+    "\n",
+    "This very simple Python code reflects well the abstract mathematical framework needed for variational formulations.\n",
+    "The structure of the code is as follows,\n",
+    "\n",
+    "1. Create a domain.\n",
+    "2. Create a space of *scalar* functions over the domain.\n",
+    "3. Create elements from this function space. These elements will denote the test and trial functions.\n",
+    "4. Create the Bilinear and Linear forms, $a$ and $l$ respectively.\n",
+    "5. Create Essential Boundary Conditions.\n",
+    "6. Create the variational problem.\n",
+    "\n",
+    "Most of the time, you will need to follow the same steps, with some minor variants depending on the problem you're considering."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f983ece",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51095918",
+   "metadata": {},
+   "source": [
+    "We shall need the **discretize** function from **PsyDAC**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "id": "a2a0a2a1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from psydac.api.discretization import discretize"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "id": "00e54163",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2]\n",
+    "ncells = [8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "id": "5999c62b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b29fbcf",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "id": "541192ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "uh = equation_h.solve()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5174c4b5",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm\n",
+    "\n",
+    "When the analytical solution is available, you might be interested in computing the $L^2$ norm or $H^1_0$ semi-norm.\n",
+    "SymPDE allows you to do so, by creating the **Norm** object.\n",
+    "In this example, the analytical solution is given by\n",
+    "\n",
+    "$$\n",
+    "u_e = \\sin(\\pi x) \\sin(\\pi y)\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a31c46f",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "id": "5925c6cd",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.49756716724664e-07\n"
+     ]
+    }
+   ],
+   "source": [
+    "ue = sin(pi*x)*sin(pi*y)\n",
+    "\n",
+    "u = element_of(V, name='u')\n",
+    "\n",
+    "# create the formal Norm object\n",
+    "l2norm = Norm(u - ue, domain, kind='l2')\n",
+    "\n",
+    "# discretize the norm\n",
+    "l2norm_h = discretize(l2norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "l2_error = l2norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(l2_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6cbfeae",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ semi-norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "id": "e5c1a8b8",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.768702410721585e-05\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "h1norm = SemiNorm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3c09131c",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "id": "d829e410",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "9.769164089397408e-05\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "h1norm = Norm(u - ue, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9250897b",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/stokes-v1.ipynb b/_sources/chapter2/stokes-v1.ipynb
new file mode 100644
index 0000000..785159e
--- /dev/null
+++ b/_sources/chapter2/stokes-v1.ipynb
@@ -0,0 +1,268 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "bee50521",
+   "metadata": {},
+   "source": [
+    "# First mixed formulation of the Stokes problem\n",
+    "\n",
+    "For the variational formulation, we take the dot product of the first equation with $v$ and integrate over the whole domain\n",
+    "\n",
+    "$$\n",
+    "\\int_{\\Omega} (-\\Delta \\mathbf{u} + \\nabla p)\\cdot \\mathbf{v} ~\\mathrm{d} \\mathbf{x} = \\int_{\\Omega}\\nabla \\mathbf{u}:\\nabla \\mathbf{v} ~\\mathrm{d} \\mathbf{x} + \\int_{\\Omega} \\nabla p \\cdot \\mathbf{v} ~\\mathrm{d} \\mathbf{x} = \\int_{\\Omega} \\mathbf{f}\\cdot \\mathbf{v} ~\\mathrm{d} \\mathbf{x}\n",
+    "$$\n",
+    "\n",
+    "The integration by parts is performed component by component. We impose the essential boundary condition $\\mathbf{v}=0$ on $\\partial\\Omega$, and we denote by\n",
+    "\n",
+    "$$\n",
+    "\\int_{\\Omega}\\nabla \\mathbf{u}:\\nabla \\mathbf{v} ~\\mathrm{d} \\mathbf{x} =\\sum_{i=1}^3 \\int_{\\Omega}\\nabla u_i\\cdot\\nabla v_i ~\\mathrm{d} \\mathbf{x} =\\sum_{i,j=1}^3 \\int_{\\Omega}\\partial_j u_i\\partial_j v_i ~\\mathrm{d} \\mathbf{x} .\n",
+    "$$\n",
+    "\n",
+    "We now need to deal with the constraint $\\nabla\\cdot \\mathbf{u}=0$. The theoretical framework for saddle point problems requires that the corresponding bilinear form is the same as the second one appearing in the first part of the variational formulation. To this aim we multiply  $\\nabla\\cdot \\mathbf{u}=0$ by a scalar test function (which will be associated to $p$) and integrate on the full domain, with an integration by parts in order to get the same bilinear form as in the first equation\n",
+    "\n",
+    "$$\n",
+    "\\int_{\\Omega} \\nabla\\cdot \\mathbf{u} ~ q ~\\mathrm{d} \\mathbf{x}= - \\int_{\\Omega} \\mathbf{u}\\cdot \\nabla q~\\mathrm{d} \\mathbf{x}=0,\n",
+    "$$\n",
+    "\n",
+    "using that $q=0$ on the boundary as an essential boundary condition.\n",
+    "We finally obtain the mixed variational formulation: \n",
+    "\n",
+    "Find $( \\mathbf{u},p)\\in H^1_0(\\Omega)^3\\times H^1_0(\\Omega)$ such that\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\left\\{\n",
+    "  \\begin{array}{llll}\n",
+    "    \\int_{\\Omega}\\nabla \\mathbf{u}:\\nabla \\mathbf{v} ~\\mathrm{d} \\mathbf{x} &+ \\int_{\\Omega} \\nabla p \\cdot \\mathbf{v} ~\\mathrm{d} \\mathbf{x}\n",
+    "    &= \\int_{\\Omega} \\mathbf{f}\\cdot \\mathbf{v} ~\\mathrm{d} \\mathbf{x}, & \\forall \\mathbf{v}\\in H_0^1(\\Omega)^3 \\\\\n",
+    "    \\int_{\\Omega} \\mathbf{u}\\cdot \\nabla q~\\mathrm{d} \\mathbf{x} & &=0, & \\forall p\\in H^1_0(\\Omega)\n",
+    "  \\end{array}\n",
+    "  \\right.\n",
+    "\\end{align}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a958607",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "d742586c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import (ScalarFunctionSpace, VectorFunctionSpace, Square,\n",
+    "                             elements_of)\n",
+    "from sympde.calculus import grad, dot, div, inner\n",
+    "from sympde.core     import Constant\n",
+    "\n",
+    "from psydac.api.discretization import discretize\n",
+    "\n",
+    "from sympy import pi, sin, cos, Tuple\n",
+    "\n",
+    "domain = Square()\n",
+    "x, y   = domain.coordinates\n",
+    "\n",
+    "V1 = VectorFunctionSpace('V1', domain, kind='H1')\n",
+    "V2 = ScalarFunctionSpace('V2', domain, kind='H1')\n",
+    "X  = V1*V2\n",
+    "\n",
+    "# rhs\n",
+    "fx = -x**2*(x - 1)**2*(24*y - 12) - 4*y*(x**2 + 4*x*(x - 1) + (x - 1)**2)*(2*y**2 - 3*y + 1) - 2*pi*cos(2*pi*x)\n",
+    "fy = 4*x*(2*x**2 - 3*x + 1)*(y**2 + 4*y*(y - 1) + (y - 1)**2) + y**2*(24*x - 12)*(y - 1)**2 + 2*pi*cos(2*pi*y)\n",
+    "f  = Tuple(fx, fy)\n",
+    "\n",
+    "# exact solution\n",
+    "ue1 = x**2*(-x + 1)**2*(4*y**3 - 6*y**2 + 2*y)\n",
+    "ue2 =-y**2*(-y + 1)**2*(4*x**3 - 6*x**2 + 2*x)\n",
+    "ue = Tuple(ue1, ue2)\n",
+    "pe = -sin(2*pi*x) + sin(2*pi*y)\n",
+    "\n",
+    "u, v = elements_of(V1, names='u, v')\n",
+    "p, q = elements_of(V2, names='p, q')\n",
+    "\n",
+    "# bilinear form\n",
+    "a  = BilinearForm(((u, p), (v, q)), integral(domain, inner(grad(u), grad(v)) + dot(grad(p), v) + dot(u, grad(q)) ) )\n",
+    "\n",
+    "# linear form\n",
+    "l  = LinearForm((v, q), integral(domain, dot(f, v)))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = EssentialBC(u, 0, domain.boundary)\n",
+    "\n",
+    "equation = find((u, p), forall=(v, q), lhs=a((u, p), (v, q)), rhs=l(v, q), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f983ece",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "00e54163",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2]\n",
+    "ncells = [8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "5999c62b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "V1h = discretize(V1, domain_h, degree=degree)\n",
+    "V2h = discretize(V2, domain_h, degree=degree)\n",
+    "Xh  = discretize(X, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Xh, Xh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b29fbcf",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "id": "6230e0ac",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "equation_h.set_solver('gmres', info=False, tol=1e-8)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "541192ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#uh, ph = equation_h.solve()\n",
+    "\n",
+    "from psydac.fem.basic          import FemField\n",
+    "\n",
+    "phi_h = equation_h.solve()\n",
+    "\n",
+    "uh = FemField(V1h)\n",
+    "uh.coeffs[0][:] = phi_h.coeffs[0][:]\n",
+    "uh.coeffs[1][:] = phi_h.coeffs[1][:]\n",
+    "\n",
+    "ph = FemField(V2h)\n",
+    "ph.coeffs[:] = phi_h.coeffs[2][:]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5174c4b5",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a31c46f",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "5925c6cd",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# L2 error norm of the velocity field\n",
+    "error_u   = [ue[0]-u[0], ue[1]-u[1]]\n",
+    "l2norm_u  = Norm(error_u, domain, kind='l2')\n",
+    "l2norm_uh = discretize(l2norm_u, domain_h, V1h)\n",
+    "\n",
+    "# L2 error norm of the pressure, after removing the average value from the field\n",
+    "l2norm_p  = Norm(pe - p, domain, kind='l2')\n",
+    "l2norm_ph = discretize(l2norm_p, domain_h, V2h)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "d829e410",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      ">>> norm-l2 uh =  5.852908166372369e-05\n",
+      ">>> norm-l2 ph =  0.01411215698700372\n"
+     ]
+    }
+   ],
+   "source": [
+    "l2norm = l2norm_uh.assemble(u=uh)\n",
+    "print('>>> norm-l2 uh = ', l2norm)\n",
+    "\n",
+    "l2norm = l2norm_ph.assemble(p=ph)\n",
+    "print('>>> norm-l2 ph = ', l2norm)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9250897b",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/stokes-v2.ipynb b/_sources/chapter2/stokes-v2.ipynb
new file mode 100644
index 0000000..7bd746e
--- /dev/null
+++ b/_sources/chapter2/stokes-v2.ipynb
@@ -0,0 +1,255 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "48ac8740",
+   "metadata": {},
+   "source": [
+    "# Second mixed formulation of the Stokes problem\n",
+    "\n",
+    "Another possibility to obtained a well posed variational formulation, is to integrate by parts the\n",
+    "$\\int_{\\Omega} \\nabla p \\cdot \\mathbf{v} ~\\mathrm{d} \\mathbf{x}$ term in the first formulation:\n",
+    "\n",
+    "$$ \n",
+    "\\int_{\\Omega} \\nabla p \\cdot \\mathbf{v} ~\\mathrm{d} \\mathbf{x} = - \\int_{\\Omega} p \\nabla \\cdot \\mathbf{v} ~\\mathrm{d} \\mathbf{x} + \\int_{\\partial\\Omega} p \\mathbf{v} \\cdot \\mathbf{n}~\\mathrm{d} \\sigma= -\\int_{\\Omega} p ~ \\nabla \\cdot \\mathbf{v} ~\\mathrm{d} \\mathbf{x} ,\n",
+    "$$\n",
+    "\n",
+    " using here $p=0$ as a natural boundary condition. Note that in the other variational formulation the same boundary condition was essential. In this case, for the second variational formulation, we just multiply $\\nabla\\cdot \\mathbf{u}=0$ by $q$ and integrate. No integration by parts is needed in this case.\n",
+    "\n",
+    "$$\n",
+    "\\int_{\\Omega} \\nabla \\cdot \\mathbf{u} ~ q ~\\mathrm{d} \\mathbf{x} =0.\n",
+    "$$\n",
+    "\n",
+    "This then leads to the following variational formulation:\n",
+    "\n",
+    "Find $( \\mathbf{u},p)\\in H^1(\\Omega)^3\\times L^2(\\Omega)$ such that \n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\left\\{\n",
+    "  \\begin{array}{llll}\n",
+    "    \\int_{\\Omega}\\nabla \\mathbf{u}:\\nabla \\mathbf{v} ~\\mathrm{d} \\mathbf{x} &- \\int_{\\Omega}  p ~ \\nabla\\cdot \\mathbf{v} ~\\mathrm{d} \\mathbf{x}\n",
+    "    &= \\int_{\\Omega} \\mathbf{f}\\cdot \\mathbf{v} ~\\mathrm{d} \\mathbf{x}, &\\forall \\mathbf{v}\\in H^1(\\Omega)^3\n",
+    "    \\\\\n",
+    "    \\int_{\\Omega}  \\nabla\\cdot\\mathbf{u} ~ q~\\mathrm{d} \\mathbf{x} & &=0,  &\\forall q\\in L^2(\\Omega)\n",
+    "  \\end{array}\n",
+    "  \\right.\n",
+    "\\end{align}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a958607",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "d742586c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import (ScalarFunctionSpace, VectorFunctionSpace, Square,\n",
+    "                             elements_of)\n",
+    "from sympde.calculus import grad, dot, div, inner\n",
+    "from sympde.core     import Constant\n",
+    "\n",
+    "from psydac.api.discretization import discretize\n",
+    "\n",
+    "from sympy import pi, sin, cos, Tuple\n",
+    "\n",
+    "domain = Square()\n",
+    "x, y   = domain.coordinates\n",
+    "\n",
+    "V1 = VectorFunctionSpace('V1', domain, kind='H1')\n",
+    "V2 = ScalarFunctionSpace('V2', domain, kind='L2')\n",
+    "X = V1*V2\n",
+    "\n",
+    "# rhs\n",
+    "fx = -x**2*(x - 1)**2*(24*y - 12) - 4*y*(x**2 + 4*x*(x - 1) + (x - 1)**2)*(2*y**2 - 3*y + 1) - 2*pi*cos(2*pi*x)\n",
+    "fy = 4*x*(2*x**2 - 3*x + 1)*(y**2 + 4*y*(y - 1) + (y - 1)**2) + y**2*(24*x - 12)*(y - 1)**2 + 2*pi*cos(2*pi*y)\n",
+    "f  = Tuple(fx, fy)\n",
+    "\n",
+    "# exact solution\n",
+    "ue1 = x**2*(-x + 1)**2*(4*y**3 - 6*y**2 + 2*y)\n",
+    "ue2 =-y**2*(-y + 1)**2*(4*x**3 - 6*x**2 + 2*x)\n",
+    "ue = Tuple(ue1, ue2)\n",
+    "pe = -sin(2*pi*x) + sin(2*pi*y)\n",
+    "\n",
+    "u, v = elements_of(V1, names='u, v')\n",
+    "p, q = elements_of(V2, names='p, q')\n",
+    "\n",
+    "# bilinear form\n",
+    "a  = BilinearForm(((u, p), (v, q)), integral(domain, inner(grad(u), grad(v)) - div(u)*q - p*div(v)) )\n",
+    "\n",
+    "# linear form\n",
+    "l  = LinearForm((v, q), integral(domain, dot(f, v)))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = EssentialBC(u, 0, domain.boundary)\n",
+    "\n",
+    "equation = find((u, p), forall=(v, q), lhs=a((u, p), (v, q)), rhs=l(v, q), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f983ece",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "id": "00e54163",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2]\n",
+    "ncells = [8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "5999c62b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "V1h = discretize(V1, domain_h, degree=degree)\n",
+    "V2h = discretize(V2, domain_h, degree=degree)\n",
+    "Xh  = discretize(X, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Xh, Xh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b29fbcf",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "541192ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "equation_h.set_solver('gmres', info=False, tol=1e-8)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "504a182d",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "#uh, ph = equation_h.solve()\n",
+    "\n",
+    "from psydac.fem.basic          import FemField\n",
+    "\n",
+    "phi_h = equation_h.solve()\n",
+    "\n",
+    "uh = FemField(V1h)\n",
+    "uh.coeffs[0][:] = phi_h.coeffs[0][:]\n",
+    "uh.coeffs[1][:] = phi_h.coeffs[1][:]\n",
+    "\n",
+    "ph = FemField(V2h)\n",
+    "ph.coeffs[:] = phi_h.coeffs[2][:]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5174c4b5",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a31c46f",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "5925c6cd",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# L2 error norm of the velocity field\n",
+    "error_u   = [ue[0]-u[0], ue[1]-u[1]]\n",
+    "l2norm_u  = Norm(error_u, domain, kind='l2')\n",
+    "l2norm_uh = discretize(l2norm_u, domain_h, V1h)\n",
+    "\n",
+    "# L2 error norm of the pressure, after removing the average value from the field\n",
+    "l2norm_p  = Norm(pe - p, domain, kind='l2')\n",
+    "l2norm_ph = discretize(l2norm_p, domain_h, V2h)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "e5c1a8b8",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      ">>> norm-l2 uh =  1.860724404008467e-05\n",
+      ">>> norm-l2 ph =  0.024428172461472355\n"
+     ]
+    }
+   ],
+   "source": [
+    "l2norm = l2norm_uh.assemble(u=uh)\n",
+    "print('>>> norm-l2 uh = ', l2norm)\n",
+    "\n",
+    "l2norm = l2norm_ph.assemble(p=ph)\n",
+    "print('>>> norm-l2 ph = ', l2norm)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/stokes.ipynb b/_sources/chapter2/stokes.ipynb
new file mode 100644
index 0000000..04d45a5
--- /dev/null
+++ b/_sources/chapter2/stokes.ipynb
@@ -0,0 +1,47 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9f28f9af",
+   "metadata": {},
+   "source": [
+    "# Mixed FEM for the Stokes problem \n",
+    "\n",
+    "We consider now the Stokes problem for the steady-state modelling of an incompressible fluid\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\left\\{\n",
+    "    \\begin{array}{rl}\n",
+    "      - \\nabla^2 \\mathbf{u} + \\nabla p = \\mathbf{f} & \\mbox{in} ~ \\Omega,  \\\\\n",
+    "      \\nabla \\cdot \\mathbf{u}          = 0   & \\mbox{in} ~ \\Omega,  \\\\\n",
+    "      \\mathbf{u}               = 0   & \\mbox{on} ~ \\partial \\Omega,\n",
+    "    \\end{array}\n",
+    "    \\right.\n",
+    "\\end{align}\n",
+    "$$"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter2/vector-poisson.ipynb b/_sources/chapter2/vector-poisson.ipynb
new file mode 100644
index 0000000..e7ed0a2
--- /dev/null
+++ b/_sources/chapter2/vector-poisson.ipynb
@@ -0,0 +1,320 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "9f28f9af",
+   "metadata": {},
+   "source": [
+    "# Vector Poisson equation \n",
+    "\n",
+    "In this example we consider the vector Poisson equation with homogeneous Dirichlet boundary conditions:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  - \\nabla^2 \\mathbf{u} = \\mathbf{f} \\quad \\mbox{in} ~ \\Omega, \\quad \\quad \n",
+    "  \\mathbf{u} = 0            \\quad \\mbox{on} ~ \\partial \\Omega.\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "## The Variational Formulation\n",
+    "\n",
+    "The corresponding variational formulation, using $\\mathbf{H}^1$ formulation, *i.e.* all components are in $H^1$, reads \n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "  \\text{find $\\mathbf{u} \\in V$ such that} \\quad \n",
+    "  a(\\mathbf{u},\\mathbf{v}) = l(\\mathbf{v}) \\quad \\forall \\mathbf{v} \\in V,\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where \n",
+    "\n",
+    "- $V \\subset \\mathbf{H}_0^1(\\Omega)$, \n",
+    "- $a(\\mathbf{u},\\mathbf{v}) := \\int_{\\Omega} \\nabla \\mathbf{u} : \\nabla \\mathbf{v} ~ d\\Omega$,\n",
+    "- $l(\\mathbf{v}) := \\int_{\\Omega} \\mathbf{f} \\cdot \\mathbf{v} ~ d\\Omega$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5a958607",
+   "metadata": {},
+   "source": [
+    "## Formal Model"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "d742586c",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.expr import BilinearForm, LinearForm, integral\n",
+    "from sympde.expr     import find, EssentialBC, Norm, SemiNorm\n",
+    "from sympde.topology import VectorFunctionSpace, Cube, element_of\n",
+    "from sympde.calculus import grad, inner, dot\n",
+    "\n",
+    "from sympy import pi, sin, Tuple, Matrix\n",
+    "\n",
+    "domain = Cube()\n",
+    "\n",
+    "V = VectorFunctionSpace('V', domain)\n",
+    "\n",
+    "x,y,z = domain.coordinates\n",
+    "\n",
+    "u,v = [element_of(V, name=i) for i in ['u', 'v']]\n",
+    "\n",
+    "# bilinear form\n",
+    "a = BilinearForm((u,v), integral(domain , inner(grad(v), grad(u))))\n",
+    "\n",
+    "# linear form\n",
+    "f1 = 3*pi**2*sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "f2 = 3*pi**2*sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "f3 = 3*pi**2*sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "f = Tuple(f1, f2, f3)\n",
+    "\n",
+    "l = LinearForm(v, integral(domain, dot(f,v)))\n",
+    "\n",
+    "# Dirichlet boundary conditions\n",
+    "bc = [EssentialBC(u, 0, domain.boundary)]\n",
+    "\n",
+    "# Variational problem\n",
+    "equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4f983ece",
+   "metadata": {},
+   "source": [
+    "## Discretization"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "51095918",
+   "metadata": {},
+   "source": [
+    "We shall need the **discretize** function from **PsyDAC**."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "a2a0a2a1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from psydac.api.discretization import discretize"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "00e54163",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "degree = [2,2,2]\n",
+    "ncells = [8,8,8]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "id": "5999c62b",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Create computational domain from topological domain\n",
+    "domain_h = discretize(domain, ncells=ncells, comm=None)\n",
+    "\n",
+    "# Create discrete spline space\n",
+    "Vh = discretize(V, domain_h, degree=degree)\n",
+    "\n",
+    "# Discretize equation\n",
+    "equation_h = discretize(equation, domain_h, [Vh, Vh])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7b29fbcf",
+   "metadata": {},
+   "source": [
+    "## Solving the PDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "541192ee",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "uh = equation_h.solve()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5174c4b5",
+   "metadata": {},
+   "source": [
+    "## Computing the error norm\n",
+    "\n",
+    "When the analytical solution is available, you might be interested in computing the $L^2$ norm or $H^1_0$ semi-norm.\n",
+    "SymPDE allows you to do so, by creating the **Norm** object.\n",
+    "In this example, the analytical solution is given by\n",
+    "\n",
+    "$$\n",
+    "u_e = \\sin(\\pi x) \\sin(\\pi y) \\sin(\\pi z)\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3a31c46f",
+   "metadata": {},
+   "source": [
+    "### Computing the $L^2$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "5925c6cd",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.0003267962376349343\n"
+     ]
+    }
+   ],
+   "source": [
+    "ue1 = sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "ue2 = sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "ue3 = sin(pi*x)*sin(pi*y)*sin(pi*z)\n",
+    "ue = Tuple(ue1, ue2, ue3)\n",
+    "\n",
+    "u = element_of(V, name='u')\n",
+    "\n",
+    "error = Matrix([u[0]-ue[0], u[1]-ue[1], u[2]-ue[2]])\n",
+    "\n",
+    "# create the formal Norm object\n",
+    "l2norm = Norm(error, domain, kind='l2')\n",
+    "\n",
+    "# discretize the norm\n",
+    "l2norm_h = discretize(l2norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "l2_error = l2norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(l2_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a6cbfeae",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ semi-norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "e5c1a8b8",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.01956220393797005\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "h1norm = SemiNorm(error, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "3c09131c",
+   "metadata": {},
+   "source": [
+    "### Computing the $H^1$ norm"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "d829e410",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.01956493339348906\n"
+     ]
+    }
+   ],
+   "source": [
+    "# create the formal Norm object\n",
+    "h1norm = Norm(error, domain, kind='h1')\n",
+    "\n",
+    "# discretize the norm\n",
+    "h1norm_h = discretize(h1norm, domain_h, Vh)\n",
+    "\n",
+    "# assemble the norm\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "\n",
+    "# print the result\n",
+    "print(h1_error)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9250897b",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter3/burgers.md b/_sources/chapter3/burgers.md
new file mode 100644
index 0000000..d5070f7
--- /dev/null
+++ b/_sources/chapter3/burgers.md
@@ -0,0 +1,154 @@
+# 1D Burgers equation
+
+We consider the 1d Burgers equation
+
+$$
+\partial_t u + u \partial_x u = \nu \frac{\partial ^2u}{\partial x^2}
+$$
+
+$u_0(x) := u(x,t)$ denotes the initial condition.
+We choose homogeneous neumann boundary conditions in this example, i.e.
+$$\partial_n u = 0, \partial \Omega$$ with $\Omega = (0,1)$
+
+## Time scheme
+We shall use a $\theta$-scheme in this case and consider the following problem
+
+$$
+\begin{align}
+\frac{u^{n+1}-u^n}{\Delta t} + 
+\theta~ u^{n+1} \partial_x u^{n+1} + (1-\theta)~ u^n \partial_x u^n = \theta~\nu \frac{\partial ^2u^{n+1}}{\partial x^2} + (1-\theta)~\nu \frac{\partial ^2u^{n}}{\partial x^2}
+\end{align}
+$$
+
+hence
+
+$$
+\begin{align}
+u^{n+1} + \Delta t ~ \theta~ u^{n+1} \partial_x u^{n+1} - \Delta t ~ \theta~\nu \frac{\partial ^2u^{n+1}}{\partial x^2} =
+u^{n} - \Delta t ~ (1-\theta)~ u^{n} \partial_x u^{n} + \Delta t ~ (1-\theta)~\nu \frac{\partial ^2u^{n}}{\partial x^2}
+\end{align}
+$$
+
+from now on, we shall denote by $f^n$ the right hand side of the previous equation
+
+$$f^n := u^{n} - \Delta t ~ (1-\theta)~ u^{n} \partial_x u^{n} + \Delta t ~ (1-\theta)~\nu \frac{\partial ^2u^{n}}{\partial x^2}$$
+
+## Weak formulation
+
+Let $v \in \mathcal{V}$ be a function test, we have by integrating by parts the highest order term:
+
+$$
+\begin{align}
+\langle v, u^{n+1}  \rangle + \Delta t ~ \theta~ \langle v, u^{n+1} \partial_x u^{n+1}  \rangle + \Delta t ~ \theta~\nu \langle \frac{\partial v}{\partial x}, \frac{\partial u^{n+1}}{\partial x}  \rangle = \langle v, f^n \rangle
+\end{align}
+$$
+
+The previous weak formulation is still nonlinear with respect to $u^{n+1}$. We shall then follow the same strategy as for the previous chapter on nonlinear Poisson problem.
+
+The strategy is to define the left hand side as a **LinearForm** with respect to $v$, then linearize it around $u^{n+1}$. We therefor can use either Picard or Newton method to treat the nonlinearity.
+
+We consider the following linear form
+
+$$
+\begin{align}
+G(v;u,w) := \langle v, u  \rangle + \Delta t ~ \theta~ \langle v, w \partial_x u  \rangle + \Delta t ~ \theta~\nu \langle \frac{\partial v}{\partial x}, \frac{\partial u}{\partial x}  \rangle , \quad \forall u,v,w \in \mathcal{V}
+\end{align}
+$$
+
+Our problem is then
+
+$$
+\begin{align}
+\mbox{Find } u^{n+1} \in \mathcal{V}, \mbox{such that}\\
+G(v;u^{n+1},u^{n+1}) = l(v), \quad \forall v \in \mathcal{V}
+\end{align}
+$$
+
+where
+
+$$
+l(v) := \int_{\Omega} f^n v ~d\Omega, \quad \forall v \in \mathcal{V}
+$$
+
+## SymPDE code
+
+```python
+domain = Line()
+
+V = ScalarFunctionSpace('V', domain)
+u  = element_of(V, name='u')
+v  = element_of(V, name='v')
+w  = element_of(V, name='w')
+un  = element_of(V, name='un') # time iteration
+uk  = element_of(V, name='uk') # nonlinear solver iteration
+
+x = domain.coordinates
+
+nu = Constant('nu')
+theta = Constant('theta')
+dt = Constant('dt')
+```
+
+### Defining the Linear form $G$
+
+```python
+# Linear form g: V --> R
+expr = v * u + dt*theta*v*w*dx(u) + dt*theta*nu*dx(v)*dx(u)
+g = LinearForm(v, integral(domain, expr))
+```
+
+### Defining the Linear form $l$
+
+```python
+# Linear form l: V --> R
+expr = v * un - dt*theta*v*un*dx(un) - dt*theta*nu*dx(v)*dx(un)
+l = LinearForm(v, integral(domain, expr))
+```
+
+
+### Picard Method
+
+$$
+\begin{align}
+\mbox{Find } u_{n+1} \in \mathcal{V}\_h, \mbox{such that}\\
+G(v;u_{n+1},u_n) = l(v), \quad \forall v \in \mathcal{V}\_h
+\end{align}
+$$
+
+#### Picard iteration
+
+```python
+# Variational problem
+picard = find(u, forall=v, lhs=g(v, u=u,w=uk), rhs=l(v))
+```
+
+### Newton Method
+
+Let's define 
+
+$$
+F(v;u) := G(v;u,u) -l(v), \quad \forall v \in \mathcal{V}
+$$
+
+Newton method writes
+
+$$
+\begin{align}
+\mbox{Find } u_{n+1} \in \mathcal{V}\_h, \mbox{such that}\\
+F^{\prime}(\delta u,v; u_n) = - F(v;u_n), \quad \forall v \in \mathcal{V} \\
+u_{n+1} := u_{n} + \delta u, \quad \delta u \in \mathcal{V}
+\end{align}
+$$
+
+#### Newton iteration
+
+```python
+F = LinearForm(v, g(v,w=u)-l(v))
+du  = element_of(V, name='du')
+
+Fprime = linearize(F, u, trials=du)
+
+# Variational problem
+newton = find(du, forall=v, lhs=Fprime(du, v,u=uk), rhs=-F(v,u=uk))
+```
+
diff --git a/_sources/chapter3/navier-stokes-steady-streamfunction-velocity.md b/_sources/chapter3/navier-stokes-steady-streamfunction-velocity.md
new file mode 100644
index 0000000..2569a83
--- /dev/null
+++ b/_sources/chapter3/navier-stokes-steady-streamfunction-velocity.md
@@ -0,0 +1,36 @@
+# The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids
+
+When $\Omega$ is a simply connected 2D domain, there exists a unique function $\psi$ such that $\mathbf{u} = \boldsymbol{\nabla} \times \psi:= \left( \partial_y \psi, - \partial_x \psi \right)$;
+substituting this expression for $\mathbf{u}$ into \eqref{eq:steady-navier-stokes} leads to the so-called ``streamfunction-velocity formulation'' of the steady-state Navier-Stokes equations for an incompressible fluid.
+For more details we refer the reader to \cite{TAGLIABUE2014277}.
+The variational formulation is then given by:
+
+$$
+\begin{align}
+  \text{find $\psi \in V$ such that} \quad 
+  a(\psi,\phi) + b(\psi,\phi;\psi) = l(\phi)
+  \quad \forall \phi \in V,
+\end{align}
+$$
+
+where $V \subset H^2(\Omega)$ and  
+
+$$
+\begin{align*}
+  a(\psi, \phi) := \int_{\Omega} D\left( \boldsymbol{\nabla} \times \psi \right) : D\left(\boldsymbol{\nabla} \times \phi \right) ~d\Omega,  
+  \quad
+  b(\psi, \phi; \xi) := \int_{\Omega} \left( \left(\boldsymbol{\nabla} \times \xi \cdot \nabla \right) \boldsymbol{\nabla} \times \psi \right) \cdot \boldsymbol{\nabla} \times \phi ~d\Omega,
+  \quad
+  l(\phi) := \int_{\Omega} \mathbf{f} \cdot \boldsymbol{\nabla} \times \phi ~d\Omega.
+\end{align*}
+$$
+
+%As in the previous example, one can use a Picard method, by considering $\xi$ as a \texttt{Field} in Eq. \ref{eq:steady-streamfunction-velocity-vf-a}, which will lead to the Python code described in \ref{code:steady-streamfunction-velocity-picard}. %, or consider $a$ as a \texttt{LinearForm} where $\xi = \psi$ are \texttt{Field} objects, and use the \texttt{NewtonIteration} constructor to get a \texttt{BilinearForm}, as described in the Python code \ref{code:steady-streamfunction-velocity-newton}.
+
+
+## Linearization of the streamfunction-velocity linear form
+
+In the sequel, we use SymPDE to linearize the streamfunction-velocity formulation, starting from a \texttt{LinearForm}.
+As presented in the previous example, we shall write our weak formulation as a linear form where the velocity $\mathbf{u}$ (\textit{i.e.} \texttt{u}) is considered as a \textit{field}, which is implemented in Python code \ref{code:steady-navier-stokes-nl}.
+Using the function \texttt{linearize} allows us to construct the bilinear form associated to the linearization of the given linear form, around the field \texttt{psi}, see Python code \ref{code:steady-streamfunction-velocity-lin}.
+To get an \texttt{Equation}, the user can add the left-hand-side for the Navier-Stokes equation to the linear form, the appropriate essential boundary conditions, then use the \texttt{NewtonIteration} as presented in the previous example.
diff --git a/_sources/chapter3/navier-stokes-steady.md b/_sources/chapter3/navier-stokes-steady.md
new file mode 100644
index 0000000..60e6a93
--- /dev/null
+++ b/_sources/chapter3/navier-stokes-steady.md
@@ -0,0 +1,41 @@
+# the steady-state Navier-Stokes equations for incompressible fluids
+
+The steady-state Navier Stokes problem for an incompressible fluid, with homogeneous Dirichlet boundary conditions (``no slip'' condition), is defined as
+
+$$
+\begin{equation}
+\left\{
+\begin{aligned}
+        \left( \mathbf{u} \cdot \nabla \right) \mathbf{u} + \nabla p - 2 \nabla \cdot \left( \frac{1}{R_e} D\left( \mathbf{u} \right) \right) &= \mathbf{f} && \text{in $\Omega$},
+  \\
+        \nabla \cdot \mathbf{u} &= 0 && \text{in $\Omega$},
+  \\
+        \mathbf{u} &= 0 && \text{on $\partial\Omega$},
+\end{aligned}
+\right.
+\end{equation}
+$$
+
+where $R_e$ is the Reynolds number and $D\left( \mathbf{u} \right) := \frac{1}{2}\left( \nabla \mathbf{u} + \nabla \mathbf{u}^T \right)$ is the strain rate tensor.
+The weak formulation reads
+
+$$
+\begin{align}
+  \text{find $\mathbf{u} \in W$ such that} \quad 
+  a(\mathbf{u},\mathbf{v}) + b(\mathbf{u},\mathbf{v};\mathbf{u}) = l(\mathbf{v}) \quad
+  \forall \mathbf{v} \in W
+\end{align}
+$$
+
+where $W \subset \{ \mathbf{v} \in \mathbf{H}_0^1(\Omega): \nabla \cdot \mathbf{v} = 0 \}$ and
+
+$$
+\begin{align*}
+  a(\mathbf{u}, \mathbf{v}) := \frac{2}{R_e} \int_{\Omega} D\left( \mathbf{u} \right) : D\left(\mathbf{v} \right) ~d\Omega,
+  \quad
+  b(\mathbf{u}, \mathbf{v}; \mathbf{w}) := \int_{\Omega} \left( \left( \mathbf{w} \cdot \nabla \right) \mathbf{u} \right) \cdot \mathbf{v} ~d\Omega,
+  \quad
+  l(\mathbf{v}) := \int_{\Omega} \mathbf{f} \cdot \mathbf{v} ~d\Omega.
+\end{align*}
+$$
+
diff --git a/_sources/chapter3/poisson.md b/_sources/chapter3/poisson.md
new file mode 100644
index 0000000..5f4671c
--- /dev/null
+++ b/_sources/chapter3/poisson.md
@@ -0,0 +1,145 @@
+# Nonlinear Poisson in 2D
+
+In this section, we consider the non-linear Poisson problem:
+
+$$
+\begin{align}
+-\nabla \cdot \left( (1+u^2) \nabla u \right) &= f, \Omega
+\\
+u &= 0, \partial \Omega
+\end{align}
+$$
+
+where $\Omega$ denotes the unit square.
+
+For testing, we shall take a function $u$ that fulfills the boundary condition, the compute $f$ as
+
+$$
+f(x,y) = -\nabla^2 u - F(u)
+$$
+
+The weak formulation is
+
+$$
+\int_{\Omega} (1+u^2) \nabla u \cdot \nabla v ~ d\Omega = \int_{\Omega} f v ~d\Omega, \quad \forall v \in \mathcal{V}
+$$
+
+For the sack of generality, we shall consider the linear form
+
+$$
+G(v;u,w) := \int_{\Omega} (1+w^2) \nabla u \cdot \nabla v ~ d\Omega, \quad \forall u,v,w \in \mathcal{V}
+$$
+
+Our problem is then
+
+$$
+\begin{align}
+\mbox{Find } u \in \mathcal{V}, \mbox{such that}\\
+G(v;u,u) = l(v), \quad \forall v \in \mathcal{V}
+\end{align}
+$$
+
+where
+
+$$
+l(v) := \int_{\Omega} f v ~d\Omega, \quad \forall v \in \mathcal{V}
+$$
+
+As usual, we'll follow the classical steps,
+
+1. Create a domain
+2. Create the Function space
+3. Define the different linear/bilinear forms
+4. Define the Picard/Newton iteration
+
+## 1. The topological domain
+```python
+domain = Square()
+B_dirichlet_0 = domain.boundary
+```
+## 2. Function space 
+```python
+V  = ScalarFunctionSpace('V', domain)
+```
+## 3.1. Define the Linear form $G$
+```python
+u  = element_of(V, name='u')
+v  = element_of(V, name='v')
+w  = element_of(V, name='w')
+
+# Linear form g: V --> R
+g = LinearForm(v, integral(domain, (1+w**2)*dot(grad(u), grad(v))))
+```
+## 3.2. Define the linear form $L$ 
+```python
+solution = sin(pi*x)*sin(pi*y)
+f = 2*pi**2*(sin(pi*x)**2*sin(pi*y)**2 + 1)*sin(pi*x)*sin(pi*y) - 2*pi**2*sin(pi*x)**3*sin(pi*y)*cos(pi*y)**2 - 2*pi**2*sin(pi*x)*sin(pi*y)**3*cos(pi*x)**2
+
+# Linear form l: V --> R
+l = LinearForm(v, integral(domain, f * v))
+```
+## 4.1. Picard iteration
+
+$$
+\begin{align}
+\mbox{Find } u_{n+1} \in \mathcal{V}\_h, \mbox{such that}\\ 
+G(v;u_{n+1},u_n) = l(v), \quad \forall v \in \mathcal{V}\_h
+\end{align}
+$$
+
+The Picard iteration for our problem can be created this way
+
+```python
+un  = element_of(V, name='un')
+
+# Bilinear form a: V x V --> R
+a = BilinearForm((u, v), g(v, u=u,w=un))
+
+# Dirichlet boundary conditions
+bc = [EssentialBC(u, 0, B_dirichlet_0)]
+
+# Variational problem
+equation   = find(u, forall=v, lhs=a(u, v), rhs=l(v), bc=bc)
+
+# Error norms
+error  = u - solution
+l2norm = Norm(error, domain, kind='l2')
+```
+## 4.2. Newton iteration
+Let's define 
+
+$$
+F(v;u) := G(v;u,u) -l(v), \quad \forall v \in \mathcal{V}
+$$
+
+Newton method writes
+
+$$
+\begin{align}
+\mbox{Find } u_{n+1} \in \mathcal{V}\_h, \mbox{such that} \\
+F^{\prime}(\delta u,v; u_n) = - F(v;u_n), \quad \forall v \in \mathcal{V}, \\ 
+u_{n+1} := u_{n} + \delta u, \quad \delta u \in \mathcal{V}
+\end{align}
+$$
+
+### Computing $F^{\prime}$ the derivative of $F$
+
+**SymPDE** allows you to linearize a linear form and get a bilinear form, using the function **linearize**
+
+```python
+F = LinearForm(v, g(v,w=u)-l(v))
+du  = element_of(V, name='du')
+
+Fprime = linearize(F, u, trials=du)
+```
+
+The Newton iteration is then defined as follows,
+
+```python
+# Dirichlet boundary conditions
+bc = [EssentialBC(du, 0, B_dirichlet_0)]
+
+# Variational problem
+equation   = find(du, forall=v, lhs=Fprime(du, v,u=un), rhs=-F(v,u=un), bc=bc)
+```
+
diff --git a/_sources/chapter4/poisson-nitsche.md b/_sources/chapter4/poisson-nitsche.md
new file mode 100644
index 0000000..a9920de
--- /dev/null
+++ b/_sources/chapter4/poisson-nitsche.md
@@ -0,0 +1,45 @@
+# The Poisson problem using Nitsche method
+
+In this section, we consider a domain $\Omega$ which a union of multiple subdomain, *i.e.*
+
+$$
+\Omega := \bigcup_{i=1}^n \Omega_i, \quad \Omega_i \bigcap \Omega_j = \emptyset, ~ i \neq j  
+$$
+
+We shall denote by $\mathcal{I}$ the set of all internal interfaces of $\Omega$.
+
+We shall also need the following operators
+
+- The jump of the function $u$, defined as $[\![ u ]\!] := u|\_{\Omega_{i_1}} -  u|\_{\Omega_{i_2}}$ for two adjacent subdomains $\Omega_{i_1}$ and $\Omega_{i_2}$
+- The average of the function $u$, defined as $\{u\} := \frac{1}{2} \left( u|\_{\Omega_{i_1}} +  u|\_{\Omega_{i_2}} \right)$ for two adjacent subdomains $\Omega_{i_1}$ and $\Omega_{i_2}$
+
+We consider the Poisson equation
+
+$$
+\begin{align}
+  - \nabla^2 u = f \quad \text{in~$\Omega$}, \quad \quad 
+  u = 0            \quad \text{on~$\partial \Omega$}, \quad \quad 
+   [\![ u ]\!] = 0 \quad \text{on~$\mathcal{I}$}, \quad \quad
+   [\![\partial_n u ]\!] = 0 \quad \text{on~$\mathcal{I}$}.
+\end{align}
+$$
+
+The variational formulation reads
+
+$$
+\begin{align}
+  \text{find $u \in V$ such that} \quad a(u,v) = l(v) \quad \forall v \in V,
+\end{align}
+$$
+
+where 
+
+- $V \subset H^1(\Omega)$, 
+- $a(u,v) := \int_{\Omega} \nabla u \cdot \nabla v ~ d\Omega + \int_{\mathcal{I}} \left( \kappa [\![ u ]\!] ~ [\![ v ]\!] - [\![ u ]\!] ~ \partial_n v - \partial_n u ~ [\![ v ]\!] \right) ~ d\mathcal{I} $, 
+- $l(v) := \int_{\Omega} f v ~ d\Omega ~ d\Gamma$.
+
+## Examples
+
+- [Poisson on a square with two subdomains and Homogeneous Boundary Conditions](https://github.com/pyccel/sympde/blob/devel-documentation/docs/examples/2d_poisson_subdomains_nitsche_dir0.py)
+
+
diff --git a/_sources/chapter4/subdomains.md b/_sources/chapter4/subdomains.md
new file mode 100644
index 0000000..a6dcdc5
--- /dev/null
+++ b/_sources/chapter4/subdomains.md
@@ -0,0 +1 @@
+# Subdomains
diff --git a/_sources/chapter5/bingham_plastic_flow_in_a_pipe.ipynb b/_sources/chapter5/bingham_plastic_flow_in_a_pipe.ipynb
new file mode 100644
index 0000000..3d1717f
--- /dev/null
+++ b/_sources/chapter5/bingham_plastic_flow_in_a_pipe.ipynb
@@ -0,0 +1,78 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "1f0b7959",
+   "metadata": {},
+   "source": [
+    "# Bingham Plastic Flow in a Pipe\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the Bingham plastic model for non-Newtonian fluids with yield stress:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\frac{\\partial u}{\\partial t} + u\\frac{\\partial u}{\\partial x} &= -\\frac{1}{\\rho}\\frac{\\partial p}{\\partial x} + \\frac{\\tau_0}{\\rho}\\frac{\\partial^2 u}{\\partial y^2} \\\\\n",
+    "\\frac{\\partial^2 u}{\\partial y^2} &\\geq 0 \\quad \\text{(Yield stress condition)}\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $u$ is the velocity, $p$ is the pressure, $\\rho$ is the density, $\\tau_0$ is the yield stress, and $y$ is the radial coordinate.\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet: Specify velocity or pressure conditions.\n",
+    "- Outlet: Specify outflow conditions.\n",
+    "- Pipe walls: No-slip conditions.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Specify initial conditions for velocity and pressure.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $u \\in H_0^1(\\Omega)$ and $p \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\frac{\\partial u}{\\partial t}v \\, d\\Omega + \\int_{\\Omega} u\\frac{\\partial u}{\\partial x}v \\, d\\Omega + \\int_{\\Omega} \\frac{1}{\\rho}\\frac{\\partial p}{\\partial x}v \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\frac{\\tau_0}{\\rho}\\frac{\\partial^2 u}{\\partial y^2}\\frac{\\partial v}{\\partial y} \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $v \\in H_0^1(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "3104c066",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/buoyancy-driven_natural_convection.ipynb b/_sources/chapter5/buoyancy-driven_natural_convection.ipynb
new file mode 100644
index 0000000..767c6a4
--- /dev/null
+++ b/_sources/chapter5/buoyancy-driven_natural_convection.ipynb
@@ -0,0 +1,81 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "315d2f4f",
+   "metadata": {},
+   "source": [
+    "# Buoyancy-Driven Natural Convection\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the Boussinesq approximation for buoyancy-driven natural convection:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\frac{\\partial \\mathbf{v}}{\\partial t} + (\\mathbf{v} \\cdot \\nabla)\\mathbf{v} &= -\\frac{1}{\\rho_0}\\nabla p + \\nu \\nabla^2 \\mathbf{v} + \\beta g (T - T_0)\\mathbf{k} \\\\\n",
+    "\\frac{\\partial T}{\\partial t} + \\mathbf{v} \\cdot \\nabla T &= \\alpha \\nabla^2 T\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Vertical walls: $\\mathbf{v} \\cdot \\mathbf{t} = 0$ and $\\frac{\\partial T}{\\partial n} = 0$.\n",
+    "\n",
+    "- Inlet and outlet: Specify velocity or temperature conditions.\n",
+    "\n",
+    "- Top and bottom: No-slip and adiabatic conditions.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Start with an initial velocity and temperature distribution.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $\\mathbf{v} \\in H_0^1(\\Omega)$ and $T \\in H^1(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\left(\\frac{\\partial \\mathbf{v}}{\\partial t} + (\\mathbf{v} \\cdot \\nabla)\\mathbf{v}\\right) \\cdot \\mathbf{w} \\, d\\Omega + \\int_{\\Omega} \\frac{1}{\\rho_0}\\nabla p \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\nu \\nabla \\mathbf{v} : \\nabla \\mathbf{w} \\, d\\Omega - \\beta g (T - T_0)\\mathbf{k} \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\left(\\frac{\\partial T}{\\partial t} + \\mathbf{v} \\cdot \\nabla T - \\alpha \\nabla^2 T\\right) \\psi \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $\\mathbf{w}, \\psi \\in H_0^1(\\Omega)$.\n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "dc53cb64",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/casson_fluid_flow_in_a_channel.ipynb b/_sources/chapter5/casson_fluid_flow_in_a_channel.ipynb
new file mode 100644
index 0000000..9c7b3b3
--- /dev/null
+++ b/_sources/chapter5/casson_fluid_flow_in_a_channel.ipynb
@@ -0,0 +1,74 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "e9f1cb0e",
+   "metadata": {},
+   "source": [
+    "# Casson Fluid Flow in a Channel\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the Casson model for non-Newtonian fluids:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\sigma_{ij} &= -p\\delta_{ij} + \\sqrt{\\frac{\\tau_0}{\\rho}}\\sqrt{\\frac{\\tau_0}{\\rho} + \\frac{1}{2}\\rho\\left(\\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i}\\right)^2} \\\\\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\sigma_{ij}$ is the stress tensor, $p$ is the pressure, $\\rho$ is the density, $u_i$ is the velocity, and $\\tau_0$ is the yield stress.\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet and outlet: Specify velocity or pressure conditions.\n",
+    "- Channel walls: No-slip conditions.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "Start with an initial guess for velocity and pressure.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $u_i \\in H_0^1(\\Omega)$ and $p \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\sigma_{ij} \\left(\\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i}\\right) \\, d\\Omega + \\int_{\\Omega} \\frac{\\partial p}{\\partial x_i}u_i \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $u_i \\in H_0^1(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "52879066",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/cfd.md b/_sources/chapter5/cfd.md
new file mode 100644
index 0000000..c156b86
--- /dev/null
+++ b/_sources/chapter5/cfd.md
@@ -0,0 +1 @@
+# Computational Fluid Dynamics
diff --git a/_sources/chapter5/compressible_flow_in_a_nozzle.ipynb b/_sources/chapter5/compressible_flow_in_a_nozzle.ipynb
new file mode 100644
index 0000000..d5b4e52
--- /dev/null
+++ b/_sources/chapter5/compressible_flow_in_a_nozzle.ipynb
@@ -0,0 +1,80 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "896cf22a",
+   "metadata": {},
+   "source": [
+    "# Compressible Flow in a Nozzle\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the compressible Euler equations for flow in a converging-diverging nozzle:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\mathbf{v}) &= 0 \\\\\n",
+    "\\frac{\\partial (\\rho \\mathbf{v})}{\\partial t} + \\nabla \\cdot (\\rho \\mathbf{v} \\otimes \\mathbf{v} + p\\mathbf{I}) &= 0 \\\\\n",
+    "\\frac{\\partial E}{\\partial t} + \\nabla \\cdot \\left[(E + p) \\mathbf{v}\\right] &= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet: Specify inflow conditions ($\\rho$, $\\mathbf{v}$, $p$).\n",
+    "\n",
+    "- Outlet: Specify outflow conditions or use characteristic-based boundary conditions.\n",
+    "\n",
+    "- Nozzle walls: $\\mathbf{v} \\cdot \\mathbf{n} = 0$ (no penetration).\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Specify initial conditions for density, velocity, and pressure.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $\\rho, \\mathbf{v}, p \\in H^1(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\left(\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho \\mathbf{v})\\right) \\phi \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\left(\\frac{\\partial (\\rho \\mathbf{v})}{\\partial t} + \\nabla \\cdot (\\rho \\mathbf{v} \\otimes \\mathbf{v} + p\\mathbf{I})\\right) \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\left(\\frac{\\partial E}{\\partial t} + \\nabla \\cdot \\left[(E + p) \\mathbf{v}\\right]\\right) \\psi \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $\\phi, \\mathbf{w}, \\psi \\in H^1(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "8a3bde01",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/convection-diffusion_equation_in_a_channel.ipynb b/_sources/chapter5/convection-diffusion_equation_in_a_channel.ipynb
new file mode 100644
index 0000000..d283e61
--- /dev/null
+++ b/_sources/chapter5/convection-diffusion_equation_in_a_channel.ipynb
@@ -0,0 +1,78 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "6889a63e",
+   "metadata": {},
+   "source": [
+    "# Convection-Diffusion Equation in a Channel\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the 2D convection-diffusion equation in a channel:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\rho u \\frac{\\partial c}{\\partial x} + \\rho v \\frac{\\partial c}{\\partial y} - \\nabla \\cdot (D \\nabla c) = f\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet: $c = c_{\\text{inlet}}$ with \\$\\mathbf{v} \\cdot \\mathbf{n} < 0$ (inward flow).\n",
+    "\n",
+    "- Outlet: $\\frac{\\partial c}{\\partial x} = \\frac{\\partial c}{\\partial y} = 0$ (zero gradient).\n",
+    "\n",
+    "- Lateral faces: $-D \\nabla c \\cdot \\mathbf{n} = 0$ (insulated).\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Assume an initial concentration distribution $c_0(x, y)$ at $t = 0$.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $c \\in H_0^1(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\left(\\rho u \\frac{\\partial c}{\\partial x} + \\rho v \\frac{\\partial c}{\\partial y}\\right) \\phi \\, d\\Omega - \\int_{\\Omega} D \\nabla c \\cdot \\nabla \\phi \\, d\\Omega \\\\\n",
+    "&= \\int_{\\Omega} f \\phi \\, d\\Omega\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $\\phi \\in H_0^1(\\Omega)$.\n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9d3c6c22",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/cross_power_law_fluid_flow_in_a_channel.ipynb b/_sources/chapter5/cross_power_law_fluid_flow_in_a_channel.ipynb
new file mode 100644
index 0000000..13fb81b
--- /dev/null
+++ b/_sources/chapter5/cross_power_law_fluid_flow_in_a_channel.ipynb
@@ -0,0 +1,77 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "45606393",
+   "metadata": {},
+   "source": [
+    "# Cross Power Law Fluid Flow in a Channel\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the cross power-law model for non-Newtonian fluids:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\sigma_{ij} &= -p\\delta_{ij} + \\lambda \\varepsilon_{kk}\\delta_{ij} + 2\\mu\\left(\\varepsilon_{ij} + \\alpha\\varepsilon_{ij}^2\\right) \\\\\n",
+    "\\varepsilon_{ij} &= \\frac{1}{2}\\left(\\frac{\\partial v_i}{\\partial x_j} + \\frac{\\partial v_j}{\\partial x_i}\\right) \\\\\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\sigma_{ij}$ is the stress tensor, $p$ is the pressure, $\\lambda$ and $\\mu$ are material constants, $v_i$ is the velocity, $\\varepsilon_{ij}$ is the strain rate, and $\\alpha$ is a power-law index.\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet and outlet: Specify velocity or pressure conditions.\n",
+    "\n",
+    "- Channel walls: No-slip conditions.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Start with an initial guess for velocity and pressure.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $v_i \\in H_0^1(\\Omega)$ and $p \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\sigma_{ij} \\varepsilon_{ij} \\, d\\Omega + \\int_{\\Omega} \\frac{\\partial p}{\\partial x_i}v_i \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $v_i \\in H_0^1(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "85946710",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/electromagnetics.md b/_sources/chapter5/electromagnetics.md
new file mode 100644
index 0000000..9f793c7
--- /dev/null
+++ b/_sources/chapter5/electromagnetics.md
@@ -0,0 +1,56 @@
+# Electromagnetics
+
+Electromagnetic problems are commonly described by Maxwell's equations, which govern the behavior of electric and magnetic fields. The Finite Element Method (FEM) provides a powerful numerical approach for solving these equations in complex geometries. This section provides a concise overview of the mathematical formulation for electromagnetic problems using finite elements.
+
+## Maxwell's Equations
+
+The time-harmonic Maxwell's equations in free space are given by:
+
+$$
+\begin{align}
+    \nabla \times \mathbf{E} &= -j\omega \mu \mathbf{H}, \label{eq:maxwell1} \\
+    \nabla \times \mathbf{H} &= j\omega \varepsilon \mathbf{E}, \label{eq:maxwell2} \\
+    \nabla \cdot \mathbf{B} &= 0, \label{eq:maxwell3} \\
+    \nabla \cdot \mathbf{D} &= 0, \label{eq:maxwell4}
+\end{align}
+$$
+
+where $\mathbf{E}$ is the electric field, $\mathbf{H}$ is the magnetic field, $\mathbf{B}$ is the magnetic flux density, $\mathbf{D}$ is the electric displacement field, $\omega$ is the angular frequency, $\mu$ is the permeability, and $\varepsilon$ is the permittivity.
+
+## Weak Formulation
+
+The weak form of Maxwell's equations is obtained by multiplying each equation with suitable test functions and integrating over the domain $\Omega$. For the electric field $\mathbf{E}$, the weak form is given by:
+
+$$
+\begin{equation}
+    \int_{\Omega} \nabla \times \mathbf{E} \cdot \nabla \times \mathbf{v} \, d\Omega + \omega^2 \mu \varepsilon \mathbf{E} \cdot \mathbf{v} \, d\Omega = 0,
+\end{equation}
+$$
+
+where $\mathbf{v}$ is a test function belonging to the function space $H(\text{curl}; \Omega)$.
+
+Similarly, for the magnetic field $\mathbf{H}$, the weak form is given by:
+
+$$
+\begin{equation}
+    \int_{\Omega} \nabla \times \mathbf{H} \cdot \nabla \times \mathbf{u} \, d\Omega - \omega^2 \mu \varepsilon \mathbf{H} \cdot \mathbf{u} \, d\Omega = 0,
+\end{equation}
+$$
+
+where $\mathbf{u}$ is a test function belonging to the function space $H(\text{curl}; \Omega)$.
+
+## Finite Element Discretization
+
+To apply the finite element method, the domain $\Omega$ is discretized into elements, and the electromagnetic fields are approximated using piecewise basis functions. The discretized weak form leads to a system of linear equations, which can be solved numerically to obtain the finite element solution.
+
+## References
+
+The following references provide in-depth coverage of the mathematical background for electromagnetism using finite elements:
+
+\begin{enumerate}
+    \item \cite{jin2015finite} 
+    \item \cite{bossavit1998whitney}
+    \item \cite{monk2003finite}
+\end{enumerate}
+
+This mathematical background serves as the foundation for implementing finite element simulations of electromagnetic problems. Researchers interested in detailed mathematical derivations and computational techniques are encouraged to explore the referenced works.
diff --git a/_sources/chapter5/free_surface_flow.ipynb b/_sources/chapter5/free_surface_flow.ipynb
new file mode 100644
index 0000000..d482cab
--- /dev/null
+++ b/_sources/chapter5/free_surface_flow.ipynb
@@ -0,0 +1,80 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "fea980ee",
+   "metadata": {},
+   "source": [
+    "# Free Surface Flow (Navier-Stokes with Free Surface)\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the Navier-Stokes equations coupled with a level set or volume-of-fluid method for tracking a free surface:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\rho \\frac{\\partial \\mathbf{v}}{\\partial t} + \\rho (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} &= -\\nabla p + \\mu \\nabla^2 \\mathbf{v} + \\rho \\mathbf{g} \\\\\n",
+    "\\nabla \\cdot \\mathbf{v} &= 0 \\\\\n",
+    "\\frac{\\partial \\phi}{\\partial t} + \\mathbf{v} \\cdot \\nabla \\phi &= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\phi$ is the level set or volume fraction representing the free surface.\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet and outlet: Specify velocity or pressure conditions.\n",
+    "- No-slip conditions on solid boundaries.\n",
+    "- Implement the level set or volume fraction boundary conditions.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Specify initial conditions for velocity, pressure, and the level set or volume fraction.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $\\mathbf{v} \\in H_0^1(\\Omega)$, $p \\in L^2(\\Omega)$, and $\\phi \\in H^1(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\rho \\frac{\\partial \\mathbf{v}}{\\partial t} \\cdot \\mathbf{w} \\, d\\Omega + \\int_{\\Omega} \\rho (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\mu \\nabla \\mathbf{v} : \\nabla \\mathbf{w} \\, d\\Omega - \\int_{\\Omega} p \\nabla \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\nabla \\cdot \\mathbf{v} q \\, d\\Omega + \\int_{\\Omega} \\frac{\\partial \\phi}{\\partial t} q \\, d\\Omega + \\int_{\\Omega} \\mathbf{v} \\cdot \\nabla \\phi q \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $\\mathbf{w} \\in H_0^1(\\Omega)$, $q \\in L^2(\\Omega)$, and $r \\in H^1(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b3ac645a",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/fsi.md b/_sources/chapter5/fsi.md
new file mode 100644
index 0000000..fb4f1c7
--- /dev/null
+++ b/_sources/chapter5/fsi.md
@@ -0,0 +1,75 @@
+# Fluid-Structure Interaction
+
+Fluid-Structure Interaction (FSI) involves the coupled interaction between a fluid and a structure, where the motion of one influences the behavior of the other. The Finite Element Method (FEM) is a powerful tool for simulating FSI problems. This section provides an overview of the mathematical formulation for fluid-structure interaction using finite elements.
+
+## Coupled Fluid-Structure Equations
+
+The governing equations for fluid-structure interaction involve the Navier-Stokes equations for the fluid and the equations of motion for the structure. In a partitioned approach, the coupled system is given by:
+
+\begin{align}
+    \text{Fluid Domain:} \quad \rho_f \left(\frac{\partial \mathbf{u}_f}{\partial t} + (\mathbf{u}_f \cdot \nabla)\mathbf{u}_f\right) &= -\nabla p_f + \mu_f \nabla^2 \mathbf{u}_f + \mathbf{f}_f + \mathbf{F}_s, \label{eq:fsi_fluid_momentum} \\
+    \nabla \cdot \mathbf{u}_f &= 0, \label{eq:fsi_fluid_continuity} \\
+    \text{Structure Domain:} \quad \rho_s \frac{\partial^2 \mathbf{u}_s}{\partial t^2} &= \nabla \cdot \mathbf{P}_s + \mathbf{f}_s, \label{eq:fsi_structure_motion}
+\end{align}
+
+where:
+
+- $\mathbf{u}_f$ is the fluid velocity,
+- $p_f$ is the fluid pressure,
+- $\rho_f$ is the fluid density,
+- $\mu_f$ is the fluid dynamic viscosity,
+- $\mathbf{f}_f$ is the fluid body force,
+- $\mathbf{F}_s$ is the fluid-structure interaction force on the fluid by the structure,
+- $\mathbf{u}_s$ is the structure displacement,
+- $\rho_s$ is the structure density,
+- $\mathbf{P}_s$ is the structure internal force,
+- $\mathbf{f}_s$ is the structure external force.
+
+## Coupling Conditions
+
+For a fluid-structure interaction problem, coupling conditions at the fluid-structure interface are essential. These conditions ensure continuity of velocities, pressures, and forces between the fluid and structure. A common approach is to enforce kinematic and dynamic conditions:
+
+$$
+\begin{align}
+    \text{Kinematic Condition:} \quad \mathbf{u}_f &= \mathbf{u}_s, \label{eq:fsi_kinematic_condition} \\
+    \text{Dynamic Condition:} \quad \mathbf{P}_f \cdot \mathbf{n} &= \mathbf{P}_s \cdot \mathbf{n}, \label{eq:fsi_dynamic_condition}
+\end{align}
+$$
+
+where $\mathbf{n}$ is the unit outward normal vector at the fluid-structure interface, and $\mathbf{P}_f$ and $\mathbf{P}_s$ are the fluid and structure stresses, respectively.
+
+## Weak Formulation
+
+The weak form of the coupled fluid-structure problem involves the variational formulation of the fluid and structure equations, incorporating the coupling conditions. For the fluid domain, the weak form is given by:
+
+$$
+\begin{equation}
+    \int_{\Omega_f} \rho_f \left(\frac{\partial \mathbf{u}_f}{\partial t} + (\mathbf{u}_f \cdot \nabla)\mathbf{u}_f\right) \cdot \mathbf{v}_f \, d\Omega_f + \int_{\Omega_f} \mu_f \nabla \mathbf{u}_f : \nabla \mathbf{v}_f \, d\Omega_f = \int_{\Omega_f} (\mathbf{f}_f + \mathbf{F}_s) \cdot \mathbf{v}_f \, d\Omega_f,
+\end{equation}
+$$
+
+where $\mathbf{v}_f$ is a test function belonging to the function space $H^1(\Omega_f)$, and $\Omega_f$ is the fluid domain.
+
+For the structure domain, the weak form is given by:
+
+$$
+\begin{equation}
+    \int_{\Omega_s} \rho_s \frac{\partial^2 \mathbf{u}_s}{\partial t^2} \cdot \mathbf{v}_s \, d\Omega_s + \int_{\Omega_s} \nabla \cdot \mathbf{P}_s \cdot \mathbf{v}_s \, d\Omega_s = \int_{\Omega_s} \mathbf{f}_s \cdot \mathbf{v}_s \, d\Omega_s,
+\end{equation}
+$$
+
+where $\mathbf{v}_s$ is a test function belonging to the function space $H^1(\Omega_s)$, and $\Omega_s$ is the structure domain.
+
+The coupling conditions, such as \eqref{eq:fsi_kinematic_condition} and \eqref{eq:fsi_dynamic_condition}, should be enforced during the assembly of the global matrices.
+
+## References
+
+The following references provide comprehensive coverage of the mathematical background for fluid-structure interaction using finite elements:
+
+\begin{enumerate}
+    \item \cite{hron2006fluid}  
+    \item \cite{bathe2014finite}
+    \item \cite{quarteroni2017fluid}
+\end{enumerate}
+
+Researchers interested in detailed mathematical derivations and computational techniques are encouraged to explore the referenced works.
diff --git a/_sources/chapter5/heat_conduction_in_a_solid.ipynb b/_sources/chapter5/heat_conduction_in_a_solid.ipynb
new file mode 100644
index 0000000..dfe455d
--- /dev/null
+++ b/_sources/chapter5/heat_conduction_in_a_solid.ipynb
@@ -0,0 +1,74 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "b3466488",
+   "metadata": {},
+   "source": [
+    "# Heat Conduction in a Solid\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the heat conduction equation in a solid domain \\(\\Omega\\):\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\rho C_p \\frac{\\partial T}{\\partial t} - \\nabla \\cdot (k \\nabla T) = Q\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Dirichlet: $T = T_{\\text{inlet}}$ on the inlet face.\n",
+    "- Neumann: $-k \\frac{\\partial T}{\\partial n} = 0$ on the outlet face.\n",
+    "- Insulated: $-k \\frac{\\partial T}{\\partial n} = 0$ on the lateral faces.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Assume an initial temperature distribution $T_0(x, y, z)$ at $t = 0$.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $T \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\int_{\\Omega} \\rho C_p \\frac{\\partial T}{\\partial t} \\phi \\, d\\Omega + \\int_{\\Omega} k \\nabla T \\cdot \\nabla \\phi \\, d\\Omega = \\int_{\\Omega} Q \\phi \\, d\\Omega\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $\\phi \\in H_0^1(\\Omega)$.\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "55ba709f",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/herschel-bulkley_fluid_flow_in_a_pipe.ipynb b/_sources/chapter5/herschel-bulkley_fluid_flow_in_a_pipe.ipynb
new file mode 100644
index 0000000..0313108
--- /dev/null
+++ b/_sources/chapter5/herschel-bulkley_fluid_flow_in_a_pipe.ipynb
@@ -0,0 +1,79 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "a0c1f909",
+   "metadata": {},
+   "source": [
+    "# Herschel-Bulkley Fluid Flow in a Pipe\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the Herschel-Bulkley model for non-Newtonian fluids with yield stress:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\frac{\\partial u}{\\partial t} + u\\frac{\\partial u}{\\partial x} &= -\\frac{1}{\\rho}\\frac{\\partial p}{\\partial x} + \\frac{\\tau_0}{\\rho}\\left(\\frac{\\partial u}{\\partial x}\\right)^{n-1} \\\\\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $u$ is the velocity, $p$ is the pressure, $\\rho$ is the density, $\\tau_0$ is the yield stress, and $n$ is the flow behavior index.\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet: Specify velocity or pressure conditions.\n",
+    "\n",
+    "- Outlet: Specify outflow conditions.\n",
+    "\n",
+    "- Pipe walls: No-slip conditions.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Specify initial conditions for velocity and pressure.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $u \\in H_0^1(\\Omega)$ and $p \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\frac{\\partial u}{\\partial t}v \\, d\\Omega + \\int_{\\Omega} u\\frac{\\partial u}{\\partial x}v \\, d\\Omega + \\int_{\\Omega} \\frac{1}{\\rho}\\frac{\\partial p}{\\partial x}v \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\frac{\\tau_0}{\\rho}\\left(\\frac{\\partial u}{\\partial x}\\right)^{n-1}v \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $v \\in H_0^1(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "1f098463",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/incompressible_flow_past_a_cylinder.ipynb b/_sources/chapter5/incompressible_flow_past_a_cylinder.ipynb
new file mode 100644
index 0000000..71188b8
--- /dev/null
+++ b/_sources/chapter5/incompressible_flow_past_a_cylinder.ipynb
@@ -0,0 +1,75 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "0b948ce3",
+   "metadata": {},
+   "source": [
+    "# Incompressible Flow Past a Cylinder\n",
+    "\n",
+    "## Mathematical Model:\n",
+    "\n",
+    "Consider the steady, incompressible Navier-Stokes equations for fluid flow around a cylinder:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\rho (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} &= -\\nabla p + \\mu \\nabla^2 \\mathbf{v} + \\rho \\mathbf{g} \\\\\n",
+    "\\nabla \\cdot \\mathbf{v} &= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet: $\\mathbf{v} = (U, 0)$ (uniform velocity).\n",
+    "- Outlet: $\\frac{\\partial \\mathbf{v}}{\\partial x} = 0$ (zero gradient).\n",
+    "- Cylinder surface: $\\mathbf{v} \\cdot \\mathbf{n} = 0$ (no-slip condition).\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Assume a quiescent fluid, i.e., $\\mathbf{v} = (0, 0)$ at $t = 0$.\n",
+    "\n",
+    "### Weak Formulation\n",
+    "\n",
+    "Find $\\mathbf{v} \\in H_0^1(\\Omega)$ and $p \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\int_{\\Omega} \\rho (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} \\cdot \\mathbf{w} \\, d\\Omega & = -\\int_{\\Omega} p \\nabla \\cdot \\mathbf{w} \\, d\\Omega + \\mu \\int_{\\Omega} \\nabla \\mathbf{v} : \\nabla \\mathbf{w} \\, d\\Omega + \\int_{\\Omega} \\rho \\mathbf{g} \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "\\int_{\\Omega} \\nabla \\cdot \\mathbf{v} q \\, d\\Omega & = 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\mathbf{w} \\in H_0^1(\\Omega)$ and $q \\in L^2(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ed14fcc5",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/magnetohydrodynamics_flow.ipynb b/_sources/chapter5/magnetohydrodynamics_flow.ipynb
new file mode 100644
index 0000000..6b0f078
--- /dev/null
+++ b/_sources/chapter5/magnetohydrodynamics_flow.ipynb
@@ -0,0 +1,81 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "6c15c8b4",
+   "metadata": {},
+   "source": [
+    "# Magnetohydrodynamics (MHD) Flow\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the MHD equations coupling the Navier-Stokes equations with Maxwell's equations:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\rho \\frac{\\partial \\mathbf{v}}{\\partial t} + \\rho (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} &= -\\nabla p + \\mu \\nabla^2 \\mathbf{v} + \\mathbf{J} \\times \\mathbf{B} \\\\\n",
+    "\\frac{\\partial \\mathbf{B}}{\\partial t} &= \\nabla \\times (\\mathbf{v} \\times \\mathbf{B} - \\eta \\nabla \\times \\mathbf{B}) \\\\\n",
+    "\\nabla \\cdot \\mathbf{v} &= 0 \\\\\n",
+    "\\nabla \\cdot \\mathbf{B} &= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\mathbf{J} = \\nabla \\times \\mathbf{B}$ is the current density.\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet and outlet: Specify velocity or pressure conditions.\n",
+    "- Magnetic field conditions depending on the problem.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Specify initial conditions for velocity, pressure, and the magnetic field.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $\\mathbf{v} \\in H_0^1(\\Omega)$, $p \\in L^2(\\Omega)$, and $\\mathbf{B} \\in H(\\text{curl}; \\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\rho \\frac{\\partial \\mathbf{v}}{\\partial t} \\cdot \\mathbf{w} \\, d\\Omega + \\int_{\\Omega} \\rho (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\mu \\nabla \\mathbf{v} : \\nabla \\mathbf{w} \\, d\\Omega - \\int_{\\Omega} p \\nabla \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\frac{\\partial \\mathbf{B}}{\\partial t} \\cdot \\boldsymbol{\\psi} \\, d\\Omega - \\int_{\\Omega} \\nabla \\times (\\mathbf{v} \\times \\mathbf{B} - \\eta \\nabla \\times \\mathbf{B}) \\cdot \\boldsymbol{\\psi} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\nabla \\cdot \\mathbf{v} q \\, d\\Omega + \\int_{\\Omega} \\nabla \\cdot \\mathbf{B} r \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $\\mathbf{w} \\in H_0^1(\\Omega)$, $q \\in L^2(\\Omega)$, $\\boldsymbol{\\psi} \\in H(\\text{curl}; \\Omega)$, and $r \\in L^2(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "850e45f1",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/material-science.md b/_sources/chapter5/material-science.md
new file mode 100644
index 0000000..1d2e5f8
--- /dev/null
+++ b/_sources/chapter5/material-science.md
@@ -0,0 +1 @@
+# Material Science
diff --git a/_sources/chapter5/mhd.md b/_sources/chapter5/mhd.md
new file mode 100644
index 0000000..cf5f17c
--- /dev/null
+++ b/_sources/chapter5/mhd.md
@@ -0,0 +1,55 @@
+# MHD
+
+In magnetohydrodynamics (MHD), finite element analysis can be applied to solve a variety of problems related to the behavior of electrically conducting fluids (plasmas or liquid metals) in the presence of magnetic fields. Here are some common problems in magnetohydrodynamics that can be addressed using finite element methods.
+
+Finite element analysis provides a powerful and flexible framework for addressing these complex problems in magnetohydrodynamics by discretizing the governing equations and solving them numerically. The specific application will determine the details of the problem setup and the required numerical techniques.
+
+## MHD Equations
+
+Magnetohydrodynamics (MHD) is a branch of physics and fluid dynamics that studies the magnetic properties and behavior of electrically conducting fluids, such as plasmas, liquid metals, and saltwater. The MHD equations describe the coupled interactions between the magnetic field, fluid flow, and electric current. Here are the mathematical models for the MHD equations:
+
+1. MHD Continuity Equation:
+   $$
+   \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0
+   $$
+   where:
+   - $\rho$ is the fluid density,
+   - $\mathbf{v}$ is the fluid velocity.
+
+2. MHD Momentum Equation:
+   $$
+   \rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \rho \mathbf{g} + \nabla \cdot \boldsymbol{\tau} + \mathbf{J} \times \mathbf{B}
+   $$
+   where:
+   - $\frac{D\mathbf{v}}{Dt}$ is the material derivative of velocity,
+   - $p$ is the pressure,
+   - $\mathbf{g}$ is the gravitational acceleration,
+   - $\boldsymbol{\tau}$ is the stress tensor,
+   - $\mathbf{J}$ is the current density,
+   - $\mathbf{B}$ is the magnetic field.
+
+3. MHD Induction Equation:
+   $$
+   \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B})
+   $$
+   where:
+   - $\eta$ is the magnetic diffusivity.
+
+4. MHD Ohm's Law:
+   $$
+   \mathbf{J} = \sigma (\mathbf{E} + \mathbf{v} \times \mathbf{B})
+   $$
+   where:
+   - $\sigma$ is the electrical conductivity,
+   - $\mathbf{E}$ is the electric field.
+
+5. MHD Energy Equation:
+   $$
+   \rho C_p \frac{DT}{Dt} = -p \nabla \cdot \mathbf{v} + \nabla \cdot (k \nabla T) + \frac{1}{\sigma}(\mathbf{J} \cdot \mathbf{E})
+   $$
+   where:
+   - $C_p$ is the specific heat at constant pressure,
+   - $T$ is the temperature,
+   - $k$ is the thermal conductivity.
+
+These equations, when solved together with appropriate boundary conditions, describe the behavior of magnetized fluids in the presence of electric and magnetic fields. Keep in mind that the specific form of these equations may vary based on the assumptions made and the type of fluid being considered (e.g., ideal MHD vs. resistive MHD).
diff --git a/_sources/chapter5/multiphysics.md b/_sources/chapter5/multiphysics.md
new file mode 100644
index 0000000..3748ec9
--- /dev/null
+++ b/_sources/chapter5/multiphysics.md
@@ -0,0 +1 @@
+# Multiphysics 
diff --git a/_sources/chapter5/non-newtonian-fluids.md b/_sources/chapter5/non-newtonian-fluids.md
new file mode 100644
index 0000000..9ae636c
--- /dev/null
+++ b/_sources/chapter5/non-newtonian-fluids.md
@@ -0,0 +1 @@
+# Non-Newtonian Fluids
diff --git a/_sources/chapter5/oldroyd-b_fluid_flow_in_a_channel.ipynb b/_sources/chapter5/oldroyd-b_fluid_flow_in_a_channel.ipynb
new file mode 100644
index 0000000..b7a8440
--- /dev/null
+++ b/_sources/chapter5/oldroyd-b_fluid_flow_in_a_channel.ipynb
@@ -0,0 +1,79 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "aea3782b",
+   "metadata": {},
+   "source": [
+    "# Oldroyd-B Fluid Flow in a Channel\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the Oldroyd-B model for viscoelastic fluids:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\frac{\\partial \\mathbf{u}}{\\partial t} + \\mathbf{u}\\cdot\\nabla\\mathbf{u} &= -\\frac{1}{\\rho}\\nabla p + \\nabla\\cdot\\boldsymbol{\\tau} \\\\\n",
+    "\\frac{\\partial \\boldsymbol{\\tau}}{\\partial t} + \\mathbf{u}\\cdot\\nabla\\boldsymbol{\\tau} &= \\boldsymbol{\\tau}\\cdot\\nabla\\mathbf{u} - \\frac{\\boldsymbol{\\tau}}{\\lambda} + 2\\mu\\left(\\nabla\\mathbf{u} - \\frac{\\boldsymbol{\\tau}}{\\lambda}\\right) \\\\\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\mathbf{u}$ is the velocity, $p$ is the pressure, $\\rho$ is the density, $\\boldsymbol{\\tau}$ is the extra stress tensor, $\\lambda$ is the relaxation time, and $\\mu$ is the viscosity.\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet and outlet: Specify velocity or pressure conditions.\n",
+    "- Channel walls: No-slip conditions.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Specify initial conditions for velocity and pressure.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $\\mathbf{u} \\in H_0^1(\\Omega)$ and $p \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\frac{\\partial \\mathbf{u}}{\\partial t}\\cdot\\mathbf{w} \\, d\\Omega + \\int_{\\Omega} \\mathbf{u}\\cdot\\nabla\\mathbf{u}\\cdot\\mathbf{w} \\, d\\Omega + \\int_{\\Omega} \\frac{1}{\\rho}\\nabla p\\cdot\\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\frac{\\partial \\boldsymbol{\\tau}}{\\partial t}:\\mathbf{v} \\, d\\Omega + \\int_{\\Omega} \\mathbf{u}\\cdot\\nabla\\boldsymbol{\\tau}:\\mathbf{v} \\, d\\Omega \\\\\n",
+    "&- \\int_{\\Omega} \\boldsymbol{\\tau}\\cdot\\nabla\\mathbf{u}:\\mathbf{v} \\, d\\Omega + \\int_{\\Omega} \\frac{\\boldsymbol{\\tau}}{\\lambda}:\\mathbf{v} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} 2\\mu\\left(\\nabla\\mathbf{u} - \\frac{\\boldsymbol{\\tau}}{\\lambda}\\right):\\nabla\\mathbf{w} \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $\\mathbf{w} \\in H_0^1(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b949a16a",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/papanastasiou_fluid_flow_in_a_channel.ipynb b/_sources/chapter5/papanastasiou_fluid_flow_in_a_channel.ipynb
new file mode 100644
index 0000000..c0a4bd3
--- /dev/null
+++ b/_sources/chapter5/papanastasiou_fluid_flow_in_a_channel.ipynb
@@ -0,0 +1,76 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "d8b8b5bd",
+   "metadata": {},
+   "source": [
+    "# Papanastasiou Fluid Flow in a Channel\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the Papanastasiou model for non-Newtonian fluids:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\sigma_{ij} &= -p\\delta_{ij} + 2\\mu\\left(\\varepsilon_{ij} + \\frac{\\tau_0}{2\\mu}\\right)\\left(\\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i}\\right) \\\\\n",
+    "\\varepsilon_{ij} &= \\frac{1}{2}\\left(\\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i}\\right) \\\\\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\sigma_{ij}$ is the stress tensor, $p$ is the pressure, $\\mu$ is the viscosity, $u_i$ is the velocity, $\\varepsilon_{ij}$ is the strain rate, and $\\tau_0$ is the yield stress.\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet and outlet: Specify velocity or pressure conditions.\n",
+    "- Channel walls: No-slip conditions.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Start with an initial guess for velocity and pressure.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $u_i \\in H_0^1(\\Omega)$ and $p \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\sigma_{ij} \\left(\\frac{\\partial u_i}{\\partial x_j} + \\frac{\\partial u_j}{\\partial x_i}\\right) \\, d\\Omega + \\int_{\\Omega} \\frac{\\partial p}{\\partial x_i}u_i \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $u_i \\in H_0^1(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "02de07e4",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/particle-laden_flow.ipynb b/_sources/chapter5/particle-laden_flow.ipynb
new file mode 100644
index 0000000..eaf2574
--- /dev/null
+++ b/_sources/chapter5/particle-laden_flow.ipynb
@@ -0,0 +1,79 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "b4a03bc4",
+   "metadata": {},
+   "source": [
+    "# Particle-Laden Flow (Lagrangian-Eulerian Approach)\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the incompressible Navier-Stokes equations coupled with the Lagrangian motion of particles:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\rho \\frac{\\partial \\mathbf{v}}{\\partial t} + \\rho (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} &= -\\nabla p + \\mu \\nabla^2 \\mathbf{v} + \\mathbf{F}_{\\text{drag}} \\\\\n",
+    "\\nabla \\cdot \\mathbf{v} &= 0 \\\\\n",
+    "\\frac{d \\mathbf{x}}{dt} &= \\mathbf{v}\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\mathbf{F}_{\\text{drag}}$ represents the drag force on the particles.\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet and outlet: Specify velocity or pressure conditions.\n",
+    "- Solid particle boundaries: Implement no-slip conditions.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Specify initial conditions for velocity, pressure, and particle positions.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $\\mathbf{v} \\in H_0^1(\\Omega)$ and $p \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\rho \\frac{\\partial \\mathbf{v}}{\\partial t} \\cdot \\mathbf{w} \\, d\\Omega + \\int_{\\Omega} \\rho (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\mu \\nabla \\mathbf{v} : \\nabla \\mathbf{w} \\, d\\Omega - \\int_{\\Omega} p \\nabla \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\nabla \\cdot \\mathbf{v} q \\, d\\Omega \\\\\n",
+    "&= -\\int_{\\Omega} \\mathbf{F}_{\\text{drag}} \\cdot \\mathbf{w} \\, d\\Omega\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $\\mathbf{w} \\in H_0^1(\\Omega)$ and $q \\in L^2(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "fe99e28f",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/power_law_fluid_flow_in_a_channel.ipynb b/_sources/chapter5/power_law_fluid_flow_in_a_channel.ipynb
new file mode 100644
index 0000000..5170b3e
--- /dev/null
+++ b/_sources/chapter5/power_law_fluid_flow_in_a_channel.ipynb
@@ -0,0 +1,76 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "33aa7908",
+   "metadata": {},
+   "source": [
+    "# Power Law Fluid Flow in a Channel\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the steady-state power-law fluid model for non-Newtonian fluids:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\sigma_{ij} &= -p\\delta_{ij} + \\lambda \\varepsilon_{kk}\\delta_{ij} + 2\\mu \\varepsilon_{ij} \\\\\n",
+    "\\varepsilon_{ij} &= \\frac{1}{2}\\left(\\frac{\\partial v_i}{\\partial x_j} + \\frac{\\partial v_j}{\\partial x_i}\\right) \\\\\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\sigma_{ij}$ is the stress tensor, $p$ is the pressure, $\\lambda$ and $\\mu$ are material constants, $v_i$ is the velocity, and $\\varepsilon_{ij}$ is the strain rate.\n",
+    "\n",
+    "### Boundary Conditions:\n",
+    "\n",
+    "- Inlet and outlet: Specify velocity or pressure conditions.\n",
+    "- Channel walls: No-slip conditions.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Start with an initial guess for velocity and pressure.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $v_i \\in H_0^1(\\Omega)$ and $p \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\sigma_{ij} \\varepsilon_{ij} \\, d\\Omega + \\int_{\\Omega} \\frac{\\partial p}{\\partial x_i}v_i \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $v_i \\in H_0^1(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f5664e1e",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/stokes_flow_in_a_lid-driven_cavity.ipynb b/_sources/chapter5/stokes_flow_in_a_lid-driven_cavity.ipynb
new file mode 100644
index 0000000..13f8d51
--- /dev/null
+++ b/_sources/chapter5/stokes_flow_in_a_lid-driven_cavity.ipynb
@@ -0,0 +1,78 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "fbe877e5",
+   "metadata": {},
+   "source": [
+    "# Stokes Flow in a Lid-Driven Cavity\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the steady-state Stokes equations for flow in a 2D lid-driven cavity:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "- \\nabla p + \\mu \\nabla^2 \\mathbf{v} &= 0 \\\\\n",
+    "\\nabla \\cdot \\mathbf{v} &= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Lid: $\\mathbf{v} = (1, 0)$ (lid-driven at constant velocity).\n",
+    "\n",
+    "- Other boundaries: $\\mathbf{v} = (0, 0)$ (no-slip condition).\n",
+    "\n",
+    "- Outlet: $\\frac{\\partial p}{\\partial n} = 0$ (zero-pressure gradient).\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Start with an initial guess for velocity and pressure.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $\\mathbf{v} \\in H_0^1(\\Omega)$ and $p \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\mu \\nabla \\mathbf{v} : \\nabla \\mathbf{w} \\, d\\Omega - \\int_{\\Omega} p \\nabla \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $\\mathbf{w} \\in H_0^1(\\Omega)$.\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "ce2a125a",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/chapter5/two-phase_flow.ipynb b/_sources/chapter5/two-phase_flow.ipynb
new file mode 100644
index 0000000..a425d84
--- /dev/null
+++ b/_sources/chapter5/two-phase_flow.ipynb
@@ -0,0 +1,78 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "94e72217",
+   "metadata": {},
+   "source": [
+    "# Two-Phase Flow (Immersed Boundary Method)\n",
+    "\n",
+    "## Mathematical Model\n",
+    "\n",
+    "Consider the incompressible Navier-Stokes equations with the Immersed Boundary Method for simulating two-phase flows:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\rho \\frac{\\partial \\mathbf{v}}{\\partial t} + \\rho (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} &= -\\nabla p + \\mu \\nabla^2 \\mathbf{v} + \\mathbf{f}_{\\text{IB}} \\\\\n",
+    "\\nabla \\cdot \\mathbf{v} &= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "where $\\mathbf{f}_{\\text{IB}}$ is the immersed boundary force.\n",
+    "\n",
+    "### Boundary Conditions\n",
+    "\n",
+    "- Inlet and outlet: Specify velocity or pressure conditions.\n",
+    "- Immersed boundary conditions to model the interface.\n",
+    "\n",
+    "### Initial Conditions\n",
+    "\n",
+    "Specify initial conditions for velocity, pressure, and the immersed boundary.\n",
+    "\n",
+    "## Weak Formulation\n",
+    "\n",
+    "Find $\\mathbf{v} \\in H_0^1(\\Omega)$ and $p \\in L^2(\\Omega)$ such that:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "&\\int_{\\Omega} \\rho \\frac{\\partial \\mathbf{v}}{\\partial t} \\cdot \\mathbf{w} \\, d\\Omega + \\int_{\\Omega} \\rho (\\mathbf{v} \\cdot \\nabla) \\mathbf{v} \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\mu \\nabla \\mathbf{v} : \\nabla \\mathbf{w} \\, d\\Omega - \\int_{\\Omega} p \\nabla \\cdot \\mathbf{w} \\, d\\Omega + \\int_{\\Omega} \\mathbf{f}_{\\text{IB}} \\cdot \\mathbf{w} \\, d\\Omega \\\\\n",
+    "&+ \\int_{\\Omega} \\nabla \\cdot \\mathbf{v} q \\, d\\Omega \\\\\n",
+    "&= 0\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "for all test functions $\\mathbf{w} \\in H_0^1(\\Omega)$ and $q \\in L^2(\\Omega)$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "2542c254",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": ".iga-python",
+   "language": "python",
+   "name": ".iga-python"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.basic.rst b/_sources/documentation/_autosummary/psydac.api.ast.basic.rst
new file mode 100644
index 0000000..f93a56d
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.basic.rst
@@ -0,0 +1,29 @@
+psydac.api.ast.basic
+====================
+
+.. automodule:: psydac.api.ast.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      SplBasic
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.evaluation.rst b/_sources/documentation/_autosummary/psydac.api.ast.evaluation.rst
new file mode 100644
index 0000000..2872e4c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.evaluation.rst
@@ -0,0 +1,30 @@
+psydac.api.ast.evaluation
+=========================
+
+.. automodule:: psydac.api.ast.evaluation
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      EvalArrayField
+      EvalArrayMapping
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.expr.rst b/_sources/documentation/_autosummary/psydac.api.ast.expr.rst
new file mode 100644
index 0000000..e82e76c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.expr.rst
@@ -0,0 +1,38 @@
+psydac.api.ast.expr
+===================
+
+.. automodule:: psydac.api.ast.expr
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_atoms_expr
+      is_scalar_field
+      is_vector_field
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      ExprInterface
+      ExprKernel
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.fem.rst b/_sources/documentation/_autosummary/psydac.api.ast.fem.rst
new file mode 100644
index 0000000..617af23
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.fem.rst
@@ -0,0 +1,44 @@
+psydac.api.ast.fem
+==================
+
+.. automodule:: psydac.api.ast.fem
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_diag_len
+      convert
+      expand
+      expand_hdiv_hcurl
+      get_degrees
+      get_multiplicity
+      get_nquads
+      regroup
+      toInteger
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      AST
+      DefNode
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.glt.rst b/_sources/documentation/_autosummary/psydac.api.ast.glt.rst
new file mode 100644
index 0000000..a4f3a14
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.glt.rst
@@ -0,0 +1,30 @@
+psydac.api.ast.glt
+==================
+
+.. automodule:: psydac.api.ast.glt
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      GltInterface
+      GltKernel
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.linalg.rst b/_sources/documentation/_autosummary/psydac.api.ast.linalg.rst
new file mode 100644
index 0000000..fbc186f
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.linalg.rst
@@ -0,0 +1,39 @@
+psydac.api.ast.linalg
+=====================
+
+.. automodule:: psydac.api.ast.linalg
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      Mod
+      compute_diag_len
+      toInteger
+      variable_to_sympy
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      LinearOperatorDot
+      VectorDot
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.nodes.rst b/_sources/documentation/_autosummary/psydac.api.ast.nodes.rst
new file mode 100644
index 0000000..7cfacea
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.nodes.rst
@@ -0,0 +1,189 @@
+psydac.api.ast.nodes
+====================
+
+.. automodule:: psydac.api.ast.nodes
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      construct_itergener
+      construct_logical_expressions
+      get_number_derivatives
+      random_string
+      toInteger
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      AddNode
+      Allocate
+      AndNode
+      ArityType
+      Array
+      ArrayNode
+      AtomicNode
+      BaseNode
+      BasisAtom
+      BilinearArity
+      Block
+      BlockLinearOperatorNode
+      BlockScalarLocalBasis
+      BlockStencilMatrixGlobalBasis
+      BlockStencilMatrixLocalBasis
+      BlockStencilVectorGlobalBasis
+      BlockStencilVectorLocalBasis
+      CoefficientBasis
+      ComputeKernelExpr
+      ComputeLogical
+      ComputeLogicalBasis
+      ComputeNode
+      ComputePhysical
+      ComputePhysicalBasis
+      Element
+      ElementOf
+      EqNode
+      EvalField
+      EvalMapping
+      Evaluation
+      ExprNode
+      Expression
+      FieldEvaluation
+      FloorDiv
+      FunctionalArity
+      GeneratorBase
+      GeometryAtom
+      GeometryExpr
+      GeometryExpressions
+      GlobalElementBasis
+      GlobalSpanArray
+      GlobalTensorQuadratureBasis
+      GlobalTensorQuadratureGrid
+      GlobalTensorQuadratureTestBasis
+      GlobalTensorQuadratureTrialBasis
+      GlobalThreadEnds
+      GlobalThreadSizes
+      GlobalThreadSpanArray
+      GlobalThreadStarts
+      Grid
+      IfNode
+      IndexDerivative
+      IndexDof
+      IndexDofTest
+      IndexDofTrial
+      IndexElement
+      IndexInnerDofTest
+      IndexNode
+      IndexOuterDofTest
+      IndexQuadrature
+      IntDivNode
+      IteratorBase
+      LengthDof
+      LengthDofTest
+      LengthDofTrial
+      LengthElement
+      LengthInnerDofTest
+      LengthNode
+      LengthOuterDofTest
+      LengthQuadrature
+      LinearArity
+      LocalElementBasis
+      LocalIndexElement
+      LocalSpanArray
+      LocalTensorQuadratureBasis
+      LocalTensorQuadratureGrid
+      LocalTensorQuadratureTestBasis
+      LocalTensorQuadratureTrialBasis
+      LocalThreadEnds
+      LocalThreadStarts
+      LogicalBasisValue
+      LogicalGeometryValue
+      LogicalValueNode
+      Loop
+      MappingEvaluation
+      Mask
+      MatrixCoordsFromRank
+      MatrixGlobalBasis
+      MatrixLocalBasis
+      MatrixNode
+      MatrixQuadrature
+      MatrixRankFromCoords
+      Max
+      Min
+      MulNode
+      NeighbourThreadCoordinates
+      NotNode
+      NumThreads
+      Pads
+      ParallelBlock
+      Pattern
+      PhysicalBasisValue
+      PhysicalGeometryValue
+      PhysicalValueNode
+      PlusGlobalTensorQuadratureGrid
+      PlusLocalTensorQuadratureGrid
+      PlusTensorQuadrature
+      ProductGenerator
+      ProductIteration
+      ProductIterator
+      RAT
+      RankDimension
+      RankNode
+      Reduce
+      Reduction
+      Reset
+      ScalarLocalBasis
+      ScalarNode
+      Span
+      SpanArray
+      SplitArray
+      StencilMatrixGlobalBasis
+      StencilMatrixLocalBasis
+      StencilVectorGlobalBasis
+      StencilVectorLocalBasis
+      StrictLessThanNode
+      TensorAdd
+      TensorAssignExpr
+      TensorBasis
+      TensorExpression
+      TensorGenerator
+      TensorIntDiv
+      TensorInteger
+      TensorIteration
+      TensorIterator
+      TensorMax
+      TensorMul
+      TensorQuadrature
+      TensorQuadratureBasis
+      TensorQuadratureTestBasis
+      TensorQuadratureTrialBasis
+      TensorTestBasis
+      TensorTrialBasis
+      ThreadCoordinates
+      ThreadId
+      ValueNode
+      VectorAssign
+      WeightedVolumeQuadrature
+      WhileLoop
+      Zeros
+      ZerosLike
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.parser.rst b/_sources/documentation/_autosummary/psydac.api.ast.parser.rst
new file mode 100644
index 0000000..fe59652
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.parser.rst
@@ -0,0 +1,38 @@
+psydac.api.ast.parser
+=====================
+
+.. automodule:: psydac.api.ast.parser
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      is_scalar_array
+      parse
+      random_string
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Parser
+      Shape
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.rst b/_sources/documentation/_autosummary/psydac.api.ast.rst
new file mode 100644
index 0000000..8fe22e4
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.rst
@@ -0,0 +1,40 @@
+psydac.api.ast
+==============
+
+.. automodule:: psydac.api.ast
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.api.ast.basic
+   psydac.api.ast.evaluation
+   psydac.api.ast.expr
+   psydac.api.ast.fem
+   psydac.api.ast.glt
+   psydac.api.ast.linalg
+   psydac.api.ast.nodes
+   psydac.api.ast.parser
+   psydac.api.ast.tests
+   psydac.api.ast.utilities
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.tests.boundary.rst b/_sources/documentation/_autosummary/psydac.api.ast.tests.boundary.rst
new file mode 100644
index 0000000..62e65ae
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.tests.boundary.rst
@@ -0,0 +1,29 @@
+psydac.api.ast.tests.boundary
+=============================
+
+.. automodule:: psydac.api.ast.tests.boundary
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_codegen
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.tests.rst b/_sources/documentation/_autosummary/psydac.api.ast.tests.rst
new file mode 100644
index 0000000..c73be3a
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.tests.rst
@@ -0,0 +1,36 @@
+psydac.api.ast.tests
+====================
+
+.. automodule:: psydac.api.ast.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.api.ast.tests.boundary
+   psydac.api.ast.tests.poisson
+   psydac.api.ast.tests.system_1
+   psydac.api.ast.tests.system_2
+   psydac.api.ast.tests.system_3
+   psydac.api.ast.tests.test_nodes
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.tests.system_1.rst b/_sources/documentation/_autosummary/psydac.api.ast.tests.system_1.rst
new file mode 100644
index 0000000..cc6bb36
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.tests.system_1.rst
@@ -0,0 +1,29 @@
+psydac.api.ast.tests.system\_1
+==============================
+
+.. automodule:: psydac.api.ast.tests.system_1
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_codegen
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.tests.system_2.rst b/_sources/documentation/_autosummary/psydac.api.ast.tests.system_2.rst
new file mode 100644
index 0000000..415fa0c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.tests.system_2.rst
@@ -0,0 +1,29 @@
+psydac.api.ast.tests.system\_2
+==============================
+
+.. automodule:: psydac.api.ast.tests.system_2
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_codegen
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.tests.system_3.rst b/_sources/documentation/_autosummary/psydac.api.ast.tests.system_3.rst
new file mode 100644
index 0000000..948dd80
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.tests.system_3.rst
@@ -0,0 +1,29 @@
+psydac.api.ast.tests.system\_3
+==============================
+
+.. automodule:: psydac.api.ast.tests.system_3
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_codegen
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.tests.test_nodes.rst b/_sources/documentation/_autosummary/psydac.api.ast.tests.test_nodes.rst
new file mode 100644
index 0000000..2aa1eef
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.tests.test_nodes.rst
@@ -0,0 +1,51 @@
+psydac.api.ast.tests.test\_nodes
+================================
+
+.. automodule:: psydac.api.ast.tests.test_nodes
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      assembly_bilinear_form_2d_1
+      assembly_linear_form_2d_1
+      basis_atom_2d_1
+      basis_atom_2d_2
+      eval_field_2d_1
+      geometry_atom_2d_1
+      global_quad_basis_span_2d_1
+      global_quad_basis_span_2d_2
+      global_quad_basis_span_2d_matrix_1
+      global_quad_basis_span_2d_matrix_2
+      global_quad_basis_span_2d_vector_1
+      global_quad_basis_span_2d_vector_2
+      global_quad_span_2d_1
+      global_span_2d_1
+      loop_global_local_quad_2d_1
+      loop_local_dof_quad_2d_1
+      loop_local_dof_quad_2d_2
+      loop_local_dof_quad_2d_3
+      loop_local_dof_quad_2d_4
+      loop_local_quad_2d_1
+      loop_local_quad_geometry_2d_1
+      teardown_function
+      teardown_module
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.ast.utilities.rst b/_sources/documentation/_autosummary/psydac.api.ast.utilities.rst
new file mode 100644
index 0000000..4a519d5
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.ast.utilities.rst
@@ -0,0 +1,47 @@
+psydac.api.ast.utilities
+========================
+
+.. automodule:: psydac.api.ast.utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      build_pyccel_type_annotations
+      build_pythran_types_header
+      compute_atoms_expr
+      compute_atoms_expr_field
+      compute_atoms_expr_mapping
+      compute_boundary_jacobian
+      compute_normal_vector
+      compute_tangent_vector
+      filter_loops
+      filter_product
+      fusion_loops
+      get_name
+      is_mapping
+      logical2physical
+      math_atoms_as_str
+      random_string
+      rationalize_eval_mapping
+      select_loops
+      variables
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.basic.rst b/_sources/documentation/_autosummary/psydac.api.basic.rst
new file mode 100644
index 0000000..065fbf9
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.basic.rst
@@ -0,0 +1,30 @@
+psydac.api.basic
+================
+
+.. automodule:: psydac.api.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BasicCodeGen
+      BasicDiscrete
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.discretization.rst b/_sources/documentation/_autosummary/psydac.api.discretization.rst
new file mode 100644
index 0000000..6b974d6
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.discretization.rst
@@ -0,0 +1,34 @@
+psydac.api.discretization
+=========================
+
+.. automodule:: psydac.api.discretization
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      change_dtype
+      discretize
+      discretize_derham
+      discretize_domain
+      discretize_space
+      reduce_space_degrees
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.equation.rst b/_sources/documentation/_autosummary/psydac.api.equation.rst
new file mode 100644
index 0000000..32a66cb
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.equation.rst
@@ -0,0 +1,36 @@
+psydac.api.equation
+===================
+
+.. automodule:: psydac.api.equation
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      l2_boundary_projection
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteEquation
+      LinearSystem
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.essential_bc.rst b/_sources/documentation/_autosummary/psydac.api.essential_bc.rst
new file mode 100644
index 0000000..13ca82e
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.essential_bc.rst
@@ -0,0 +1,34 @@
+psydac.api.essential\_bc
+========================
+
+.. automodule:: psydac.api.essential_bc
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      apply_essential_bc
+      apply_essential_bc_BlockLinearOperator
+      apply_essential_bc_BlockVector
+      apply_essential_bc_kronecker_dense_matrix
+      apply_essential_bc_stencil
+      check_boundary_type
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.expr.rst b/_sources/documentation/_autosummary/psydac.api.expr.rst
new file mode 100644
index 0000000..993bb9d
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.expr.rst
@@ -0,0 +1,29 @@
+psydac.api.expr
+===============
+
+.. automodule:: psydac.api.expr
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteExpr
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.feec.rst b/_sources/documentation/_autosummary/psydac.api.feec.rst
new file mode 100644
index 0000000..dc6b997
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.feec.rst
@@ -0,0 +1,29 @@
+psydac.api.feec
+===============
+
+.. automodule:: psydac.api.feec
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteDerham
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.fem.rst b/_sources/documentation/_autosummary/psydac.api.fem.rst
new file mode 100644
index 0000000..af7dcd0
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.fem.rst
@@ -0,0 +1,47 @@
+psydac.api.fem
+==============
+
+.. automodule:: psydac.api.fem
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      collect_spaces
+      compute_diag_len
+      construct_quad_grids_arguments
+      construct_test_space_arguments
+      construct_trial_space_arguments
+      do_nothing
+      extract_stencil_mats
+      get_nquads
+      reset_arrays
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteBilinearForm
+      DiscreteFunctional
+      DiscreteLinearForm
+      DiscreteSesquilinearForm
+      DiscreteSumForm
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.glt.rst b/_sources/documentation/_autosummary/psydac.api.glt.rst
new file mode 100644
index 0000000..6428dc5
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.glt.rst
@@ -0,0 +1,30 @@
+psydac.api.glt
+==============
+
+.. automodule:: psydac.api.glt
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteGltExpr
+      GltBasicCodeGen
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.grid.rst b/_sources/documentation/_autosummary/psydac.api.grid.rst
new file mode 100644
index 0000000..d158c7c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.grid.rst
@@ -0,0 +1,38 @@
+psydac.api.grid
+===============
+
+.. automodule:: psydac.api.grid
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      create_collocation_basis
+      get_points_weights
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BasisValues
+      CollocationBasisValues
+      QuadratureGrid
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.postprocessing.rst b/_sources/documentation/_autosummary/psydac.api.postprocessing.rst
new file mode 100644
index 0000000..7a87d87
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.postprocessing.rst
@@ -0,0 +1,36 @@
+psydac.api.postprocessing
+=========================
+
+.. automodule:: psydac.api.postprocessing
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_grid_lines_2d
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      OutputManager
+      PostProcessManager
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.printing.pycode.rst b/_sources/documentation/_autosummary/psydac.api.printing.pycode.rst
new file mode 100644
index 0000000..c7dcf02
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.printing.pycode.rst
@@ -0,0 +1,35 @@
+psydac.api.printing.pycode
+==========================
+
+.. automodule:: psydac.api.printing.pycode
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      pycode
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      PythonCodePrinter
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.printing.rst b/_sources/documentation/_autosummary/psydac.api.printing.rst
new file mode 100644
index 0000000..1eb5091
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.printing.rst
@@ -0,0 +1,31 @@
+psydac.api.printing
+===================
+
+.. automodule:: psydac.api.printing
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.api.printing.pycode
+
diff --git a/_sources/documentation/_autosummary/psydac.api.rst b/_sources/documentation/_autosummary/psydac.api.rst
new file mode 100644
index 0000000..0cd02e0
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.rst
@@ -0,0 +1,45 @@
+psydac.api
+==========
+
+.. automodule:: psydac.api
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.api.ast
+   psydac.api.basic
+   psydac.api.discretization
+   psydac.api.equation
+   psydac.api.essential_bc
+   psydac.api.expr
+   psydac.api.feec
+   psydac.api.fem
+   psydac.api.glt
+   psydac.api.grid
+   psydac.api.postprocessing
+   psydac.api.printing
+   psydac.api.settings
+   psydac.api.tests
+   psydac.api.utilities
+
diff --git a/_sources/documentation/_autosummary/psydac.api.settings.rst b/_sources/documentation/_autosummary/psydac.api.settings.rst
new file mode 100644
index 0000000..45664be
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.settings.rst
@@ -0,0 +1,23 @@
+psydac.api.settings
+===================
+
+.. automodule:: psydac.api.settings
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.build_domain.rst b/_sources/documentation/_autosummary/psydac.api.tests.build_domain.rst
new file mode 100644
index 0000000..392175c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.build_domain.rst
@@ -0,0 +1,38 @@
+psydac.api.tests.build\_domain
+==============================
+
+.. automodule:: psydac.api.tests.build_domain
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      build_pretzel
+      create_domain
+      flip_axis
+      get_2D_rotation_mapping
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      TransposedPolarMapping
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.rst b/_sources/documentation/_autosummary/psydac.api.tests.rst
new file mode 100644
index 0000000..f0163d6
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.rst
@@ -0,0 +1,67 @@
+psydac.api.tests
+================
+
+.. automodule:: psydac.api.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.api.tests.build_domain
+   psydac.api.tests.test_2d_biharmonic
+   psydac.api.tests.test_2d_complex
+   psydac.api.tests.test_2d_laplace
+   psydac.api.tests.test_2d_mapping_biharmonic
+   psydac.api.tests.test_2d_mapping_laplace
+   psydac.api.tests.test_2d_mapping_poisson
+   psydac.api.tests.test_2d_multipatch_mapping_maxwell
+   psydac.api.tests.test_2d_multipatch_mapping_poisson
+   psydac.api.tests.test_2d_multipatch_poisson
+   psydac.api.tests.test_2d_navier_stokes
+   psydac.api.tests.test_2d_poisson
+   psydac.api.tests.test_api_1d_compatible_spaces
+   psydac.api.tests.test_api_2d_compatible_spaces
+   psydac.api.tests.test_api_2d_fields
+   psydac.api.tests.test_api_2d_scalar_analytical_mapping
+   psydac.api.tests.test_api_2d_system
+   psydac.api.tests.test_api_2d_vector
+   psydac.api.tests.test_api_2d_vector_mapping
+   psydac.api.tests.test_api_2d_vector_multipatch_mapping
+   psydac.api.tests.test_api_3d_scalar
+   psydac.api.tests.test_api_3d_scalar_mapping
+   psydac.api.tests.test_api_3d_vector
+   psydac.api.tests.test_api_3d_vector_mapping
+   psydac.api.tests.test_api_expr_2d_scalar
+   psydac.api.tests.test_api_feec_1d
+   psydac.api.tests.test_api_feec_2d
+   psydac.api.tests.test_api_feec_3d
+   psydac.api.tests.test_api_glt_2d_scalar
+   psydac.api.tests.test_api_glt_2d_scalar_mapping
+   psydac.api.tests.test_api_glt_2d_vector
+   psydac.api.tests.test_assembly
+   psydac.api.tests.test_build_derham_mapping
+   psydac.api.tests.test_equation
+   psydac.api.tests.test_postprocessing
+   psydac.api.tests.test_quadorder
+   psydac.api.tests.utils
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_2d_biharmonic.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_biharmonic.rst
new file mode 100644
index 0000000..07743dd
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_biharmonic.rst
@@ -0,0 +1,35 @@
+psydac.api.tests.test\_2d\_biharmonic
+=====================================
+
+.. automodule:: psydac.api.tests.test_2d_biharmonic
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_boundaries
+      run_biharmonic_2d_dir
+      teardown_function
+      teardown_module
+      test_biharmonic_2d_dir0_1234
+      test_biharmonic_2d_dir0_123_diri_4
+      test_biharmonic_2d_dir0_13_diri_24
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_2d_complex.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_complex.rst
new file mode 100644
index 0000000..3954a96
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_complex.rst
@@ -0,0 +1,42 @@
+psydac.api.tests.test\_2d\_complex
+==================================
+
+.. automodule:: psydac.api.tests.test_2d_complex
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_boundaries
+      run_biharmonic_2d_dir
+      run_helmholtz_2d
+      run_maxwell_2d
+      run_poisson_2d
+      teardown_function
+      teardown_module
+      test_complex_biharmonic_2d
+      test_complex_biharmonic_2d_mapping
+      test_complex_helmholtz_2d
+      test_complex_poisson_2d_multipatch
+      test_complex_poisson_2d_multipatch_mapping
+      test_maxwell_2d_2_patch_dirichlet_2
+      test_maxwell_2d_2_patch_dirichlet_parallel_0
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_2d_laplace.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_laplace.rst
new file mode 100644
index 0000000..8621f0e
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_laplace.rst
@@ -0,0 +1,33 @@
+psydac.api.tests.test\_2d\_laplace
+==================================
+
+.. automodule:: psydac.api.tests.test_2d_laplace
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_boundaries
+      run_laplace_2d
+      teardown_function
+      teardown_module
+      test_laplace_2d_neu0_1234
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_2d_mapping_biharmonic.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_mapping_biharmonic.rst
new file mode 100644
index 0000000..383719a
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_mapping_biharmonic.rst
@@ -0,0 +1,35 @@
+psydac.api.tests.test\_2d\_mapping\_biharmonic
+==============================================
+
+.. automodule:: psydac.api.tests.test_2d_mapping_biharmonic
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_boundaries
+      run_biharmonic_2d_dir
+      teardown_function
+      teardown_module
+      test_biharmonic_2d_collela_dir0_1234
+      test_biharmonic_2d_identity_dir0_1234
+      test_biharmonic_2d_quarter_annulus_dir0_1234
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_2d_mapping_laplace.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_mapping_laplace.rst
new file mode 100644
index 0000000..af2f5f8
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_mapping_laplace.rst
@@ -0,0 +1,34 @@
+psydac.api.tests.test\_2d\_mapping\_laplace
+===========================================
+
+.. automodule:: psydac.api.tests.test_2d_mapping_laplace
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_boundaries
+      run_laplace_2d
+      teardown_function
+      teardown_module
+      test_laplace_2d_collela_neu0_1234
+      test_laplace_2d_identity_neu0_1234
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_2d_mapping_poisson.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_mapping_poisson.rst
new file mode 100644
index 0000000..b4631e4
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_mapping_poisson.rst
@@ -0,0 +1,65 @@
+psydac.api.tests.test\_2d\_mapping\_poisson
+===========================================
+
+.. automodule:: psydac.api.tests.test_2d_mapping_poisson
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_boundaries
+      run_poisson_2d
+      teardown_function
+      teardown_module
+      test_poisson_2d_circle_dir0
+      test_poisson_2d_collela_dir0_1234
+      test_poisson_2d_collela_dir0_123_diri_4
+      test_poisson_2d_collela_dir0_123_neu0_4
+      test_poisson_2d_collela_dir0_123_neui_4
+      test_poisson_2d_collela_dir0_124_neu0_3
+      test_poisson_2d_collela_dir0_124_neui_3
+      test_poisson_2d_collela_dir0_134_neu0_2
+      test_poisson_2d_collela_dir0_134_neui_2
+      test_poisson_2d_collela_dir0_13_diri_24
+      test_poisson_2d_collela_dir0_234_neu0_1
+      test_poisson_2d_collela_dir0_234_neui_1
+      test_poisson_2d_collela_diri_1234
+      test_poisson_2d_identity_dir0_1234
+      test_poisson_2d_identity_dir0_1234_parallel
+      test_poisson_2d_identity_dir0_123_diri_4
+      test_poisson_2d_identity_dir0_123_neu0_4
+      test_poisson_2d_identity_dir0_123_neui_4
+      test_poisson_2d_identity_dir0_124_neu0_3
+      test_poisson_2d_identity_dir0_124_neui_3
+      test_poisson_2d_identity_dir0_134_neu0_2
+      test_poisson_2d_identity_dir0_134_neui_2
+      test_poisson_2d_identity_dir0_13_diri_24
+      test_poisson_2d_identity_dir0_234_neu0_1
+      test_poisson_2d_identity_dir0_234_neui_1
+      test_poisson_2d_identity_dir0_24_neu0_13
+      test_poisson_2d_identity_dir0_4_neu0_123
+      test_poisson_2d_pipe_dir_1234
+      test_poisson_2d_quarter_annulus_dir0_1234
+      test_poisson_2d_quarter_annulus_dir0_12_diri_34
+      test_poisson_2d_quarter_annulus_diri_1234
+      test_poisson_2d_quarter_annulus_diri_12_neui_34
+      test_poisson_2d_quarter_annulus_diri_34_neui_12
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_2d_multipatch_mapping_maxwell.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_multipatch_mapping_maxwell.rst
new file mode 100644
index 0000000..f4fab44
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_multipatch_mapping_maxwell.rst
@@ -0,0 +1,36 @@
+psydac.api.tests.test\_2d\_multipatch\_mapping\_maxwell
+=======================================================
+
+.. automodule:: psydac.api.tests.test_2d_multipatch_mapping_maxwell
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_maxwell_2d
+      teardown_function
+      teardown_module
+      test_maxwell_2d_2_patch_dirichlet_0
+      test_maxwell_2d_2_patch_dirichlet_1
+      test_maxwell_2d_2_patch_dirichlet_2
+      test_maxwell_2d_2_patch_dirichlet_parallel_0
+      test_maxwell_2d_2_patch_dirichlet_parallel_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_2d_multipatch_mapping_poisson.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_multipatch_mapping_poisson.rst
new file mode 100644
index 0000000..7300617
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_multipatch_mapping_poisson.rst
@@ -0,0 +1,42 @@
+psydac.api.tests.test\_2d\_multipatch\_mapping\_poisson
+=======================================================
+
+.. automodule:: psydac.api.tests.test_2d_multipatch_mapping_poisson
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_poisson_2d
+      teardown_function
+      teardown_module
+      test_poisson_2d_2_patches_dirichlet_0
+      test_poisson_2d_2_patches_dirichlet_1
+      test_poisson_2d_2_patches_dirichlet_2
+      test_poisson_2d_2_patches_dirichlet_3
+      test_poisson_2d_2_patches_dirichlet_4
+      test_poisson_2d_2_patches_dirichlet_5
+      test_poisson_2d_2_patches_dirichlet_6
+      test_poisson_2d_2_patches_dirichlet_parallel_0
+      test_poisson_2d_2_patches_dirichlet_parallel_1
+      test_poisson_2d_4_patch_dirichlet_0
+      test_poisson_2d_4_patches_dirichlet_parallel_0
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_2d_multipatch_poisson.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_multipatch_poisson.rst
new file mode 100644
index 0000000..9cdd721
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_multipatch_poisson.rst
@@ -0,0 +1,36 @@
+psydac.api.tests.test\_2d\_multipatch\_poisson
+==============================================
+
+.. automodule:: psydac.api.tests.test_2d_multipatch_poisson
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_poisson_2d
+      teardown_function
+      teardown_module
+      test_poisson_2d_2_patch_dirichlet_0
+      test_poisson_2d_2_patch_dirichlet_1
+      test_poisson_2d_2_patch_dirichlet_2
+      test_poisson_2d_2_patch_dirichlet_3
+      test_poisson_2d_2_patch_dirichlet_4
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_2d_navier_stokes.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_navier_stokes.rst
new file mode 100644
index 0000000..6de4076
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_navier_stokes.rst
@@ -0,0 +1,38 @@
+psydac.api.tests.test\_2d\_navier\_stokes
+=========================================
+
+.. automodule:: psydac.api.tests.test_2d_navier_stokes
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_boundaries
+      psydac_solver
+      run_steady_state_navier_stokes_2d
+      run_time_dependent_navier_stokes_2d
+      scipy_solver
+      teardown_function
+      teardown_module
+      test_navier_stokes_2d
+      test_st_navier_stokes_2d
+      test_st_navier_stokes_2d_parallel
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_2d_poisson.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_poisson.rst
new file mode 100644
index 0000000..d902671
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_2d_poisson.rst
@@ -0,0 +1,48 @@
+psydac.api.tests.test\_2d\_poisson
+==================================
+
+.. automodule:: psydac.api.tests.test_2d_poisson
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_boundaries
+      run_poisson_2d
+      teardown_function
+      teardown_module
+      test_poisson_2d_dir0_1234
+      test_poisson_2d_dir0_1234_parallel
+      test_poisson_2d_dir0_1234_user_function
+      test_poisson_2d_dir0_123_diri_4
+      test_poisson_2d_dir0_123_neu0_4
+      test_poisson_2d_dir0_123_neui_4
+      test_poisson_2d_dir0_124_neu0_3
+      test_poisson_2d_dir0_124_neui_3
+      test_poisson_2d_dir0_134_neu0_2
+      test_poisson_2d_dir0_134_neui_2
+      test_poisson_2d_dir0_13_diri_24
+      test_poisson_2d_dir0_13_neu0_24
+      test_poisson_2d_dir0_234_neu0_1
+      test_poisson_2d_dir0_234_neui_1
+      test_poisson_2d_dir0_24_neu0_13
+      test_poisson_2d_dir0_4_neu0_123
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_1d_compatible_spaces.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_1d_compatible_spaces.rst
new file mode 100644
index 0000000..0f2d0a2
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_1d_compatible_spaces.rst
@@ -0,0 +1,30 @@
+psydac.api.tests.test\_api\_1d\_compatible\_spaces
+==================================================
+
+.. automodule:: psydac.api.tests.test_api_1d_compatible_spaces
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_system_1_1d_dir
+      test_api_system_1_1d_dir_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_compatible_spaces.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_compatible_spaces.rst
new file mode 100644
index 0000000..179570a
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_compatible_spaces.rst
@@ -0,0 +1,39 @@
+psydac.api.tests.test\_api\_2d\_compatible\_spaces
+==================================================
+
+.. automodule:: psydac.api.tests.test_api_2d_compatible_spaces
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_maxwell_time_harmonic_2d_dir
+      run_poisson_mixed_form_2d_dir
+      run_stokes_2d_dir
+      run_stokes_2d_dir_petsc
+      teardown_function
+      teardown_module
+      test_maxwell_time_harmonic_2d_dir_1
+      test_poisson_mixed_form_2d_dir_1
+      test_stokes_2d_dir_homogeneous
+      test_stokes_2d_dir_non_homogeneous
+      test_stokes_2d_dir_non_homogeneous_petsc
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_fields.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_fields.rst
new file mode 100644
index 0000000..921a75c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_fields.rst
@@ -0,0 +1,39 @@
+psydac.api.tests.test\_api\_2d\_fields
+======================================
+
+.. automodule:: psydac.api.tests.test_api_2d_fields
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_boundary_field_test
+      run_field_test
+      run_non_linear_poisson
+      teardown_function
+      teardown_module
+      test_boundary_field_identity
+      test_field_collela
+      test_field_identity_1
+      test_field_identity_2
+      test_field_quarter_annulus
+      test_nonlinear_poisson_circle
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_scalar_analytical_mapping.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_scalar_analytical_mapping.rst
new file mode 100644
index 0000000..c70558c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_scalar_analytical_mapping.rst
@@ -0,0 +1,32 @@
+psydac.api.tests.test\_api\_2d\_scalar\_analytical\_mapping
+===========================================================
+
+.. automodule:: psydac.api.tests.test_api_2d_scalar_analytical_mapping
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_poisson_2d
+      teardown_function
+      teardown_module
+      test_poisson_2d_analytical_mapping_0
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_system.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_system.rst
new file mode 100644
index 0000000..57bb6dc
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_system.rst
@@ -0,0 +1,32 @@
+psydac.api.tests.test\_api\_2d\_system
+======================================
+
+.. automodule:: psydac.api.tests.test_api_2d_system
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_system_1_2d_dir
+      teardown_function
+      teardown_module
+      test_api_system_1_2d_dir_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_vector.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_vector.rst
new file mode 100644
index 0000000..317e002
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_vector.rst
@@ -0,0 +1,32 @@
+psydac.api.tests.test\_api\_2d\_vector
+======================================
+
+.. automodule:: psydac.api.tests.test_api_2d_vector
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_vector_poisson_2d_dir
+      teardown_function
+      teardown_module
+      test_api_vector_poisson_2d_dir_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_vector_mapping.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_vector_mapping.rst
new file mode 100644
index 0000000..13ac329
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_vector_mapping.rst
@@ -0,0 +1,33 @@
+psydac.api.tests.test\_api\_2d\_vector\_mapping
+===============================================
+
+.. automodule:: psydac.api.tests.test_api_2d_vector_mapping
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_vector_poisson_2d_dir
+      teardown_function
+      teardown_module
+      test_api_vector_poisson_2d_dir_collela
+      test_api_vector_poisson_2d_dir_identity
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_vector_multipatch_mapping.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_vector_multipatch_mapping.rst
new file mode 100644
index 0000000..a093b40
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_2d_vector_multipatch_mapping.rst
@@ -0,0 +1,32 @@
+psydac.api.tests.test\_api\_2d\_vector\_multipatch\_mapping
+===========================================================
+
+.. automodule:: psydac.api.tests.test_api_2d_vector_multipatch_mapping
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_vector_poisson_2d_dir
+      teardown_function
+      teardown_module
+      test_api_vector_poisson_2d_dir_identity
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_scalar.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_scalar.rst
new file mode 100644
index 0000000..f94b3be
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_scalar.rst
@@ -0,0 +1,37 @@
+psydac.api.tests.test\_api\_3d\_scalar
+======================================
+
+.. automodule:: psydac.api.tests.test_api_3d_scalar
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_poisson_3d_dir
+      run_poisson_3d_dirneu
+      teardown_function
+      teardown_module
+      test_api_poisson_3d_dir_1
+      test_api_poisson_3d_dir_1_parallel
+      test_api_poisson_3d_dirneu_13
+      test_api_poisson_3d_dirneu_2
+      test_api_poisson_3d_dirneu_24
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_scalar_mapping.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_scalar_mapping.rst
new file mode 100644
index 0000000..a62afbb
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_scalar_mapping.rst
@@ -0,0 +1,40 @@
+psydac.api.tests.test\_api\_3d\_scalar\_mapping
+===============================================
+
+.. automodule:: psydac.api.tests.test_api_3d_scalar_mapping
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_laplace_3d_neu
+      run_poisson_3d_dir
+      run_poisson_3d_dirneu
+      teardown_function
+      teardown_module
+      test_api_laplace_3d_neu_identity
+      test_api_poisson_3d_dir_collela
+      test_api_poisson_3d_dirneu_collela_2
+      test_api_poisson_3d_dirneu_collela_24
+      test_api_poisson_3d_dirneu_identity_13
+      test_api_poisson_3d_dirneu_identity_2
+      test_api_poisson_3d_dirneu_identity_24
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_vector.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_vector.rst
new file mode 100644
index 0000000..ce2fb37
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_vector.rst
@@ -0,0 +1,32 @@
+psydac.api.tests.test\_api\_3d\_vector
+======================================
+
+.. automodule:: psydac.api.tests.test_api_3d_vector
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_vector_poisson_3d_dir
+      teardown_function
+      teardown_module
+      test_api_vector_poisson_3d_dir_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_vector_mapping.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_vector_mapping.rst
new file mode 100644
index 0000000..179d89b
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_3d_vector_mapping.rst
@@ -0,0 +1,33 @@
+psydac.api.tests.test\_api\_3d\_vector\_mapping
+===============================================
+
+.. automodule:: psydac.api.tests.test_api_3d_vector_mapping
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_vector_poisson_3d_dir
+      teardown_function
+      teardown_module
+      test_api_vector_poisson_3d_dir_collela
+      test_api_vector_poisson_3d_dir_identity
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_expr_2d_scalar.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_expr_2d_scalar.rst
new file mode 100644
index 0000000..cb5100a
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_expr_2d_scalar.rst
@@ -0,0 +1,30 @@
+psydac.api.tests.test\_api\_expr\_2d\_scalar
+============================================
+
+.. automodule:: psydac.api.tests.test_api_expr_2d_scalar
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_poisson_2d_dir
+      test_api_expr_2d_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_feec_1d.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_feec_1d.rst
new file mode 100644
index 0000000..036c0af
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_feec_1d.rst
@@ -0,0 +1,38 @@
+psydac.api.tests.test\_api\_feec\_1d
+====================================
+
+.. automodule:: psydac.api.tests.test_api_feec_1d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      make_plot
+      run_maxwell_1d
+      test_maxwell_1d_dirichlet_penalization
+      test_maxwell_1d_dirichlet_penalization_par
+      test_maxwell_1d_dirichlet_strong
+      test_maxwell_1d_dirichlet_strong_par
+      test_maxwell_1d_periodic
+      test_maxwell_1d_periodic_mult
+      test_maxwell_1d_periodic_par
+      update_plot
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_feec_2d.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_feec_2d.rst
new file mode 100644
index 0000000..6ccc4b6
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_feec_2d.rst
@@ -0,0 +1,46 @@
+psydac.api.tests.test\_api\_feec\_2d
+====================================
+
+.. automodule:: psydac.api.tests.test_api_feec_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      add_colorbar
+      plot_field_and_error
+      run_maxwell_2d_TE
+      test_maxwell_2d_dirichlet
+      test_maxwell_2d_dirichlet_par
+      test_maxwell_2d_dirichlet_spline_mapping
+      test_maxwell_2d_multiplicity
+      test_maxwell_2d_periodic
+      test_maxwell_2d_periodic_multiplicity
+      test_maxwell_2d_periodic_multiplicity_equal_deg
+      test_maxwell_2d_periodic_par
+      update_plot
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      CavitySolution
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_feec_3d.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_feec_3d.rst
new file mode 100644
index 0000000..acd2960
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_feec_3d.rst
@@ -0,0 +1,38 @@
+psydac.api.tests.test\_api\_feec\_3d
+====================================
+
+.. automodule:: psydac.api.tests.test_api_feec_3d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      evaluation_all_times
+      run_maxwell_3d_scipy
+      run_maxwell_3d_stencil
+      splitting_integrator_scipy
+      splitting_integrator_stencil
+      teardown_function
+      teardown_module
+      test_maxwell_3d_1
+      test_maxwell_3d_2
+      test_maxwell_3d_2_mult
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_glt_2d_scalar.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_glt_2d_scalar.rst
new file mode 100644
index 0000000..3c6929f
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_glt_2d_scalar.rst
@@ -0,0 +1,36 @@
+psydac.api.tests.test\_api\_glt\_2d\_scalar
+===========================================
+
+.. automodule:: psydac.api.tests.test_api_glt_2d_scalar
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_field_2d_dir
+      run_poisson_2d_dir
+      run_variable_coeff_2d_dir
+      teardown_function
+      teardown_module
+      test_api_glt_field_2d_dir_1
+      test_api_glt_poisson_2d_dir_1
+      test_api_glt_variable_coeff_2d_dir_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_glt_2d_scalar_mapping.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_glt_2d_scalar_mapping.rst
new file mode 100644
index 0000000..d0aa6bf
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_glt_2d_scalar_mapping.rst
@@ -0,0 +1,34 @@
+psydac.api.tests.test\_api\_glt\_2d\_scalar\_mapping
+====================================================
+
+.. automodule:: psydac.api.tests.test_api_glt_2d_scalar_mapping
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_poisson_2d_dir
+      teardown_function
+      teardown_module
+      test_api_glt_poisson_2d_dir_collela
+      test_api_glt_poisson_2d_dir_identity
+      test_api_glt_poisson_2d_dir_quarter_annulus
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_api_glt_2d_vector.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_api_glt_2d_vector.rst
new file mode 100644
index 0000000..8aa8495
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_api_glt_2d_vector.rst
@@ -0,0 +1,32 @@
+psydac.api.tests.test\_api\_glt\_2d\_vector
+===========================================
+
+.. automodule:: psydac.api.tests.test_api_glt_2d_vector
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_vector_poisson_2d_dir
+      teardown_function
+      teardown_module
+      test_api_glt_vector_poisson_2d_dir_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_assembly.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_assembly.rst
new file mode 100644
index 0000000..a4a13ac
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_assembly.rst
@@ -0,0 +1,40 @@
+psydac.api.tests.test\_assembly
+===============================
+
+.. automodule:: psydac.api.tests.test_assembly
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      backend
+      dtype
+      test_Norm_complex
+      test_assemble_complex_parallel
+      test_assembly_no_synchr_args
+      test_bilinearForm_complex
+      test_field_and_constant
+      test_linearForm_complex
+      test_math_imports
+      test_multiple_fields
+      test_non_symmetric_BilinearForm
+      test_non_symmetric_different_space_BilinearForm
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_build_derham_mapping.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_build_derham_mapping.rst
new file mode 100644
index 0000000..44fcf4a
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_build_derham_mapping.rst
@@ -0,0 +1,31 @@
+psydac.api.tests.test\_build\_derham\_mapping
+=============================================
+
+.. automodule:: psydac.api.tests.test_build_derham_mapping
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_build_derham_spline_mapping_id_1d
+      test_build_derham_spline_mapping_id_2d
+      test_build_derham_spline_mapping_id_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_equation.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_equation.rst
new file mode 100644
index 0000000..d8dcc5f
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_equation.rst
@@ -0,0 +1,30 @@
+psydac.api.tests.test\_equation
+===============================
+
+.. automodule:: psydac.api.tests.test_equation
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      backend
+      test_field_and_constant
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_postprocessing.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_postprocessing.rst
new file mode 100644
index 0000000..92fe415
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_postprocessing.rst
@@ -0,0 +1,41 @@
+psydac.api.tests.test\_postprocessing
+=====================================
+
+.. automodule:: psydac.api.tests.test_postprocessing
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      build_2_cubes
+      build_2_mapped_squares
+      build_2_squares
+      test_add_spaces
+      test_export_fields_parallel
+      test_export_fields_serial
+      test_incorrect_arg_export_to_vtk
+      test_parallel_export_discrete_domain
+      test_parallel_export_topological_domain
+      test_reconstruct_DerhamSequence_discrete_domain
+      test_reconstruct_DerhamSequence_topological_domain
+      test_reconstruct_multipatch
+      test_reconstruct_spaces_topological_domain
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.test_quadorder.rst b/_sources/documentation/_autosummary/psydac.api.tests.test_quadorder.rst
new file mode 100644
index 0000000..9f0b2ab
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.test_quadorder.rst
@@ -0,0 +1,29 @@
+psydac.api.tests.test\_quadorder
+================================
+
+.. automodule:: psydac.api.tests.test_quadorder
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_custom_nquads
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.tests.utils.rst b/_sources/documentation/_autosummary/psydac.api.tests.utils.rst
new file mode 100644
index 0000000..5b5b1a8
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.tests.utils.rst
@@ -0,0 +1,29 @@
+psydac.api.tests.utils
+======================
+
+.. automodule:: psydac.api.tests.utils
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      assert_identical_coo
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.api.utilities.rst b/_sources/documentation/_autosummary/psydac.api.utilities.rst
new file mode 100644
index 0000000..f3bb455
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.api.utilities.rst
@@ -0,0 +1,33 @@
+psydac.api.utilities
+====================
+
+.. automodule:: psydac.api.utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      flatten
+      mkdir_p
+      random_string
+      touch_init_file
+      write_code
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.cad.cad.rst b/_sources/documentation/_autosummary/psydac.cad.cad.rst
new file mode 100644
index 0000000..c0c12ae
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.cad.cad.rst
@@ -0,0 +1,32 @@
+psydac.cad.cad
+==============
+
+.. automodule:: psydac.cad.cad
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      elevate
+      random_string
+      refine
+      translate
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.cad.gallery.rst b/_sources/documentation/_autosummary/psydac.cad.gallery.rst
new file mode 100644
index 0000000..cf49a7d
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.cad.gallery.rst
@@ -0,0 +1,31 @@
+psydac.cad.gallery
+==================
+
+.. automodule:: psydac.cad.gallery
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      annulus
+      circle
+      quart_circle
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.cad.geometry.rst b/_sources/documentation/_autosummary/psydac.cad.geometry.rst
new file mode 100644
index 0000000..e7a7da8
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.cad.geometry.rst
@@ -0,0 +1,38 @@
+psydac.cad.geometry
+===================
+
+.. automodule:: psydac.cad.geometry
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      export_nurbs_to_hdf5
+      import_geopdes_to_nurbs
+      refine_knots
+      refine_nurbs
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Geometry
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.cad.multipatch.rst b/_sources/documentation/_autosummary/psydac.cad.multipatch.rst
new file mode 100644
index 0000000..283b511
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.cad.multipatch.rst
@@ -0,0 +1,29 @@
+psydac.cad.multipatch
+=====================
+
+.. automodule:: psydac.cad.multipatch
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      export_multipatch_nurbs_to_hdf5
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.cad.rst b/_sources/documentation/_autosummary/psydac.cad.rst
new file mode 100644
index 0000000..36a0e68
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.cad.rst
@@ -0,0 +1,36 @@
+psydac.cad
+==========
+
+.. automodule:: psydac.cad
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.cad.cad
+   psydac.cad.gallery
+   psydac.cad.geometry
+   psydac.cad.multipatch
+   psydac.cad.tests
+   psydac.cad.utils
+
diff --git a/_sources/documentation/_autosummary/psydac.cad.tests.rst b/_sources/documentation/_autosummary/psydac.cad.tests.rst
new file mode 100644
index 0000000..a750fcd
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.cad.tests.rst
@@ -0,0 +1,31 @@
+psydac.cad.tests
+================
+
+.. automodule:: psydac.cad.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.cad.tests.test_geometry
+
diff --git a/_sources/documentation/_autosummary/psydac.cad.tests.test_geometry.rst b/_sources/documentation/_autosummary/psydac.cad.tests.test_geometry.rst
new file mode 100644
index 0000000..9f9f7b7
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.cad.tests.test_geometry.rst
@@ -0,0 +1,37 @@
+psydac.cad.tests.test\_geometry
+===============================
+
+.. automodule:: psydac.cad.tests.test_geometry
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_export_nurbs_to_hdf5
+      test_geometry_1
+      test_geometry_2d_1
+      test_geometry_2d_2
+      test_geometry_2d_3
+      test_geometry_2d_4
+      test_import_geopdes_to_nurbs
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.cad.utils.rst b/_sources/documentation/_autosummary/psydac.cad.utils.rst
new file mode 100644
index 0000000..df344e5
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.cad.utils.rst
@@ -0,0 +1,29 @@
+psydac.cad.utils
+================
+
+.. automodule:: psydac.cad.utils
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      plot_mapping
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.cmd.mesh.rst b/_sources/documentation/_autosummary/psydac.cmd.mesh.rst
new file mode 100644
index 0000000..8dab4ad
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.cmd.mesh.rst
@@ -0,0 +1,31 @@
+psydac.cmd.mesh
+===============
+
+.. automodule:: psydac.cmd.mesh
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      export_analytical_mapping
+      export_caid_geometry
+      main
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.cmd.rst b/_sources/documentation/_autosummary/psydac.cmd.rst
new file mode 100644
index 0000000..5b4ad8c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.cmd.rst
@@ -0,0 +1,31 @@
+psydac.cmd
+==========
+
+.. automodule:: psydac.cmd
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.cmd.mesh
+
diff --git a/_sources/documentation/_autosummary/psydac.core.bsplines.rst b/_sources/documentation/_autosummary/psydac.core.bsplines.rst
new file mode 100644
index 0000000..34dd342
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.core.bsplines.rst
@@ -0,0 +1,47 @@
+psydac.core.bsplines
+====================
+
+.. automodule:: psydac.core.bsplines
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      basis_ders_on_irregular_grid
+      basis_ders_on_quad_grid
+      basis_funs
+      basis_funs_1st_der
+      basis_funs_all_ders
+      basis_funs_array
+      basis_integrals
+      breakpoints
+      cell_index
+      collocation_matrix
+      elements_spans
+      elevate_knots
+      find_span
+      find_spans
+      greville
+      histopolation_matrix
+      hrefinement_matrix
+      make_knots
+      quadrature_grid
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.core.bsplines_kernels.rst b/_sources/documentation/_autosummary/psydac.core.bsplines_kernels.rst
new file mode 100644
index 0000000..f4b770d
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.core.bsplines_kernels.rst
@@ -0,0 +1,47 @@
+psydac.core.bsplines\_kernels
+=============================
+
+.. automodule:: psydac.core.bsplines_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      basis_ders_on_irregular_grid_p
+      basis_ders_on_quad_grid_p
+      basis_funs_1st_der_p
+      basis_funs_all_ders_p
+      basis_funs_array_p
+      basis_funs_p
+      basis_integrals_p
+      breakpoints_p
+      cell_index_p
+      collocation_matrix_p
+      elements_spans_p
+      elevate_knots_p
+      find_span_p
+      find_spans_p
+      greville_p
+      histopolation_matrix_p
+      make_knots_p
+      merge_sort
+      quadrature_grid_p
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.core.field_evaluation_kernels.rst b/_sources/documentation/_autosummary/psydac.core.field_evaluation_kernels.rst
new file mode 100644
index 0000000..a79c0b4
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.core.field_evaluation_kernels.rst
@@ -0,0 +1,66 @@
+psydac.core.field\_evaluation\_kernels
+======================================
+
+.. automodule:: psydac.core.field_evaluation_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      eval_fields_2d_irregular_no_weights
+      eval_fields_2d_irregular_weighted
+      eval_fields_2d_no_weights
+      eval_fields_2d_weighted
+      eval_fields_3d_irregular_no_weights
+      eval_fields_3d_irregular_weighted
+      eval_fields_3d_no_weights
+      eval_fields_3d_weighted
+      eval_jac_det_2d
+      eval_jac_det_2d_weights
+      eval_jac_det_3d
+      eval_jac_det_3d_weights
+      eval_jac_det_irregular_2d
+      eval_jac_det_irregular_2d_weights
+      eval_jac_det_irregular_3d
+      eval_jac_det_irregular_3d_weights
+      eval_jacobians_2d
+      eval_jacobians_2d_weights
+      eval_jacobians_3d
+      eval_jacobians_3d_weights
+      eval_jacobians_inv_2d
+      eval_jacobians_inv_2d_weights
+      eval_jacobians_inv_3d
+      eval_jacobians_inv_3d_weights
+      eval_jacobians_inv_irregular_2d
+      eval_jacobians_inv_irregular_2d_weights
+      eval_jacobians_inv_irregular_3d
+      eval_jacobians_inv_irregular_3d_weights
+      eval_jacobians_irregular_2d
+      eval_jacobians_irregular_2d_weights
+      eval_jacobians_irregular_3d
+      eval_jacobians_irregular_3d_weights
+      pushforward_2d_hcurl
+      pushforward_2d_hdiv
+      pushforward_2d_l2
+      pushforward_3d_hcurl
+      pushforward_3d_hdiv
+      pushforward_3d_l2
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.core.rst b/_sources/documentation/_autosummary/psydac.core.rst
new file mode 100644
index 0000000..e73e403
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.core.rst
@@ -0,0 +1,34 @@
+psydac.core
+===========
+
+.. automodule:: psydac.core
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.core.bsplines
+   psydac.core.bsplines_kernels
+   psydac.core.field_evaluation_kernels
+   psydac.core.tests
+
diff --git a/_sources/documentation/_autosummary/psydac.core.tests.rst b/_sources/documentation/_autosummary/psydac.core.tests.rst
new file mode 100644
index 0000000..57dc238
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.core.tests.rst
@@ -0,0 +1,33 @@
+psydac.core.tests
+=================
+
+.. automodule:: psydac.core.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.core.tests.test_bsplines
+   psydac.core.tests.test_bsplines_pyccel
+   psydac.core.tests.test_field_evaluation_kernel
+
diff --git a/_sources/documentation/_autosummary/psydac.core.tests.test_bsplines.rst b/_sources/documentation/_autosummary/psydac.core.tests.test_bsplines.rst
new file mode 100644
index 0000000..fa77e4f
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.core.tests.test_bsplines.rst
@@ -0,0 +1,35 @@
+psydac.core.tests.test\_bsplines
+================================
+
+.. automodule:: psydac.core.tests.test_bsplines
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_basis_funs
+      test_basis_funs_1st_der
+      test_basis_funs_all_ders
+      test_cell_index
+      test_collocation_matrix
+      test_find_span
+      test_histopolation_matrix
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.core.tests.test_bsplines_pyccel.rst b/_sources/documentation/_autosummary/psydac.core.tests.test_bsplines_pyccel.rst
new file mode 100644
index 0000000..3be3ff5
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.core.tests.test_bsplines_pyccel.rst
@@ -0,0 +1,56 @@
+psydac.core.tests.test\_bsplines\_pyccel
+========================================
+
+.. automodule:: psydac.core.tests.test_bsplines_pyccel
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      basis_ders_on_quad_grid_true
+      basis_funs_1st_der_true
+      basis_funs_all_ders_true
+      basis_funs_true
+      basis_integrals_true
+      breakpoints_true
+      collocation_matrix_true
+      elements_spans_true
+      elevate_knots_true
+      find_span_true
+      greville_true
+      histopolation_matrix_true
+      make_knots_true
+      quadrature_grid_true
+      test_basis_ders_on_quad_grid
+      test_basis_funs
+      test_basis_funs_1st_der
+      test_basis_funs_all_ders
+      test_basis_integrals
+      test_breakpoints
+      test_collocation_matrix
+      test_elements_spans
+      test_elevate_knots
+      test_find_span
+      test_greville
+      test_histopolation_matrix
+      test_make_knots
+      test_quadrature_grid
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.core.tests.test_field_evaluation_kernel.rst b/_sources/documentation/_autosummary/psydac.core.tests.test_field_evaluation_kernel.rst
new file mode 100644
index 0000000..5c2eef7
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.core.tests.test_field_evaluation_kernel.rst
@@ -0,0 +1,35 @@
+psydac.core.tests.test\_field\_evaluation\_kernel
+=================================================
+
+.. automodule:: psydac.core.tests.test_field_evaluation_kernel
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_irregular_evaluations
+      test_irregular_jacobians
+      test_pushforwards_hcurl
+      test_pushforwards_hdiv
+      test_pushforwards_l2
+      test_regular_evaluations
+      test_regular_jacobians
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.basic.rst b/_sources/documentation/_autosummary/psydac.ddm.basic.rst
new file mode 100644
index 0000000..dcd7a52
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.basic.rst
@@ -0,0 +1,29 @@
+psydac.ddm.basic
+================
+
+.. automodule:: psydac.ddm.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      CartDataExchanger
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.blocking_data_exchanger.rst b/_sources/documentation/_autosummary/psydac.ddm.blocking_data_exchanger.rst
new file mode 100644
index 0000000..a673168
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.blocking_data_exchanger.rst
@@ -0,0 +1,29 @@
+psydac.ddm.blocking\_data\_exchanger
+====================================
+
+.. automodule:: psydac.ddm.blocking_data_exchanger
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BlockingCartDataExchanger
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.cart.rst b/_sources/documentation/_autosummary/psydac.ddm.cart.rst
new file mode 100644
index 0000000..59e7aac
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.cart.rst
@@ -0,0 +1,39 @@
+psydac.ddm.cart
+===============
+
+.. automodule:: psydac.ddm.cart
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      create_interfaces_cart
+      find_mpi_type
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      CartDecomposition
+      DomainDecomposition
+      InterfaceCartDecomposition
+      MultiPatchDomainDecomposition
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.interface_data_exchanger.rst b/_sources/documentation/_autosummary/psydac.ddm.interface_data_exchanger.rst
new file mode 100644
index 0000000..7a877b5
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.interface_data_exchanger.rst
@@ -0,0 +1,29 @@
+psydac.ddm.interface\_data\_exchanger
+=====================================
+
+.. automodule:: psydac.ddm.interface_data_exchanger
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      InterfaceCartDataExchanger
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.nonblocking_data_exchanger.rst b/_sources/documentation/_autosummary/psydac.ddm.nonblocking_data_exchanger.rst
new file mode 100644
index 0000000..889445a
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.nonblocking_data_exchanger.rst
@@ -0,0 +1,29 @@
+psydac.ddm.nonblocking\_data\_exchanger
+=======================================
+
+.. automodule:: psydac.ddm.nonblocking_data_exchanger
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      NonBlockingCartDataExchanger
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.partition.rst b/_sources/documentation/_autosummary/psydac.ddm.partition.rst
new file mode 100644
index 0000000..597f2ce
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.partition.rst
@@ -0,0 +1,32 @@
+psydac.ddm.partition
+====================
+
+.. automodule:: psydac.ddm.partition
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_dims
+      compute_dims_general
+      compute_dims_uniform
+      partition_procs_per_patch
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.petsc.rst b/_sources/documentation/_autosummary/psydac.ddm.petsc.rst
new file mode 100644
index 0000000..860fc86
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.petsc.rst
@@ -0,0 +1,29 @@
+psydac.ddm.petsc
+================
+
+.. automodule:: psydac.ddm.petsc
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      PetscCart
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.rst b/_sources/documentation/_autosummary/psydac.ddm.rst
new file mode 100644
index 0000000..8f40258
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.rst
@@ -0,0 +1,39 @@
+psydac.ddm
+==========
+
+.. automodule:: psydac.ddm
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.ddm.basic
+   psydac.ddm.blocking_data_exchanger
+   psydac.ddm.cart
+   psydac.ddm.interface_data_exchanger
+   psydac.ddm.nonblocking_data_exchanger
+   psydac.ddm.partition
+   psydac.ddm.petsc
+   psydac.ddm.tests
+   psydac.ddm.utilities
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.tests.rst b/_sources/documentation/_autosummary/psydac.ddm.tests.rst
new file mode 100644
index 0000000..db8054a
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.tests.rst
@@ -0,0 +1,35 @@
+psydac.ddm.tests
+================
+
+.. automodule:: psydac.ddm.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.ddm.tests.test_cart_1d
+   psydac.ddm.tests.test_cart_2d
+   psydac.ddm.tests.test_cart_3d
+   psydac.ddm.tests.test_multicart_2d
+   psydac.ddm.tests.test_partition
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.tests.test_cart_1d.rst b/_sources/documentation/_autosummary/psydac.ddm.tests.test_cart_1d.rst
new file mode 100644
index 0000000..fe6355e
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.tests.test_cart_1d.rst
@@ -0,0 +1,30 @@
+psydac.ddm.tests.test\_cart\_1d
+===============================
+
+.. automodule:: psydac.ddm.tests.test_cart_1d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_cart_1d
+      test_cart_1d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.tests.test_cart_2d.rst b/_sources/documentation/_autosummary/psydac.ddm.tests.test_cart_2d.rst
new file mode 100644
index 0000000..fd91d53
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.tests.test_cart_2d.rst
@@ -0,0 +1,32 @@
+psydac.ddm.tests.test\_cart\_2d
+===============================
+
+.. automodule:: psydac.ddm.tests.test_cart_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_cart_2d
+      test_cart_2d
+      test_cart_2d_reverse_axis_0
+      test_cart_2d_reverse_axis_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.tests.test_cart_3d.rst b/_sources/documentation/_autosummary/psydac.ddm.tests.test_cart_3d.rst
new file mode 100644
index 0000000..8714ee9
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.tests.test_cart_3d.rst
@@ -0,0 +1,30 @@
+psydac.ddm.tests.test\_cart\_3d
+===============================
+
+.. automodule:: psydac.ddm.tests.test_cart_3d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_cart_3d
+      test_cart_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.tests.test_multicart_2d.rst b/_sources/documentation/_autosummary/psydac.ddm.tests.test_multicart_2d.rst
new file mode 100644
index 0000000..b4ae1e5
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.tests.test_multicart_2d.rst
@@ -0,0 +1,31 @@
+psydac.ddm.tests.test\_multicart\_2d
+====================================
+
+.. automodule:: psydac.ddm.tests.test_multicart_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_minus_starts_ends
+      get_plus_starts_ends
+      run_carts_2d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.tests.test_partition.rst b/_sources/documentation/_autosummary/psydac.ddm.tests.test_partition.rst
new file mode 100644
index 0000000..ae7b061
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.tests.test_partition.rst
@@ -0,0 +1,33 @@
+psydac.ddm.tests.test\_partition
+================================
+
+.. automodule:: psydac.ddm.tests.test_partition
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_partition_1d_general
+      test_partition_1d_uniform
+      test_partition_2d_dims_mask
+      test_partition_3d
+      test_partition_3d_dims_mask
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.ddm.utilities.rst b/_sources/documentation/_autosummary/psydac.ddm.utilities.rst
new file mode 100644
index 0000000..730edcb
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.ddm.utilities.rst
@@ -0,0 +1,29 @@
+psydac.ddm.utilities
+====================
+
+.. automodule:: psydac.ddm.utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_data_exchanger
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.derivatives.rst b/_sources/documentation/_autosummary/psydac.feec.derivatives.rst
new file mode 100644
index 0000000..d67b661
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.derivatives.rst
@@ -0,0 +1,44 @@
+psydac.feec.derivatives
+=======================
+
+.. automodule:: psydac.feec.derivatives
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      block_tostencil
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Curl_3D
+      Derivative_1D
+      DiffOperator
+      DirectionalDerivativeOperator
+      Divergence_2D
+      Divergence_3D
+      Gradient_2D
+      Gradient_3D
+      ScalarCurl_2D
+      VectorCurl_2D
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.global_projectors.rst b/_sources/documentation/_autosummary/psydac.feec.global_projectors.rst
new file mode 100644
index 0000000..1c3aa14
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.global_projectors.rst
@@ -0,0 +1,51 @@
+psydac.feec.global\_projectors
+==============================
+
+.. automodule:: psydac.feec.global_projectors
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      evaluate_dofs_1d_0form
+      evaluate_dofs_1d_1form
+      evaluate_dofs_2d_0form
+      evaluate_dofs_2d_1form_hcurl
+      evaluate_dofs_2d_1form_hdiv
+      evaluate_dofs_2d_2form
+      evaluate_dofs_2d_vec
+      evaluate_dofs_3d_0form
+      evaluate_dofs_3d_1form
+      evaluate_dofs_3d_2form
+      evaluate_dofs_3d_3form
+      evaluate_dofs_3d_vec
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      GlobalProjector
+      Projector_H1
+      Projector_H1vec
+      Projector_Hcurl
+      Projector_Hdiv
+      Projector_L2
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.api.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.api.rst
new file mode 100644
index 0000000..4dbdfb9
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.api.rst
@@ -0,0 +1,36 @@
+psydac.feec.multipatch.api
+==========================
+
+.. automodule:: psydac.feec.multipatch.api
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      discretize
+      discretize_derham_multipatch
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteDerhamMultipatch
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.h1_source_pbms_conga_2d.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.h1_source_pbms_conga_2d.rst
new file mode 100644
index 0000000..d675b94
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.h1_source_pbms_conga_2d.rst
@@ -0,0 +1,29 @@
+psydac.feec.multipatch.examples.h1\_source\_pbms\_conga\_2d
+===========================================================
+
+.. automodule:: psydac.feec.multipatch.examples.h1_source_pbms_conga_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      solve_h1_source_pbm
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d.rst
new file mode 100644
index 0000000..eb83df6
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d.rst
@@ -0,0 +1,30 @@
+psydac.feec.multipatch.examples.hcurl\_eigen\_pbms\_conga\_2d
+=============================================================
+
+.. automodule:: psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_eigenvalues
+      hcurl_solve_eigen_pbm
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d.rst
new file mode 100644
index 0000000..d62bfb2
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d.rst
@@ -0,0 +1,29 @@
+psydac.feec.multipatch.examples.hcurl\_source\_pbms\_conga\_2d
+==============================================================
+
+.. automodule:: psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      solve_hcurl_source_pbm
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d.rst
new file mode 100644
index 0000000..1a520b2
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d.rst
@@ -0,0 +1,29 @@
+psydac.feec.multipatch.examples.mixed\_source\_pbms\_conga\_2d
+==============================================================
+
+.. automodule:: psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      solve_magnetostatic_pbm
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.ppc_test_cases.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.ppc_test_cases.rst
new file mode 100644
index 0000000..963a575
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.ppc_test_cases.rst
@@ -0,0 +1,31 @@
+psydac.feec.multipatch.examples.ppc\_test\_cases
+================================================
+
+.. automodule:: psydac.feec.multipatch.examples.ppc_test_cases
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_sol_ref_V1h
+      get_source_and_sol_for_magnetostatic_pbm
+      get_source_and_solution
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.rst
new file mode 100644
index 0000000..159f763
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.examples.rst
@@ -0,0 +1,35 @@
+psydac.feec.multipatch.examples
+===============================
+
+.. automodule:: psydac.feec.multipatch.examples
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.feec.multipatch.examples.h1_source_pbms_conga_2d
+   psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d
+   psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d
+   psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d
+   psydac.feec.multipatch.examples.ppc_test_cases
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.fem_linear_operators.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.fem_linear_operators.rst
new file mode 100644
index 0000000..8772b14
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.fem_linear_operators.rst
@@ -0,0 +1,33 @@
+psydac.feec.multipatch.fem\_linear\_operators
+=============================================
+
+.. automodule:: psydac.feec.multipatch.fem_linear_operators
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      ComposedLinearOperator
+      FemLinearOperator
+      IdLinearOperator
+      MultLinearOperator
+      SumLinearOperator
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.multipatch_domain_utilities.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.multipatch_domain_utilities.rst
new file mode 100644
index 0000000..1e9ba3b
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.multipatch_domain_utilities.rst
@@ -0,0 +1,39 @@
+psydac.feec.multipatch.multipatch\_domain\_utilities
+====================================================
+
+.. automodule:: psydac.feec.multipatch.multipatch_domain_utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      build_multipatch_domain
+      create_domain
+      flip_axis
+      get_2D_rotation_mapping
+      get_ref_eigenvalues
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      TransposedPolarMapping
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.operators.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.operators.rst
new file mode 100644
index 0000000..1b2766d
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.operators.rst
@@ -0,0 +1,50 @@
+psydac.feec.multipatch.operators
+================================
+
+.. automodule:: psydac.feec.multipatch.operators
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      allocate_interface_matrix
+      get_K0_and_K0_inv
+      get_K1_and_K1_inv
+      get_interface_from_corners
+      get_patch_index_from_face
+      get_row_col_index
+      ortho_proj_Hcurl
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BrokenGradient_2D
+      BrokenScalarCurl_2D
+      BrokenTransposedGradient_2D
+      BrokenTransposedScalarCurl_2D
+      ConformingProjection_V0
+      ConformingProjection_V1
+      HodgeOperator
+      Multipatch_Projector_H1
+      Multipatch_Projector_Hcurl
+      Multipatch_Projector_L2
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.plotting_utilities.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.plotting_utilities.rst
new file mode 100644
index 0000000..2286fcf
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.plotting_utilities.rst
@@ -0,0 +1,37 @@
+psydac.feec.multipatch.plotting\_utilities
+==========================================
+
+.. automodule:: psydac.feec.multipatch.plotting_utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_diag_grid
+      get_grid_quad_weights
+      get_grid_vals
+      get_patch_knots_gridlines
+      get_plotting_grid
+      is_vector_valued
+      my_small_plot
+      my_small_streamplot
+      plot_field
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.rst
new file mode 100644
index 0000000..f9fd14e
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.rst
@@ -0,0 +1,38 @@
+psydac.feec.multipatch
+======================
+
+.. automodule:: psydac.feec.multipatch
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.feec.multipatch.api
+   psydac.feec.multipatch.examples
+   psydac.feec.multipatch.fem_linear_operators
+   psydac.feec.multipatch.multipatch_domain_utilities
+   psydac.feec.multipatch.operators
+   psydac.feec.multipatch.plotting_utilities
+   psydac.feec.multipatch.utilities
+   psydac.feec.multipatch.utils_conga_2d
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.utilities.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.utilities.rst
new file mode 100644
index 0000000..2401a19
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.utilities.rst
@@ -0,0 +1,35 @@
+psydac.feec.multipatch.utilities
+================================
+
+.. automodule:: psydac.feec.multipatch.utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      error_fn
+      get_fem_name
+      get_load_dir
+      get_method_name
+      sol_ref_fn
+      source_name
+      time_count
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.multipatch.utils_conga_2d.rst b/_sources/documentation/_autosummary/psydac.feec.multipatch.utils_conga_2d.rst
new file mode 100644
index 0000000..74499f6
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.multipatch.utils_conga_2d.rst
@@ -0,0 +1,23 @@
+psydac.feec.multipatch.utils\_conga\_2d
+=======================================
+
+.. automodule:: psydac.feec.multipatch.utils_conga_2d
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.pull_push.rst b/_sources/documentation/_autosummary/psydac.feec.pull_push.rst
new file mode 100644
index 0000000..bbb3922
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.pull_push.rst
@@ -0,0 +1,50 @@
+psydac.feec.pull\_push
+======================
+
+.. automodule:: psydac.feec.pull_push
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      pull_1d_h1
+      pull_1d_l2
+      pull_2d_h1
+      pull_2d_h1vec
+      pull_2d_hcurl
+      pull_2d_hdiv
+      pull_2d_l2
+      pull_3d_h1
+      pull_3d_h1vec
+      pull_3d_hcurl
+      pull_3d_hdiv
+      pull_3d_l2
+      push_1d_h1
+      push_1d_l2
+      push_2d_h1
+      push_2d_hcurl
+      push_2d_hdiv
+      push_2d_l2
+      push_3d_h1
+      push_3d_hcurl
+      push_3d_hdiv
+      push_3d_l2
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.pushforward.rst b/_sources/documentation/_autosummary/psydac.feec.pushforward.rst
new file mode 100644
index 0000000..e465165
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.pushforward.rst
@@ -0,0 +1,29 @@
+psydac.feec.pushforward
+=======================
+
+.. automodule:: psydac.feec.pushforward
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Pushforward
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.rst b/_sources/documentation/_autosummary/psydac.feec.rst
new file mode 100644
index 0000000..7bab4c8
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.rst
@@ -0,0 +1,36 @@
+psydac.feec
+===========
+
+.. automodule:: psydac.feec
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.feec.derivatives
+   psydac.feec.global_projectors
+   psydac.feec.multipatch
+   psydac.feec.pull_push
+   psydac.feec.pushforward
+   psydac.feec.tests
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.tests.rst b/_sources/documentation/_autosummary/psydac.feec.tests.rst
new file mode 100644
index 0000000..16382ea
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.tests.rst
@@ -0,0 +1,36 @@
+psydac.feec.tests
+=================
+
+.. automodule:: psydac.feec.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.feec.tests.test_axis_projection
+   psydac.feec.tests.test_commuting_projections
+   psydac.feec.tests.test_commuting_projections_dual
+   psydac.feec.tests.test_differentiation_matrices
+   psydac.feec.tests.test_global_projectors
+   psydac.feec.tests.test_projections_parallel
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.tests.test_axis_projection.rst b/_sources/documentation/_autosummary/psydac.feec.tests.test_axis_projection.rst
new file mode 100644
index 0000000..99dadb3
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.tests.test_axis_projection.rst
@@ -0,0 +1,29 @@
+psydac.feec.tests.test\_axis\_projection
+========================================
+
+.. automodule:: psydac.feec.tests.test_axis_projection
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_axis_projection
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.tests.test_commuting_projections.rst b/_sources/documentation/_autosummary/psydac.feec.tests.test_commuting_projections.rst
new file mode 100644
index 0000000..2e9094f
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.tests.test_commuting_projections.rst
@@ -0,0 +1,36 @@
+psydac.feec.tests.test\_commuting\_projections
+==============================================
+
+.. automodule:: psydac.feec.tests.test_commuting_projections
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_1d_commuting_pro_1
+      test_2d_commuting_pro_1
+      test_2d_commuting_pro_2
+      test_2d_commuting_pro_3
+      test_2d_commuting_pro_4
+      test_3d_commuting_pro_1
+      test_3d_commuting_pro_2
+      test_3d_commuting_pro_3
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.tests.test_commuting_projections_dual.rst b/_sources/documentation/_autosummary/psydac.feec.tests.test_commuting_projections_dual.rst
new file mode 100644
index 0000000..6030036
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.tests.test_commuting_projections_dual.rst
@@ -0,0 +1,31 @@
+psydac.feec.tests.test\_commuting\_projections\_dual
+====================================================
+
+.. automodule:: psydac.feec.tests.test_commuting_projections_dual
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_transpose_curl_3d
+      test_transpose_div_3d
+      test_transpose_grad_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.tests.test_differentiation_matrices.rst b/_sources/documentation/_autosummary/psydac.feec.tests.test_differentiation_matrices.rst
new file mode 100644
index 0000000..d1f40d2
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.tests.test_differentiation_matrices.rst
@@ -0,0 +1,48 @@
+psydac.feec.tests.test\_differentiation\_matrices
+=================================================
+
+.. automodule:: psydac.feec.tests.test_differentiation_matrices
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compare_diff_operators_by_matrixassembly
+      run_directional_derivative_operator
+      test_Curl_3D
+      test_Derivative_1D
+      test_Divergence_2D
+      test_Divergence_3D
+      test_Gradient_2D
+      test_Gradient_3D
+      test_ScalarCurl_2D
+      test_VectorCurl_2D
+      test_directional_derivative_operator_1d_par
+      test_directional_derivative_operator_1d_ser
+      test_directional_derivative_operator_2d_par
+      test_directional_derivative_operator_2d_ser
+      test_directional_derivative_operator_3d_par
+      test_directional_derivative_operator_3d_ser
+      test_directional_derivative_operator_interface
+      test_directional_derivative_operator_invalid_wrongsized1
+      test_directional_derivative_operator_invalid_wrongspace2
+      test_directional_derivative_operator_transposition_correctness
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.tests.test_global_projectors.rst b/_sources/documentation/_autosummary/psydac.feec.tests.test_global_projectors.rst
new file mode 100644
index 0000000..9a0541c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.tests.test_global_projectors.rst
@@ -0,0 +1,34 @@
+psydac.feec.tests.test\_global\_projectors
+==========================================
+
+.. automodule:: psydac.feec.tests.test_global_projectors
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_H1_projector_1d
+      test_L2_projector_1d
+      test_derham_projector_2d_hcurl
+      test_derham_projector_2d_hdiv
+      test_derham_projector_2d_hdiv_2
+      test_derham_projector_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.feec.tests.test_projections_parallel.rst b/_sources/documentation/_autosummary/psydac.feec.tests.test_projections_parallel.rst
new file mode 100644
index 0000000..d2726f6
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.feec.tests.test_projections_parallel.rst
@@ -0,0 +1,32 @@
+psydac.feec.tests.test\_projections\_parallel
+=============================================
+
+.. automodule:: psydac.feec.tests.test_projections_parallel
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      run_projection_comparison
+      test_projection_parallel_1d
+      test_projection_parallel_2d
+      test_projection_parallel_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.basic.rst b/_sources/documentation/_autosummary/psydac.fem.basic.rst
new file mode 100644
index 0000000..b2cdf20
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.basic.rst
@@ -0,0 +1,30 @@
+psydac.fem.basic
+================
+
+.. automodule:: psydac.fem.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      FemField
+      FemSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.context.rst b/_sources/documentation/_autosummary/psydac.fem.context.rst
new file mode 100644
index 0000000..12ec5c3
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.context.rst
@@ -0,0 +1,29 @@
+psydac.fem.context
+==================
+
+.. automodule:: psydac.fem.context
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      fem_context
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.grid.rst b/_sources/documentation/_autosummary/psydac.fem.grid.rst
new file mode 100644
index 0000000..0522e67
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.grid.rst
@@ -0,0 +1,29 @@
+psydac.fem.grid
+===============
+
+.. automodule:: psydac.fem.grid
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      FemAssemblyGrid
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.partitioning.rst b/_sources/documentation/_autosummary/psydac.fem.partitioning.rst
new file mode 100644
index 0000000..6690ee7
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.partitioning.rst
@@ -0,0 +1,35 @@
+psydac.fem.partitioning
+=======================
+
+.. automodule:: psydac.fem.partitioning
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      construct_connectivity
+      construct_interface_spaces
+      construct_reduced_interface_spaces
+      create_cart
+      get_minus_starts_ends
+      get_plus_starts_ends
+      partition_coefficients
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.projectors.rst b/_sources/documentation/_autosummary/psydac.fem.projectors.rst
new file mode 100644
index 0000000..6113083
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.projectors.rst
@@ -0,0 +1,30 @@
+psydac.fem.projectors
+=====================
+
+.. automodule:: psydac.fem.projectors
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      knot_insertion_projection_operator
+      knots_to_insert
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.rst b/_sources/documentation/_autosummary/psydac.fem.rst
new file mode 100644
index 0000000..6836aa7
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.rst
@@ -0,0 +1,39 @@
+psydac.fem
+==========
+
+.. automodule:: psydac.fem
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.fem.basic
+   psydac.fem.context
+   psydac.fem.grid
+   psydac.fem.partitioning
+   psydac.fem.projectors
+   psydac.fem.splines
+   psydac.fem.tensor
+   psydac.fem.tests
+   psydac.fem.vector
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.splines.rst b/_sources/documentation/_autosummary/psydac.fem.splines.rst
new file mode 100644
index 0000000..9813679
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.splines.rst
@@ -0,0 +1,29 @@
+psydac.fem.splines
+==================
+
+.. automodule:: psydac.fem.splines
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      SplineSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tensor.rst b/_sources/documentation/_autosummary/psydac.fem.tensor.rst
new file mode 100644
index 0000000..0100dc6
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tensor.rst
@@ -0,0 +1,29 @@
+psydac.fem.tensor
+=================
+
+.. automodule:: psydac.fem.tensor
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      TensorFemSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tests.analytical_profiles_1d.rst b/_sources/documentation/_autosummary/psydac.fem.tests.analytical_profiles_1d.rst
new file mode 100644
index 0000000..bd7f59a
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tests.analytical_profiles_1d.rst
@@ -0,0 +1,31 @@
+psydac.fem.tests.analytical\_profiles\_1d
+=========================================
+
+.. automodule:: psydac.fem.tests.analytical_profiles_1d
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      AnalyticalProfile1D_Cos
+      AnalyticalProfile1D_Poly
+      AnalyticalProfile1D_Sin
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tests.analytical_profiles_base.rst b/_sources/documentation/_autosummary/psydac.fem.tests.analytical_profiles_base.rst
new file mode 100644
index 0000000..e0dda39
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tests.analytical_profiles_base.rst
@@ -0,0 +1,29 @@
+psydac.fem.tests.analytical\_profiles\_base
+===========================================
+
+.. automodule:: psydac.fem.tests.analytical_profiles_base
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      AnalyticalProfile
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tests.rst b/_sources/documentation/_autosummary/psydac.fem.tests.rst
new file mode 100644
index 0000000..1f62cc3
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tests.rst
@@ -0,0 +1,40 @@
+psydac.fem.tests
+================
+
+.. automodule:: psydac.fem.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.fem.tests.analytical_profiles_1d
+   psydac.fem.tests.analytical_profiles_base
+   psydac.fem.tests.splines_error_bounds
+   psydac.fem.tests.test_eval_fields_parallel
+   psydac.fem.tests.test_product
+   psydac.fem.tests.test_spline_histopolation
+   psydac.fem.tests.test_spline_interpolation
+   psydac.fem.tests.test_splines
+   psydac.fem.tests.test_splines_par
+   psydac.fem.tests.utilities
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tests.splines_error_bounds.rst b/_sources/documentation/_autosummary/psydac.fem.tests.splines_error_bounds.rst
new file mode 100644
index 0000000..16b34d2
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tests.splines_error_bounds.rst
@@ -0,0 +1,33 @@
+psydac.fem.tests.splines\_error\_bounds
+=======================================
+
+.. automodule:: psydac.fem.tests.splines_error_bounds
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      spline_1d_error_bound
+      spline_1d_error_bound_on_deriv
+      spline_2d_error_bound
+      spline_2d_error_bounds_on_grad
+      tihomirov_error_bound
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tests.test_eval_fields_parallel.rst b/_sources/documentation/_autosummary/psydac.fem.tests.test_eval_fields_parallel.rst
new file mode 100644
index 0000000..7fb4140
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tests.test_eval_fields_parallel.rst
@@ -0,0 +1,30 @@
+psydac.fem.tests.test\_eval\_fields\_parallel
+=============================================
+
+.. automodule:: psydac.fem.tests.test_eval_fields_parallel
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_eval_fields_irregular
+      test_eval_fields_regular
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tests.test_product.rst b/_sources/documentation/_autosummary/psydac.fem.tests.test_product.rst
new file mode 100644
index 0000000..1f7c6f3
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tests.test_product.rst
@@ -0,0 +1,30 @@
+psydac.fem.tests.test\_product
+==============================
+
+.. automodule:: psydac.fem.tests.test_product
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_product_space_2d
+      test_product_space_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tests.test_spline_histopolation.rst b/_sources/documentation/_autosummary/psydac.fem.tests.test_spline_histopolation.rst
new file mode 100644
index 0000000..8e88688
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tests.test_spline_histopolation.rst
@@ -0,0 +1,32 @@
+psydac.fem.tests.test\_spline\_histopolation
+============================================
+
+.. automodule:: psydac.fem.tests.test_spline_histopolation
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compare_and_plot
+      histopolate_polynomial
+      test_histopolation_cosine
+      test_histopolation_exact
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tests.test_spline_interpolation.rst b/_sources/documentation/_autosummary/psydac.fem.tests.test_spline_interpolation.rst
new file mode 100644
index 0000000..923cdc8
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tests.test_spline_interpolation.rst
@@ -0,0 +1,32 @@
+psydac.fem.tests.test\_spline\_interpolation
+============================================
+
+.. automodule:: psydac.fem.tests.test_spline_interpolation
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      args_SplineInterpolation1D_cosine
+      test_SplineInterpolation1D_cosine
+      test_SplineInterpolation1D_exact
+      test_SplineInterpolation2D_parallel_exact
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tests.test_splines.rst b/_sources/documentation/_autosummary/psydac.fem.tests.test_splines.rst
new file mode 100644
index 0000000..4eff677
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tests.test_splines.rst
@@ -0,0 +1,37 @@
+psydac.fem.tests.test\_splines
+==============================
+
+.. automodule:: psydac.fem.tests.test_splines
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_1d_1
+      test_1d_2
+      test_1d_3
+      test_2d_1
+      test_2d_2
+      test_2d_3
+      test_3d_1
+      test_3d_2
+      test_3d_3
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tests.test_splines_par.rst b/_sources/documentation/_autosummary/psydac.fem.tests.test_splines_par.rst
new file mode 100644
index 0000000..6124b89
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tests.test_splines_par.rst
@@ -0,0 +1,29 @@
+psydac.fem.tests.test\_splines\_par
+===================================
+
+.. automodule:: psydac.fem.tests.test_splines_par
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_2d_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.tests.utilities.rst b/_sources/documentation/_autosummary/psydac.fem.tests.utilities.rst
new file mode 100644
index 0000000..5dfd1b4
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.tests.utilities.rst
@@ -0,0 +1,31 @@
+psydac.fem.tests.utilities
+==========================
+
+.. automodule:: psydac.fem.tests.utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      falling_factorial
+      horner
+      random_grid
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.fem.vector.rst b/_sources/documentation/_autosummary/psydac.fem.vector.rst
new file mode 100644
index 0000000..4ee18a7
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.fem.vector.rst
@@ -0,0 +1,30 @@
+psydac.fem.vector
+=================
+
+.. automodule:: psydac.fem.vector
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      ProductFemSpace
+      VectorFemSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.basic.rst b/_sources/documentation/_autosummary/psydac.linalg.basic.rst
new file mode 100644
index 0000000..e4450ff
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.basic.rst
@@ -0,0 +1,39 @@
+psydac.linalg.basic
+===================
+
+.. automodule:: psydac.linalg.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      ComposedLinearOperator
+      IdentityOperator
+      InverseLinearOperator
+      LinearOperator
+      LinearSolver
+      PowerLinearOperator
+      ScaledLinearOperator
+      SumLinearOperator
+      Vector
+      VectorSpace
+      ZeroOperator
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.block.rst b/_sources/documentation/_autosummary/psydac.linalg.block.rst
new file mode 100644
index 0000000..c31e5f6
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.block.rst
@@ -0,0 +1,31 @@
+psydac.linalg.block
+===================
+
+.. automodule:: psydac.linalg.block
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BlockLinearOperator
+      BlockVector
+      BlockVectorSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.direct_solvers.rst b/_sources/documentation/_autosummary/psydac.linalg.direct_solvers.rst
new file mode 100644
index 0000000..d0daddc
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.direct_solvers.rst
@@ -0,0 +1,30 @@
+psydac.linalg.direct\_solvers
+=============================
+
+.. automodule:: psydac.linalg.direct_solvers
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BandedSolver
+      SparseSolver
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.fft.rst b/_sources/documentation/_autosummary/psydac.linalg.fft.rst
new file mode 100644
index 0000000..64d7b3c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.fft.rst
@@ -0,0 +1,35 @@
+psydac.linalg.fft
+=================
+
+.. automodule:: psydac.linalg.fft
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DistributedDCT
+      DistributedDST
+      DistributedFFT
+      DistributedFFTBase
+      DistributedIDCT
+      DistributedIDST
+      DistributedIFFT
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.kernels.axpy_kernels.rst b/_sources/documentation/_autosummary/psydac.linalg.kernels.axpy_kernels.rst
new file mode 100644
index 0000000..a00ff1e
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.kernels.axpy_kernels.rst
@@ -0,0 +1,31 @@
+psydac.linalg.kernels.axpy\_kernels
+===================================
+
+.. automodule:: psydac.linalg.kernels.axpy_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      axpy_1d
+      axpy_2d
+      axpy_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.kernels.inner_kernels.rst b/_sources/documentation/_autosummary/psydac.linalg.kernels.inner_kernels.rst
new file mode 100644
index 0000000..7a8864a
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.kernels.inner_kernels.rst
@@ -0,0 +1,31 @@
+psydac.linalg.kernels.inner\_kernels
+====================================
+
+.. automodule:: psydac.linalg.kernels.inner_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      inner_1d
+      inner_2d
+      inner_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.kernels.matvec_kernels.rst b/_sources/documentation/_autosummary/psydac.linalg.kernels.matvec_kernels.rst
new file mode 100644
index 0000000..5f2636d
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.kernels.matvec_kernels.rst
@@ -0,0 +1,31 @@
+psydac.linalg.kernels.matvec\_kernels
+=====================================
+
+.. automodule:: psydac.linalg.kernels.matvec_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      matvec_1d
+      matvec_2d
+      matvec_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.kernels.rst b/_sources/documentation/_autosummary/psydac.linalg.kernels.rst
new file mode 100644
index 0000000..fdf0191
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.kernels.rst
@@ -0,0 +1,35 @@
+psydac.linalg.kernels
+=====================
+
+.. automodule:: psydac.linalg.kernels
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.linalg.kernels.axpy_kernels
+   psydac.linalg.kernels.inner_kernels
+   psydac.linalg.kernels.matvec_kernels
+   psydac.linalg.kernels.stencil2coo_kernels
+   psydac.linalg.kernels.transpose_kernels
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.kernels.stencil2coo_kernels.rst b/_sources/documentation/_autosummary/psydac.linalg.kernels.stencil2coo_kernels.rst
new file mode 100644
index 0000000..c0bbae1
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.kernels.stencil2coo_kernels.rst
@@ -0,0 +1,34 @@
+psydac.linalg.kernels.stencil2coo\_kernels
+==========================================
+
+.. automodule:: psydac.linalg.kernels.stencil2coo_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      stencil2coo_1d_C
+      stencil2coo_1d_F
+      stencil2coo_2d_C
+      stencil2coo_2d_F
+      stencil2coo_3d_C
+      stencil2coo_3d_F
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.kernels.transpose_kernels.rst b/_sources/documentation/_autosummary/psydac.linalg.kernels.transpose_kernels.rst
new file mode 100644
index 0000000..30e76cc
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.kernels.transpose_kernels.rst
@@ -0,0 +1,34 @@
+psydac.linalg.kernels.transpose\_kernels
+========================================
+
+.. automodule:: psydac.linalg.kernels.transpose_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      interface_transpose_1d
+      interface_transpose_2d
+      interface_transpose_3d
+      transpose_1d
+      transpose_2d
+      transpose_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.kron.rst b/_sources/documentation/_autosummary/psydac.linalg.kron.rst
new file mode 100644
index 0000000..4e030f6
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.kron.rst
@@ -0,0 +1,37 @@
+psydac.linalg.kron
+==================
+
+.. automodule:: psydac.linalg.kron
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      kronecker_solve
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      KroneckerDenseMatrix
+      KroneckerLinearSolver
+      KroneckerStencilMatrix
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.rst b/_sources/documentation/_autosummary/psydac.linalg.rst
new file mode 100644
index 0000000..9d334b5
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.rst
@@ -0,0 +1,41 @@
+psydac.linalg
+=============
+
+.. automodule:: psydac.linalg
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.linalg.basic
+   psydac.linalg.block
+   psydac.linalg.direct_solvers
+   psydac.linalg.fft
+   psydac.linalg.kernels
+   psydac.linalg.kron
+   psydac.linalg.solvers
+   psydac.linalg.stencil
+   psydac.linalg.tests
+   psydac.linalg.topetsc
+   psydac.linalg.utilities
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.solvers.rst b/_sources/documentation/_autosummary/psydac.linalg.solvers.rst
new file mode 100644
index 0000000..fe54902
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.solvers.rst
@@ -0,0 +1,42 @@
+psydac.linalg.solvers
+=====================
+
+.. automodule:: psydac.linalg.solvers
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      inverse
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BiConjugateGradient
+      BiConjugateGradientStabilized
+      ConjugateGradient
+      GMRES
+      LSMR
+      MinimumResidual
+      PBiConjugateGradientStabilized
+      PConjugateGradient
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.stencil.rst b/_sources/documentation/_autosummary/psydac.linalg.stencil.rst
new file mode 100644
index 0000000..4329521
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.stencil.rst
@@ -0,0 +1,40 @@
+psydac.linalg.stencil
+=====================
+
+.. automodule:: psydac.linalg.stencil
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_diag_len
+      flip_axis
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      StencilDiagonalMatrix
+      StencilInterfaceMatrix
+      StencilMatrix
+      StencilVector
+      StencilVectorSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.tests.rst b/_sources/documentation/_autosummary/psydac.linalg.tests.rst
new file mode 100644
index 0000000..cc508e7
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.tests.rst
@@ -0,0 +1,40 @@
+psydac.linalg.tests
+===================
+
+.. automodule:: psydac.linalg.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.linalg.tests.test_block
+   psydac.linalg.tests.test_fft
+   psydac.linalg.tests.test_kron_direct_solver
+   psydac.linalg.tests.test_kron_stencil_matrix
+   psydac.linalg.tests.test_linalg
+   psydac.linalg.tests.test_solvers
+   psydac.linalg.tests.test_stencil_interface_matrix
+   psydac.linalg.tests.test_stencil_matrix
+   psydac.linalg.tests.test_stencil_vector
+   psydac.linalg.tests.test_stencil_vector_space
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.tests.test_block.rst b/_sources/documentation/_autosummary/psydac.linalg.tests.test_block.rst
new file mode 100644
index 0000000..3c57996
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.tests.test_block.rst
@@ -0,0 +1,41 @@
+psydac.linalg.tests.test\_block
+===============================
+
+.. automodule:: psydac.linalg.tests.test_block
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_global_starts_ends
+      test_2D_block_linear_operator_serial_init
+      test_2D_block_serial_math
+      test_2D_block_vector_serial_init
+      test_2D_block_vector_space_serial_init
+      test_3D_block_serial_basic_operator
+      test_block_2d_array_to_psydac_1
+      test_block_2d_array_to_psydac_2
+      test_block_linear_operator_dot_backend
+      test_block_linear_operator_parallel_dot
+      test_block_linear_operator_serial_dot
+      test_block_matrix_operator_parallel_dot_backend
+      test_block_serial_dimension
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.tests.test_fft.rst b/_sources/documentation/_autosummary/psydac.linalg.tests.test_fft.rst
new file mode 100644
index 0000000..a2d2374
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.tests.test_fft.rst
@@ -0,0 +1,33 @@
+psydac.linalg.tests.test\_fft
+=============================
+
+.. automodule:: psydac.linalg.tests.test_fft
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_global_starts_ends
+      decode_fft_type
+      method_test
+      test_kron_fft_par
+      test_kron_fft_ser
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.tests.test_kron_direct_solver.rst b/_sources/documentation/_autosummary/psydac.linalg.tests.test_kron_direct_solver.rst
new file mode 100644
index 0000000..60a3fae
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.tests.test_kron_direct_solver.rst
@@ -0,0 +1,52 @@
+psydac.linalg.tests.test\_kron\_direct\_solver
+==============================================
+
+.. automodule:: psydac.linalg.tests.test_kron_direct_solver
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compare_solve
+      compute_global_starts_ends
+      get_M1_block_kron_solver
+      get_inverse_mass_matrices
+      kron_solve_seq_ref
+      matrix_to_bandsolver
+      matrix_to_sparse
+      random_matrix
+      random_vectordata
+      test_2d_mass_solver
+      test_3d_m1_solver
+      test_direct_solvers
+      test_kron_solver_1d_par
+      test_kron_solver_1d_ser
+      test_kron_solver_2d_par
+      test_kron_solver_2d_ser
+      test_kron_solver_2d_transposed_par
+      test_kron_solver_2d_transposed_ser
+      test_kron_solver_3d_par
+      test_kron_solver_3d_ser
+      test_kron_solver_nd_par
+      test_kron_solver_nd_ser
+      test_kron_solver_nompi
+      to_bnd
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.tests.test_kron_stencil_matrix.rst b/_sources/documentation/_autosummary/psydac.linalg.tests.test_kron_stencil_matrix.rst
new file mode 100644
index 0000000..3cb2ce7
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.tests.test_kron_stencil_matrix.rst
@@ -0,0 +1,30 @@
+psydac.linalg.tests.test\_kron\_stencil\_matrix
+===============================================
+
+.. automodule:: psydac.linalg.tests.test_kron_stencil_matrix
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_global_starts_ends
+      test_KroneckerStencilMatrix
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.tests.test_linalg.rst b/_sources/documentation/_autosummary/psydac.linalg.tests.test_linalg.rst
new file mode 100644
index 0000000..ea69d2a
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.tests.test_linalg.rst
@@ -0,0 +1,42 @@
+psydac.linalg.tests.test\_linalg
+================================
+
+.. automodule:: psydac.linalg.tests.test_linalg
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      array_equal
+      compute_global_starts_ends
+      get_StencilVectorSpace
+      get_positive_definite_stencilmatrix
+      is_pos_def
+      sparse_equal
+      test_in_place_operations
+      test_internal_storage
+      test_inverse_transpose_interaction
+      test_operator_evaluation
+      test_positive_definite_matrix
+      test_square_block_basic
+      test_square_stencil_basic
+      test_x0update
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.tests.test_solvers.rst b/_sources/documentation/_autosummary/psydac.linalg.tests.test_solvers.rst
new file mode 100644
index 0000000..aa6d536
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.tests.test_solvers.rst
@@ -0,0 +1,31 @@
+psydac.linalg.tests.test\_solvers
+=================================
+
+.. automodule:: psydac.linalg.tests.test_solvers
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      define_data
+      define_data_hermitian
+      test_solver_tridiagonal
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_interface_matrix.rst b/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_interface_matrix.rst
new file mode 100644
index 0000000..19ac4d0
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_interface_matrix.rst
@@ -0,0 +1,35 @@
+psydac.linalg.tests.test\_stencil\_interface\_matrix
+====================================================
+
+.. automodule:: psydac.linalg.tests.test_stencil_interface_matrix
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_global_starts_ends
+      get_minus_starts_ends
+      get_plus_starts_ends
+      test_stencil_interface_matrix_1d_serial_init
+      test_stencil_interface_matrix_2d_parallel_dot
+      test_stencil_interface_matrix_2d_serial_init
+      test_stencil_interface_matrix_3d_serial_init
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_matrix.rst b/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_matrix.rst
new file mode 100644
index 0000000..32d23b1
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_matrix.rst
@@ -0,0 +1,72 @@
+psydac.linalg.tests.test\_stencil\_matrix
+=========================================
+
+.. automodule:: psydac.linalg.tests.test_stencil_matrix
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_global_starts_ends
+      test_stencil_matrix_1d_parallel_backend_dot
+      test_stencil_matrix_1d_parallel_dot
+      test_stencil_matrix_1d_parallel_sync
+      test_stencil_matrix_1d_parallel_toarray
+      test_stencil_matrix_1d_parallel_transpose
+      test_stencil_matrix_1d_serial_dot
+      test_stencil_matrix_1d_serial_init
+      test_stencil_matrix_1d_serial_spurious_entries
+      test_stencil_matrix_1d_serial_toarray
+      test_stencil_matrix_1d_serial_transpose
+      test_stencil_matrix_2d_basic_ops
+      test_stencil_matrix_2d_copy
+      test_stencil_matrix_2d_math
+      test_stencil_matrix_2d_parallel_backend_dot
+      test_stencil_matrix_2d_parallel_dot
+      test_stencil_matrix_2d_parallel_sync
+      test_stencil_matrix_2d_parallel_transpose
+      test_stencil_matrix_2d_serial_backend_dot_1
+      test_stencil_matrix_2d_serial_backend_dot_2
+      test_stencil_matrix_2d_serial_backend_dot_3
+      test_stencil_matrix_2d_serial_backend_switch
+      test_stencil_matrix_2d_serial_dot_1
+      test_stencil_matrix_2d_serial_dot_2
+      test_stencil_matrix_2d_serial_dot_3
+      test_stencil_matrix_2d_serial_dot_4
+      test_stencil_matrix_2d_serial_dot_5
+      test_stencil_matrix_2d_serial_dot_6
+      test_stencil_matrix_2d_serial_init
+      test_stencil_matrix_2d_serial_spurious_entries
+      test_stencil_matrix_2d_serial_toarray
+      test_stencil_matrix_2d_serial_tosparse
+      test_stencil_matrix_2d_serial_transpose_1
+      test_stencil_matrix_2d_serial_transpose_2
+      test_stencil_matrix_2d_serial_transpose_3
+      test_stencil_matrix_2d_serial_transpose_4
+      test_stencil_matrix_2d_serial_transpose_5
+      test_stencil_matrix_2d_serial_transpose_6
+      test_stencil_matrix_2d_serial_transpose_7
+      test_stencil_matrix_2d_serial_transpose_8
+      test_stencil_matrix_2d_serial_transpose_9
+      test_stencil_matrix_2d_serial_vdot
+      test_stencil_matrix_3d_serial_init
+      test_stencil_matrix_3d_serial_transpose_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_vector.rst b/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_vector.rst
new file mode 100644
index 0000000..77f5419
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_vector.rst
@@ -0,0 +1,44 @@
+psydac.linalg.tests.test\_stencil\_vector
+=========================================
+
+.. automodule:: psydac.linalg.tests.test_stencil_vector
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_global_starts_ends
+      test_stencil_matrix_2d_serial_toarray
+      test_stencil_vector_1d_parallel_init
+      test_stencil_vector_2d_array_to_psydac
+      test_stencil_vector_2d_basic_ops
+      test_stencil_vector_2d_parallel_dot
+      test_stencil_vector_2d_parallel_init
+      test_stencil_vector_2d_parallel_toarray
+      test_stencil_vector_2d_serial_conjugate
+      test_stencil_vector_2d_serial_copy
+      test_stencil_vector_2d_serial_dot
+      test_stencil_vector_2d_serial_init
+      test_stencil_vector_2d_serial_math
+      test_stencil_vector_2d_serial_update_ghost_region_interior
+      test_stencil_vector_3d_parallel_dot
+      test_stencil_vector_3d_parallel_init
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_vector_space.rst b/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_vector_space.rst
new file mode 100644
index 0000000..e626b65
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.tests.test_stencil_vector_space.rst
@@ -0,0 +1,39 @@
+psydac.linalg.tests.test\_stencil\_vector\_space
+================================================
+
+.. automodule:: psydac.linalg.tests.test_stencil_vector_space
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_global_starts_ends
+      test_stencil_vector_space_1d_parallel_init
+      test_stencil_vector_space_1d_serial_init
+      test_stencil_vector_space_2D_parallel_parent
+      test_stencil_vector_space_2D_serial_set_interface
+      test_stencil_vector_space_2D_serial_zeros
+      test_stencil_vector_space_2d_parallel_init
+      test_stencil_vector_space_2d_serial_init
+      test_stencil_vector_space_3D_serial_parent
+      test_stencil_vector_space_3d_parallel_init
+      test_stencil_vector_space_3d_serial_init
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.topetsc.rst b/_sources/documentation/_autosummary/psydac.linalg.topetsc.rst
new file mode 100644
index 0000000..6b074b9
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.topetsc.rst
@@ -0,0 +1,31 @@
+psydac.linalg.topetsc
+=====================
+
+.. automodule:: psydac.linalg.topetsc
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      flatten_vec
+      mat_topetsc
+      vec_topetsc
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.linalg.utilities.rst b/_sources/documentation/_autosummary/psydac.linalg.utilities.rst
new file mode 100644
index 0000000..76efafb
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.linalg.utilities.rst
@@ -0,0 +1,30 @@
+psydac.linalg.utilities
+=======================
+
+.. automodule:: psydac.linalg.utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      array_to_psydac
+      petsc_to_psydac
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.mapping.discrete.rst b/_sources/documentation/_autosummary/psydac.mapping.discrete.rst
new file mode 100644
index 0000000..dbdbb90
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.mapping.discrete.rst
@@ -0,0 +1,36 @@
+psydac.mapping.discrete
+=======================
+
+.. automodule:: psydac.mapping.discrete
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      random_string
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      NurbsMapping
+      SplineMapping
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.mapping.discrete_gallery.rst b/_sources/documentation/_autosummary/psydac.mapping.discrete_gallery.rst
new file mode 100644
index 0000000..c28f3ae
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.mapping.discrete_gallery.rst
@@ -0,0 +1,35 @@
+psydac.mapping.discrete\_gallery
+================================
+
+.. automodule:: psydac.mapping.discrete_gallery
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      discrete_mapping
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Collela3D
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.mapping.rst b/_sources/documentation/_autosummary/psydac.mapping.rst
new file mode 100644
index 0000000..dc42c06
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.mapping.rst
@@ -0,0 +1,33 @@
+psydac.mapping
+==============
+
+.. automodule:: psydac.mapping
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.mapping.discrete
+   psydac.mapping.discrete_gallery
+   psydac.mapping.tests
+
diff --git a/_sources/documentation/_autosummary/psydac.mapping.tests.rst b/_sources/documentation/_autosummary/psydac.mapping.tests.rst
new file mode 100644
index 0000000..ed4c612
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.mapping.tests.rst
@@ -0,0 +1,33 @@
+psydac.mapping.tests
+====================
+
+.. automodule:: psydac.mapping.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.mapping.tests.test_discrete_mapping
+   psydac.mapping.tests.visual_test_discrete_mapping_2d
+   psydac.mapping.tests.visual_test_discrete_mapping_3d_surface
+
diff --git a/_sources/documentation/_autosummary/psydac.mapping.tests.test_discrete_mapping.rst b/_sources/documentation/_autosummary/psydac.mapping.tests.test_discrete_mapping.rst
new file mode 100644
index 0000000..a4bad88
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.mapping.tests.test_discrete_mapping.rst
@@ -0,0 +1,33 @@
+psydac.mapping.tests.test\_discrete\_mapping
+============================================
+
+.. automodule:: psydac.mapping.tests.test_discrete_mapping
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_build_mesh_i
+      test_build_mesh_reg
+      test_nurbs_circle
+      test_parallel_jacobians_irregular
+      test_parallel_jacobians_regular
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.mapping.tests.visual_test_discrete_mapping_2d.rst b/_sources/documentation/_autosummary/psydac.mapping.tests.visual_test_discrete_mapping_2d.rst
new file mode 100644
index 0000000..ee316f0
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.mapping.tests.visual_test_discrete_mapping_2d.rst
@@ -0,0 +1,30 @@
+psydac.mapping.tests.visual\_test\_discrete\_mapping\_2d
+========================================================
+
+.. automodule:: psydac.mapping.tests.visual_test_discrete_mapping_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      main
+      parse_input_arguments
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.mapping.tests.visual_test_discrete_mapping_3d_surface.rst b/_sources/documentation/_autosummary/psydac.mapping.tests.visual_test_discrete_mapping_3d_surface.rst
new file mode 100644
index 0000000..6a69b86
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.mapping.tests.visual_test_discrete_mapping_3d_surface.rst
@@ -0,0 +1,30 @@
+psydac.mapping.tests.visual\_test\_discrete\_mapping\_3d\_surface
+=================================================================
+
+.. automodule:: psydac.mapping.tests.visual_test_discrete_mapping_3d_surface
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      main
+      parse_input_arguments
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.polar.c1_cart.rst b/_sources/documentation/_autosummary/psydac.polar.c1_cart.rst
new file mode 100644
index 0000000..20bd6cc
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.polar.c1_cart.rst
@@ -0,0 +1,29 @@
+psydac.polar.c1\_cart
+=====================
+
+.. automodule:: psydac.polar.c1_cart
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      C1_Cart
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.polar.c1_linops.rst b/_sources/documentation/_autosummary/psydac.polar.c1_linops.rst
new file mode 100644
index 0000000..cdee3f2
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.polar.c1_linops.rst
@@ -0,0 +1,30 @@
+psydac.polar.c1\_linops
+=======================
+
+.. automodule:: psydac.polar.c1_linops
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      LinearOperator_DenseToStencil
+      LinearOperator_StencilToDense
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.polar.c1_projections.rst b/_sources/documentation/_autosummary/psydac.polar.c1_projections.rst
new file mode 100644
index 0000000..6298721
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.polar.c1_projections.rst
@@ -0,0 +1,29 @@
+psydac.polar.c1\_projections
+============================
+
+.. automodule:: psydac.polar.c1_projections
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      C1Projector
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.polar.c1_spaces.rst b/_sources/documentation/_autosummary/psydac.polar.c1_spaces.rst
new file mode 100644
index 0000000..63ffc55
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.polar.c1_spaces.rst
@@ -0,0 +1,29 @@
+psydac.polar.c1\_spaces
+=======================
+
+.. automodule:: psydac.polar.c1_spaces
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      new_c1_vector_space
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.polar.dense.rst b/_sources/documentation/_autosummary/psydac.polar.dense.rst
new file mode 100644
index 0000000..8db99ce
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.polar.dense.rst
@@ -0,0 +1,31 @@
+psydac.polar.dense
+==================
+
+.. automodule:: psydac.polar.dense
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DenseMatrix
+      DenseVector
+      DenseVectorSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.polar.rst b/_sources/documentation/_autosummary/psydac.polar.rst
new file mode 100644
index 0000000..434e067
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.polar.rst
@@ -0,0 +1,36 @@
+psydac.polar
+============
+
+.. automodule:: psydac.polar
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.polar.c1_cart
+   psydac.polar.c1_linops
+   psydac.polar.c1_projections
+   psydac.polar.c1_spaces
+   psydac.polar.dense
+   psydac.polar.tests
+
diff --git a/_sources/documentation/_autosummary/psydac.polar.tests.rst b/_sources/documentation/_autosummary/psydac.polar.tests.rst
new file mode 100644
index 0000000..7b43eea
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.polar.tests.rst
@@ -0,0 +1,32 @@
+psydac.polar.tests
+==================
+
+.. automodule:: psydac.polar.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.polar.tests.test_c1_linops
+   psydac.polar.tests.test_c1_projections
+
diff --git a/_sources/documentation/_autosummary/psydac.polar.tests.test_c1_linops.rst b/_sources/documentation/_autosummary/psydac.polar.tests.test_c1_linops.rst
new file mode 100644
index 0000000..f56bef3
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.polar.tests.test_c1_linops.rst
@@ -0,0 +1,29 @@
+psydac.polar.tests.test\_c1\_linops
+===================================
+
+.. automodule:: psydac.polar.tests.test_c1_linops
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_c1_linops
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.polar.tests.test_c1_projections.rst b/_sources/documentation/_autosummary/psydac.polar.tests.test_c1_projections.rst
new file mode 100644
index 0000000..be2d8fd
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.polar.tests.test_c1_projections.rst
@@ -0,0 +1,29 @@
+psydac.polar.tests.test\_c1\_projections
+========================================
+
+.. automodule:: psydac.polar.tests.test_c1_projections
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_c1_projections
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.basic.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.basic.rst
new file mode 100644
index 0000000..3fbd2a2
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.basic.rst
@@ -0,0 +1,30 @@
+psydac.pyccel.ast.basic
+=======================
+
+.. automodule:: psydac.pyccel.ast.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Basic
+      PyccelAstNode
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.builtins.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.builtins.rst
new file mode 100644
index 0000000..aa415f7
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.builtins.rst
@@ -0,0 +1,46 @@
+psydac.pyccel.ast.builtins
+==========================
+
+.. automodule:: psydac.pyccel.ast.builtins
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      python_builtin_datatype
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Enumerate
+      Len
+      List
+      Map
+      Print
+      PythonBool
+      PythonComplex
+      PythonFloat
+      PythonInt
+      PythonTuple
+      Range
+      Zip
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.core.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.core.rst
new file mode 100644
index 0000000..cdd7928
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.core.rst
@@ -0,0 +1,162 @@
+psydac.pyccel.ast.core
+======================
+
+.. automodule:: psydac.pyccel.ast.core
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      allocatable_like
+      broadcast
+      create_incremented_string
+      create_variable
+      extract_subexpressions
+      float2int
+      get_assigned_symbols
+      get_initial_value
+      get_iterable_ranges
+      handle_precedence
+      inline
+      int2float
+      is_simple_assign
+      operator
+      process_shape
+      subs
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      AddOp
+      AliasAssign
+      AnnotatedArgument
+      AnnotatedComment
+      Argument
+      AsName
+      Assert
+      Assign
+      AugAssign
+      Block
+      Break
+      ClassDef
+      CodeBlock
+      Comment
+      CommentBlock
+      Concatenate
+      Constant
+      ConstructorCall
+      Continue
+      Declare
+      Del
+      DivOp
+      Dlist
+      DoConcurrent
+      DottedName
+      DottedVariable
+      EmptyNode
+      ErrorExit
+      Eval
+      Exit
+      F2PYFunctionDef
+      For
+      ForAll
+      ForIterator
+      FunctionCall
+      FunctionDef
+      GetDefaultFunctionArg
+      If
+      IfTernaryOperator
+      Import
+      IndexedElement
+      IndexedVariable
+      Interface
+      Is
+      IsNot
+      Load
+      ModOp
+      Module
+      MulOp
+      NativeOp
+      NewLine
+      Nil
+      ParallelBlock
+      ParallelRange
+      ParserResult
+      Pass
+      Product
+      Program
+      PyccelAdd
+      PyccelAnd
+      PyccelArraySize
+      PyccelAssociativeParenthesis
+      PyccelBitAnd
+      PyccelBitOperator
+      PyccelBitOr
+      PyccelBitXor
+      PyccelBooleanOperator
+      PyccelDiv
+      PyccelEq
+      PyccelFloorDiv
+      PyccelGe
+      PyccelGt
+      PyccelInvert
+      PyccelLShift
+      PyccelLe
+      PyccelLt
+      PyccelMinus
+      PyccelMod
+      PyccelMul
+      PyccelNe
+      PyccelNot
+      PyccelOperator
+      PyccelOr
+      PyccelPow
+      PyccelRShift
+      PyccelUnary
+      PyccelUnarySub
+      PythonFunction
+      Raise
+      Random
+      Return
+      SeparatorComment
+      Slice
+      StarredArguments
+      String
+      SubOp
+      Subroutine
+      SumFunction
+      SymbolicAssign
+      SymbolicPrint
+      SympyFunction
+      Tensor
+      Tile
+      TupleImport
+      TupleVariable
+      ValuedArgument
+      ValuedVariable
+      Variable
+      VariableAddress
+      Void
+      VoidFunction
+      While
+      With
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.datatypes.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.datatypes.rst
new file mode 100644
index 0000000..c848c98
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.datatypes.rst
@@ -0,0 +1,57 @@
+psydac.pyccel.ast.datatypes
+===========================
+
+.. automodule:: psydac.pyccel.ast.datatypes
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      DataTypeFactory
+      datatype
+      get_default_value
+      is_iterable_datatype
+      is_pyccel_datatype
+      is_with_construct_datatype
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      CustomDataType
+      DataType
+      FunctionType
+      NativeBool
+      NativeComplex
+      NativeGeneric
+      NativeInteger
+      NativeNil
+      NativeParallelRange
+      NativeRange
+      NativeReal
+      NativeString
+      NativeSymbol
+      NativeTensor
+      NativeTuple
+      NativeVoid
+      UnionType
+      VariableType
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.functionalexpr.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.functionalexpr.rst
new file mode 100644
index 0000000..aa29637
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.functionalexpr.rst
@@ -0,0 +1,34 @@
+psydac.pyccel.ast.functionalexpr
+================================
+
+.. automodule:: psydac.pyccel.ast.functionalexpr
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      FunctionalFor
+      FunctionalMap
+      FunctionalMax
+      FunctionalMin
+      FunctionalSum
+      GeneratorComprehension
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.mathext.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.mathext.rst
new file mode 100644
index 0000000..82bfcbc
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.mathext.rst
@@ -0,0 +1,77 @@
+psydac.pyccel.ast.mathext
+=========================
+
+.. automodule:: psydac.pyccel.ast.mathext
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      MathAcos
+      MathAcosh
+      MathAsin
+      MathAsinh
+      MathAtan
+      MathAtan2
+      MathAtanh
+      MathCeil
+      MathCopysign
+      MathCos
+      MathCosh
+      MathDegrees
+      MathErf
+      MathErfc
+      MathExp
+      MathExpm1
+      MathFabs
+      MathFactorial
+      MathFloor
+      MathFmod
+      MathFrexp
+      MathFsum
+      MathFunctionBase
+      MathFunctionBool
+      MathFunctionFloat
+      MathFunctionInt
+      MathGamma
+      MathGcd
+      MathHypot
+      MathIsclose
+      MathIsfinite
+      MathIsinf
+      MathIsnan
+      MathLdexp
+      MathLgamma
+      MathLog
+      MathLog10
+      MathLog1p
+      MathLog2
+      MathModf
+      MathPow
+      MathRadians
+      MathRemainder
+      MathSin
+      MathSinh
+      MathSqrt
+      MathTan
+      MathTanh
+      MathTrunc
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.numbers.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.numbers.rst
new file mode 100644
index 0000000..95d812e
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.numbers.rst
@@ -0,0 +1,34 @@
+psydac.pyccel.ast.numbers
+=========================
+
+.. automodule:: psydac.pyccel.ast.numbers
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BooleanFalse
+      BooleanTrue
+      Complex
+      Float
+      ImaginaryUnit
+      Integer
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.numpyext.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.numpyext.rst
new file mode 100644
index 0000000..6ce137b
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.numpyext.rst
@@ -0,0 +1,92 @@
+psydac.pyccel.ast.numpyext
+==========================
+
+.. automodule:: psydac.pyccel.ast.numpyext
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      Shape
+      process_dtype
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Array
+      Complex128
+      Complex64
+      Cross
+      Diag
+      Empty
+      EmptyLike
+      Float32
+      Float64
+      Full
+      FullLike
+      Imag
+      Int32
+      Int64
+      Linspace
+      Matmul
+      Norm
+      NumpyAbs
+      NumpyArccos
+      NumpyArccosh
+      NumpyArcsin
+      NumpyArcsinh
+      NumpyArctan
+      NumpyArctan2
+      NumpyArctanh
+      NumpyComplex
+      NumpyCos
+      NumpyCosh
+      NumpyExp
+      NumpyFloat
+      NumpyFloor
+      NumpyHypot
+      NumpyInt
+      NumpyLog
+      NumpyMax
+      NumpyMin
+      NumpyMod
+      NumpyNewArray
+      NumpyRandint
+      NumpySin
+      NumpySinh
+      NumpySqrt
+      NumpySum
+      NumpyTan
+      NumpyTanh
+      NumpyUfuncBase
+      NumpyUfuncBinary
+      NumpyUfuncUnary
+      Ones
+      OnesLike
+      Product
+      Rand
+      Real
+      Sqrt
+      Where
+      Zeros
+      ZerosLike
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.rst
new file mode 100644
index 0000000..35ed49b
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.rst
@@ -0,0 +1,41 @@
+psydac.pyccel.ast
+=================
+
+.. automodule:: psydac.pyccel.ast
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.pyccel.ast.basic
+   psydac.pyccel.ast.builtins
+   psydac.pyccel.ast.core
+   psydac.pyccel.ast.datatypes
+   psydac.pyccel.ast.functionalexpr
+   psydac.pyccel.ast.mathext
+   psydac.pyccel.ast.numbers
+   psydac.pyccel.ast.numpyext
+   psydac.pyccel.ast.singleton
+   psydac.pyccel.ast.type_inference
+   psydac.pyccel.ast.utilities
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.singleton.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.singleton.rst
new file mode 100644
index 0000000..cd0679d
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.singleton.rst
@@ -0,0 +1,29 @@
+psydac.pyccel.ast.singleton
+===========================
+
+.. automodule:: psydac.pyccel.ast.singleton
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Singleton
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.type_inference.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.type_inference.rst
new file mode 100644
index 0000000..70b2c0c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.type_inference.rst
@@ -0,0 +1,29 @@
+psydac.pyccel.ast.type\_inference
+=================================
+
+.. automodule:: psydac.pyccel.ast.type_inference
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      str_dtype
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.ast.utilities.rst b/_sources/documentation/_autosummary/psydac.pyccel.ast.utilities.rst
new file mode 100644
index 0000000..0afc445
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.ast.utilities.rst
@@ -0,0 +1,34 @@
+psydac.pyccel.ast.utilities
+===========================
+
+.. automodule:: psydac.pyccel.ast.utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      build_types_decorator
+      builtin_function
+      builtin_import
+      collect_relevant_imports
+      get_function_from_ast
+      split_positional_keyword_arguments
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.codegen.printing.pycode.rst b/_sources/documentation/_autosummary/psydac.pyccel.codegen.printing.pycode.rst
new file mode 100644
index 0000000..689663c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.codegen.printing.pycode.rst
@@ -0,0 +1,35 @@
+psydac.pyccel.codegen.printing.pycode
+=====================================
+
+.. automodule:: psydac.pyccel.codegen.printing.pycode
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      pycode
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      PythonCodePrinter
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.codegen.printing.rst b/_sources/documentation/_autosummary/psydac.pyccel.codegen.printing.rst
new file mode 100644
index 0000000..5fa8c8b
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.codegen.printing.rst
@@ -0,0 +1,31 @@
+psydac.pyccel.codegen.printing
+==============================
+
+.. automodule:: psydac.pyccel.codegen.printing
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.pyccel.codegen.printing.pycode
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.codegen.rst b/_sources/documentation/_autosummary/psydac.pyccel.codegen.rst
new file mode 100644
index 0000000..e5f648f
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.codegen.rst
@@ -0,0 +1,31 @@
+psydac.pyccel.codegen
+=====================
+
+.. automodule:: psydac.pyccel.codegen
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.pyccel.codegen.printing
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.errors.errors.rst b/_sources/documentation/_autosummary/psydac.pyccel.errors.errors.rst
new file mode 100644
index 0000000..b635e1c
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.errors.errors.rst
@@ -0,0 +1,46 @@
+psydac.pyccel.errors.errors
+===========================
+
+.. automodule:: psydac.pyccel.errors.errors
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      Errors
+      ErrorsMode
+      make_symbol
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      ErrorInfo
+   
+   
+
+   
+   
+   .. rubric:: Exceptions
+
+   .. autosummary::
+   
+      PyccelCodegenError
+      PyccelError
+      PyccelSemanticError
+      PyccelSyntaxError
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.errors.messages.rst b/_sources/documentation/_autosummary/psydac.pyccel.errors.messages.rst
new file mode 100644
index 0000000..639dd0d
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.errors.messages.rst
@@ -0,0 +1,23 @@
+psydac.pyccel.errors.messages
+=============================
+
+.. automodule:: psydac.pyccel.errors.messages
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.errors.rst b/_sources/documentation/_autosummary/psydac.pyccel.errors.rst
new file mode 100644
index 0000000..c2ab434
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.errors.rst
@@ -0,0 +1,32 @@
+psydac.pyccel.errors
+====================
+
+.. automodule:: psydac.pyccel.errors
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.pyccel.errors.errors
+   psydac.pyccel.errors.messages
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.rst b/_sources/documentation/_autosummary/psydac.pyccel.rst
new file mode 100644
index 0000000..324c271
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.rst
@@ -0,0 +1,34 @@
+psydac.pyccel
+=============
+
+.. automodule:: psydac.pyccel
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.pyccel.ast
+   psydac.pyccel.codegen
+   psydac.pyccel.errors
+   psydac.pyccel.symbolic
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.symbolic.lambdify.rst b/_sources/documentation/_autosummary/psydac.pyccel.symbolic.lambdify.rst
new file mode 100644
index 0000000..df736ef
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.symbolic.lambdify.rst
@@ -0,0 +1,32 @@
+psydac.pyccel.symbolic.lambdify
+===============================
+
+.. automodule:: psydac.pyccel.symbolic.lambdify
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      cse
+      lambdify
+      pyccel_sum
+      set_fst
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.pyccel.symbolic.rst b/_sources/documentation/_autosummary/psydac.pyccel.symbolic.rst
new file mode 100644
index 0000000..e53e2a1
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.pyccel.symbolic.rst
@@ -0,0 +1,31 @@
+psydac.pyccel.symbolic
+======================
+
+.. automodule:: psydac.pyccel.symbolic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.pyccel.symbolic.lambdify
+
diff --git a/_sources/documentation/_autosummary/psydac.utilities.quadratures.rst b/_sources/documentation/_autosummary/psydac.utilities.quadratures.rst
new file mode 100644
index 0000000..0f93af6
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.utilities.quadratures.rst
@@ -0,0 +1,31 @@
+psydac.utilities.quadratures
+============================
+
+.. automodule:: psydac.utilities.quadratures
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      gauss_legendre
+      gauss_lobatto
+      quadrature
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.utilities.rst b/_sources/documentation/_autosummary/psydac.utilities.rst
new file mode 100644
index 0000000..76f4bc7
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.utilities.rst
@@ -0,0 +1,33 @@
+psydac.utilities
+================
+
+.. automodule:: psydac.utilities
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   psydac.utilities.quadratures
+   psydac.utilities.utils
+   psydac.utilities.vtk
+
diff --git a/_sources/documentation/_autosummary/psydac.utilities.utils.rst b/_sources/documentation/_autosummary/psydac.utilities.utils.rst
new file mode 100644
index 0000000..fa6ba81
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.utilities.utils.rst
@@ -0,0 +1,35 @@
+psydac.utilities.utils
+======================
+
+.. automodule:: psydac.utilities.utils
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      animate_field
+      is_real
+      refine_array_1d
+      roll_edges
+      split_field
+      split_space
+      unroll_edges
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/psydac.utilities.vtk.rst b/_sources/documentation/_autosummary/psydac.utilities.vtk.rst
new file mode 100644
index 0000000..1a18d65
--- /dev/null
+++ b/_sources/documentation/_autosummary/psydac.utilities.vtk.rst
@@ -0,0 +1,29 @@
+psydac.utilities.vtk
+====================
+
+.. automodule:: psydac.utilities.vtk
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      writeParallelVTKUnstructuredGrid
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.calculus.rst b/_sources/documentation/_autosummary/sympde.calculus.rst
new file mode 100644
index 0000000..695f93f
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.calculus.rst
@@ -0,0 +1,47 @@
+sympde.calculus
+===============
+
+.. automodule:: sympde.calculus
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Adjoint
+      AdjointExteriorDerivative
+      AdjointInteriorProduct
+      AdjointLieDerivative
+      BasicOperator
+      ExteriorDerivative
+      ExteriorProduct
+      Hodge
+      InteriorProduct
+      Ld
+      LieDerivative
+      PullBack
+      d
+      delta
+      hodge
+      ip
+      jp
+      ld
+      wedge
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.core.algebra.rst b/_sources/documentation/_autosummary/sympde.core.algebra.rst
new file mode 100644
index 0000000..252170c
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.core.algebra.rst
@@ -0,0 +1,39 @@
+sympde.core.algebra
+===================
+
+.. automodule:: sympde.core.algebra
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      CrossBasic
+      Cross_2d
+      Cross_3d
+      DotBasic
+      Dot_1d
+      Dot_2d
+      Dot_3d
+      InnerBasic
+      Inner_2d
+      Inner_3d
+      LinearOperator
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.core.basic.rst b/_sources/documentation/_autosummary/sympde.core.basic.rst
new file mode 100644
index 0000000..c9e1436
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.core.basic.rst
@@ -0,0 +1,32 @@
+sympde.core.basic
+=================
+
+.. automodule:: sympde.core.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BasicDerivable
+      BasicMapping
+      CalculusFunction
+      Constant
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.core.rst b/_sources/documentation/_autosummary/sympde.core.rst
new file mode 100644
index 0000000..69c55c9
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.core.rst
@@ -0,0 +1,34 @@
+sympde.core
+===========
+
+.. automodule:: sympde.core
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.core.algebra
+   sympde.core.basic
+   sympde.core.tests
+   sympde.core.utils
+
diff --git a/_sources/documentation/_autosummary/sympde.core.tests.rst b/_sources/documentation/_autosummary/sympde.core.tests.rst
new file mode 100644
index 0000000..2e85e2f
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.core.tests.rst
@@ -0,0 +1,31 @@
+sympde.core.tests
+=================
+
+.. automodule:: sympde.core.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.core.tests.test_utils
+
diff --git a/_sources/documentation/_autosummary/sympde.core.tests.test_utils.rst b/_sources/documentation/_autosummary/sympde.core.tests.test_utils.rst
new file mode 100644
index 0000000..5d441bf
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.core.tests.test_utils.rst
@@ -0,0 +1,29 @@
+sympde.core.tests.test\_utils
+=============================
+
+.. automodule:: sympde.core.tests.test_utils
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_expand_name_patterns
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.core.utils.rst b/_sources/documentation/_autosummary/sympde.core.utils.rst
new file mode 100644
index 0000000..0d8fa15
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.core.utils.rst
@@ -0,0 +1,30 @@
+sympde.core.utils
+=================
+
+.. automodule:: sympde.core.utils
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      expand_name_patterns
+      random_string
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.basic.rst b/_sources/documentation/_autosummary/sympde.expr.basic.rst
new file mode 100644
index 0000000..94cf7b7
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.basic.rst
@@ -0,0 +1,30 @@
+sympde.expr.basic
+=================
+
+.. automodule:: sympde.expr.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BasicExpr
+      BasicForm
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.equation.rst b/_sources/documentation/_autosummary/sympde.expr.equation.rst
new file mode 100644
index 0000000..e99be8a
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.equation.rst
@@ -0,0 +1,40 @@
+sympde.expr.equation
+====================
+
+.. automodule:: sympde.expr.equation
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      find
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BasicBoundaryCondition
+      BasicConstraint
+      Equation
+      EssentialBC
+      Mean
+      NewtonIteration
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.errors.rst b/_sources/documentation/_autosummary/sympde.expr.errors.rst
new file mode 100644
index 0000000..4810632
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.errors.rst
@@ -0,0 +1,34 @@
+sympde.expr.errors
+==================
+
+.. automodule:: sympde.expr.errors
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Exceptions
+
+   .. autosummary::
+   
+      UnconsistentArgumentsError
+      UnconsistentBCError
+      UnconsistentError
+      UnconsistentLhsError
+      UnconsistentLinearExpressionError
+      UnconsistentRhsError
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.evaluation.rst b/_sources/documentation/_autosummary/sympde.expr.evaluation.rst
new file mode 100644
index 0000000..ac43848
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.evaluation.rst
@@ -0,0 +1,46 @@
+sympde.expr.evaluation
+======================
+
+.. automodule:: sympde.expr.evaluation
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      is_sequence
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Advection
+      AdvectionT
+      Basic1dForm
+      Bilaplacian
+      BoundaryExpression
+      DomainExpression
+      InterfaceExpression
+      KernelExpression
+      Mass
+      Stiffness
+      TensorExpr
+      TerminalExpr
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.expr.rst b/_sources/documentation/_autosummary/sympde.expr.expr.rst
new file mode 100644
index 0000000..523d083
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.expr.rst
@@ -0,0 +1,48 @@
+sympde.expr.expr
+================
+
+.. automodule:: sympde.expr.expr
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      add_int
+      expand
+      integral
+      is_linear_expression
+      linearize
+      mul_add
+      mul_integral
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BilinearForm
+      Functional
+      IntAdd
+      Integral
+      LinearExpr
+      LinearForm
+      Norm
+      SemiNorm
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.rst b/_sources/documentation/_autosummary/sympde.expr.rst
new file mode 100644
index 0000000..59a41f0
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.rst
@@ -0,0 +1,36 @@
+sympde.expr
+===========
+
+.. automodule:: sympde.expr
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.expr.basic
+   sympde.expr.equation
+   sympde.expr.errors
+   sympde.expr.evaluation
+   sympde.expr.expr
+   sympde.expr.tests
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.tests.rst b/_sources/documentation/_autosummary/sympde.expr.tests.rst
new file mode 100644
index 0000000..6596b38
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.tests.rst
@@ -0,0 +1,36 @@
+sympde.expr.tests
+=================
+
+.. automodule:: sympde.expr.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.expr.tests.test_equation_2d
+   sympde.expr.tests.test_essential_bc
+   sympde.expr.tests.test_expr
+   sympde.expr.tests.test_find_2d
+   sympde.expr.tests.test_newton_2d
+   sympde.expr.tests.test_tensor_2d
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.tests.test_equation_2d.rst b/_sources/documentation/_autosummary/sympde.expr.tests.test_equation_2d.rst
new file mode 100644
index 0000000..cede436
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.tests.test_equation_2d.rst
@@ -0,0 +1,36 @@
+sympde.expr.tests.test\_equation\_2d
+====================================
+
+.. automodule:: sympde.expr.tests.test_equation_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_equation_2d_1
+      test_equation_2d_2
+      test_equation_2d_3
+      test_equation_2d_4
+      test_equation_2d_5
+      test_equation_2d_6
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.tests.test_essential_bc.rst b/_sources/documentation/_autosummary/sympde.expr.tests.test_essential_bc.rst
new file mode 100644
index 0000000..f7a2c03
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.tests.test_essential_bc.rst
@@ -0,0 +1,31 @@
+sympde.expr.tests.test\_essential\_bc
+=====================================
+
+.. automodule:: sympde.expr.tests.test_essential_bc
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_essential_bc_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.tests.test_expr.rst b/_sources/documentation/_autosummary/sympde.expr.tests.test_expr.rst
new file mode 100644
index 0000000..839d920
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.tests.test_expr.rst
@@ -0,0 +1,67 @@
+sympde.expr.tests.test\_expr
+============================
+
+.. automodule:: sympde.expr.tests.test_expr
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_area_2d_1
+      test_bilinear_form_2d_1
+      test_bilinear_form_2d_2
+      test_bilinear_form_2d_3
+      test_bilinear_form_2d_4
+      test_functional_2d_1
+      test_interface_2d_1
+      test_interface_integral_1
+      test_interface_integral_2
+      test_interface_integral_3
+      test_interface_integral_4
+      test_linear_expr_2d_1
+      test_linear_expr_2d_2
+      test_linear_form_2d_1
+      test_linear_form_2d_2
+      test_linearity_bilinear_form_2d_1
+      test_linearity_linear_form_2d_1
+      test_linearize_expr_2d_1
+      test_linearize_expr_2d_2
+      test_linearize_form_2d_1
+      test_linearize_form_2d_2
+      test_linearize_form_2d_3
+      test_linearize_form_2d_4
+      test_norm_2d_1
+      test_norm_2d_2
+      test_stabilization_2d_1
+      test_terminal_expr_bilinear_2d_1
+      test_terminal_expr_bilinear_2d_2
+      test_terminal_expr_bilinear_2d_3
+      test_terminal_expr_bilinear_2d_4
+      test_terminal_expr_bilinear_3d_1
+      test_terminal_expr_linear_2d_1
+      test_terminal_expr_linear_2d_2
+      test_terminal_expr_linear_2d_3
+      test_terminal_expr_linear_2d_4
+      test_terminal_expr_linear_2d_5
+      test_user_function_2d_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.tests.test_find_2d.rst b/_sources/documentation/_autosummary/sympde.expr.tests.test_find_2d.rst
new file mode 100644
index 0000000..7f4bf3f
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.tests.test_find_2d.rst
@@ -0,0 +1,31 @@
+sympde.expr.tests.test\_find\_2d
+================================
+
+.. automodule:: sympde.expr.tests.test_find_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_find_2d_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.tests.test_newton_2d.rst b/_sources/documentation/_autosummary/sympde.expr.tests.test_newton_2d.rst
new file mode 100644
index 0000000..300f857
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.tests.test_newton_2d.rst
@@ -0,0 +1,31 @@
+sympde.expr.tests.test\_newton\_2d
+==================================
+
+.. automodule:: sympde.expr.tests.test_newton_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_newton_2d_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.expr.tests.test_tensor_2d.rst b/_sources/documentation/_autosummary/sympde.expr.tests.test_tensor_2d.rst
new file mode 100644
index 0000000..274fe6a
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.expr.tests.test_tensor_2d.rst
@@ -0,0 +1,35 @@
+sympde.expr.tests.test\_tensor\_2d
+==================================
+
+.. automodule:: sympde.expr.tests.test_tensor_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_tensorize_2d_1
+      test_tensorize_2d_1_mapping
+      test_tensorize_2d_2
+      test_tensorize_2d_2_mapping
+      test_tensorize_2d_3
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.exterior.calculus.rst b/_sources/documentation/_autosummary/sympde.exterior.calculus.rst
new file mode 100644
index 0000000..91796ee
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.exterior.calculus.rst
@@ -0,0 +1,47 @@
+sympde.exterior.calculus
+========================
+
+.. automodule:: sympde.exterior.calculus
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Adjoint
+      AdjointExteriorDerivative
+      AdjointInteriorProduct
+      AdjointLieDerivative
+      BasicOperator
+      ExteriorDerivative
+      ExteriorProduct
+      Hodge
+      InteriorProduct
+      Ld
+      LieDerivative
+      PullBack
+      d
+      delta
+      hodge
+      ip
+      jp
+      ld
+      wedge
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.exterior.compiler.rst b/_sources/documentation/_autosummary/sympde.exterior.compiler.rst
new file mode 100644
index 0000000..3b0f3ea
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.exterior.compiler.rst
@@ -0,0 +1,35 @@
+sympde.exterior.compiler
+========================
+
+.. automodule:: sympde.exterior.compiler
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      augmented_expression
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      ExteriorCalculusExpr
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.exterior.datatype.rst b/_sources/documentation/_autosummary/sympde.exterior.datatype.rst
new file mode 100644
index 0000000..636edd5
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.exterior.datatype.rst
@@ -0,0 +1,43 @@
+sympde.exterior.datatype
+========================
+
+.. automodule:: sympde.exterior.datatype
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      FormTypeFactory
+      get_index_form
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      FiveFormType
+      FormType
+      FourFormType
+      OneFormType
+      SixFormType
+      ThreeFormType
+      TwoFormType
+      ZeroFormType
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.exterior.form.rst b/_sources/documentation/_autosummary/sympde.exterior.form.rst
new file mode 100644
index 0000000..de9f429
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.exterior.form.rst
@@ -0,0 +1,29 @@
+sympde.exterior.form
+====================
+
+.. automodule:: sympde.exterior.form
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DifferentialForm
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.exterior.inference.rst b/_sources/documentation/_autosummary/sympde.exterior.inference.rst
new file mode 100644
index 0000000..358b915
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.exterior.inference.rst
@@ -0,0 +1,29 @@
+sympde.exterior.inference
+=========================
+
+.. automodule:: sympde.exterior.inference
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      infere_type
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.exterior.rst b/_sources/documentation/_autosummary/sympde.exterior.rst
new file mode 100644
index 0000000..f79169e
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.exterior.rst
@@ -0,0 +1,36 @@
+sympde.exterior
+===============
+
+.. automodule:: sympde.exterior
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.exterior.calculus
+   sympde.exterior.compiler
+   sympde.exterior.datatype
+   sympde.exterior.form
+   sympde.exterior.inference
+   sympde.exterior.tests
+
diff --git a/_sources/documentation/_autosummary/sympde.exterior.tests.rst b/_sources/documentation/_autosummary/sympde.exterior.tests.rst
new file mode 100644
index 0000000..5b008e6
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.exterior.tests.rst
@@ -0,0 +1,34 @@
+sympde.exterior.tests
+=====================
+
+.. automodule:: sympde.exterior.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.exterior.tests.test_compiler
+   sympde.exterior.tests.test_datatype
+   sympde.exterior.tests.test_exterior
+   sympde.exterior.tests.test_inference
+
diff --git a/_sources/documentation/_autosummary/sympde.exterior.tests.test_compiler.rst b/_sources/documentation/_autosummary/sympde.exterior.tests.test_compiler.rst
new file mode 100644
index 0000000..9d8ba30
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.exterior.tests.test_compiler.rst
@@ -0,0 +1,34 @@
+sympde.exterior.tests.test\_compiler
+====================================
+
+.. automodule:: sympde.exterior.tests.test_compiler
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_compiler_3d_1
+      test_compiler_3d_2
+      test_compiler_3d_poisson
+      test_compiler_3d_stokes
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.exterior.tests.test_datatype.rst b/_sources/documentation/_autosummary/sympde.exterior.tests.test_datatype.rst
new file mode 100644
index 0000000..b9a7f02
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.exterior.tests.test_datatype.rst
@@ -0,0 +1,31 @@
+sympde.exterior.tests.test\_datatype
+====================================
+
+.. automodule:: sympde.exterior.tests.test_datatype
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_datatype_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.exterior.tests.test_exterior.rst b/_sources/documentation/_autosummary/sympde.exterior.tests.test_exterior.rst
new file mode 100644
index 0000000..45e3e62
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.exterior.tests.test_exterior.rst
@@ -0,0 +1,31 @@
+sympde.exterior.tests.test\_exterior
+====================================
+
+.. automodule:: sympde.exterior.tests.test_exterior
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_exterior_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.exterior.tests.test_inference.rst b/_sources/documentation/_autosummary/sympde.exterior.tests.test_inference.rst
new file mode 100644
index 0000000..218f0df
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.exterior.tests.test_inference.rst
@@ -0,0 +1,31 @@
+sympde.exterior.tests.test\_inference
+=====================================
+
+.. automodule:: sympde.exterior.tests.test_inference
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_type_inference_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.printing.latex.rst b/_sources/documentation/_autosummary/sympde.printing.latex.rst
new file mode 100644
index 0000000..55841f2
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.printing.latex.rst
@@ -0,0 +1,6 @@
+sympde.printing.latex
+=====================
+
+.. currentmodule:: sympde.printing
+
+.. autofunction:: latex
\ No newline at end of file
diff --git a/_sources/documentation/_autosummary/sympde.printing.rst b/_sources/documentation/_autosummary/sympde.printing.rst
new file mode 100644
index 0000000..f4ebd8b
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.printing.rst
@@ -0,0 +1,32 @@
+sympde.printing
+===============
+
+.. automodule:: sympde.printing
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.printing.latex
+   sympde.printing.tests
+
diff --git a/_sources/documentation/_autosummary/sympde.printing.tests.rst b/_sources/documentation/_autosummary/sympde.printing.tests.rst
new file mode 100644
index 0000000..373e2db
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.printing.tests.rst
@@ -0,0 +1,31 @@
+sympde.printing.tests
+=====================
+
+.. automodule:: sympde.printing.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.printing.tests.test_latex
+
diff --git a/_sources/documentation/_autosummary/sympde.printing.tests.test_latex.rst b/_sources/documentation/_autosummary/sympde.printing.tests.test_latex.rst
new file mode 100644
index 0000000..f9d5e7b
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.printing.tests.test_latex.rst
@@ -0,0 +1,39 @@
+sympde.printing.tests.test\_latex
+=================================
+
+.. automodule:: sympde.printing.tests.test_latex
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_latex_1d
+      test_latex_2d_1
+      test_latex_2d_2
+      test_latex_2d_3
+      test_latex_2d_4
+      test_latex_2d_5
+      test_latex_3d_1
+      test_latex_3d_2
+      test_latex_ec_3d_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.analytical_mapping.rst b/_sources/documentation/_autosummary/sympde.topology.analytical_mapping.rst
new file mode 100644
index 0000000..0ab2963
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.analytical_mapping.rst
@@ -0,0 +1,39 @@
+sympde.topology.analytical\_mapping
+===================================
+
+.. automodule:: sympde.topology.analytical_mapping
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      AffineMapping
+      CollelaMapping2D
+      CzarnyMapping
+      IdentityMapping
+      PolarMapping
+      SphericalMapping
+      TargetMapping
+      TorusMapping
+      TorusSurfaceMapping
+      TwistedTargetMapping
+      TwistedTargetSurfaceMapping
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.basic.rst b/_sources/documentation/_autosummary/sympde.topology.basic.rst
new file mode 100644
index 0000000..b55284b
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.basic.rst
@@ -0,0 +1,39 @@
+sympde.topology.basic
+=====================
+
+.. automodule:: sympde.topology.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BasicDomain
+      Boundary
+      Connectivity
+      CornerBoundary
+      CornerInterface
+      Edge
+      Interface
+      InteriorDomain
+      Interval
+      ProductDomain
+      Union
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.callable_mapping.rst b/_sources/documentation/_autosummary/sympde.topology.callable_mapping.rst
new file mode 100644
index 0000000..14a2f1e
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.callable_mapping.rst
@@ -0,0 +1,29 @@
+sympde.topology.callable\_mapping
+=================================
+
+.. automodule:: sympde.topology.callable_mapping
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      CallableMapping
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.datatype.rst b/_sources/documentation/_autosummary/sympde.topology.datatype.rst
new file mode 100644
index 0000000..4fdf910
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.datatype.rst
@@ -0,0 +1,39 @@
+sympde.topology.datatype
+========================
+
+.. automodule:: sympde.topology.datatype
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      H1RegularityType
+      H1SpaceType
+      HcurlRegularityType
+      HcurlSpaceType
+      HdivRegularityType
+      HdivSpaceType
+      L2RegularityType
+      L2SpaceType
+      RegularityType
+      SpaceType
+      UndefinedSpaceType
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.derivatives.rst b/_sources/documentation/_autosummary/sympde.topology.derivatives.rst
new file mode 100644
index 0000000..b397c8d
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.derivatives.rst
@@ -0,0 +1,96 @@
+sympde.topology.derivatives
+===========================
+
+.. automodule:: sympde.topology.derivatives
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      find_partial_derivatives
+      get_atom_derivatives
+      get_atom_logical_derivatives
+      get_index_derivatives
+      get_index_derivatives_atom
+      get_index_logical_derivatives
+      get_index_logical_derivatives_atom
+      get_max_logical_partial_derivatives
+      get_max_partial_derivatives
+      get_number_derivatives
+      sort_partial_derivatives
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BracketBasic
+      Bracket_2d
+      CrossBasic
+      Cross_2d
+      Cross_3d
+      CurlBasic
+      Curl_2d
+      Curl_3d
+      DifferentialOperator
+      DivBasic
+      Div_1d
+      Div_2d
+      Div_3d
+      DotBasic
+      Dot_1d
+      Dot_2d
+      Dot_3d
+      GradBasic
+      Grad_1d
+      Grad_2d
+      Grad_3d
+      HessianBasic
+      Hessian_1d
+      Hessian_2d
+      Hessian_3d
+      LaplaceBasic
+      Laplace_1d
+      Laplace_2d
+      Laplace_3d
+      LogicalBracket_2d
+      LogicalCurl_2d
+      LogicalCurl_3d
+      LogicalDiv_1d
+      LogicalDiv_2d
+      LogicalDiv_3d
+      LogicalGrad_1d
+      LogicalGrad_2d
+      LogicalGrad_3d
+      LogicalHessian_1d
+      LogicalHessian_2d
+      LogicalHessian_3d
+      LogicalLaplace_1d
+      LogicalLaplace_2d
+      LogicalLaplace_3d
+      LogicalRot_2d
+      Rot_2d
+      dx
+      dx1
+      dx2
+      dx3
+      dy
+      dz
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.domain.rst b/_sources/documentation/_autosummary/sympde.topology.domain.rst
new file mode 100644
index 0000000..ea7928f
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.domain.rst
@@ -0,0 +1,52 @@
+sympde.topology.domain
+======================
+
+.. automodule:: sympde.topology.domain
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      split
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Area
+      BasicArea
+      BasicGeometryOperator
+      BoundaryVector
+      Cube
+      Domain
+      DomainArea
+      ElementArea
+      ElementDomain
+      Line
+      MinusNormalVector
+      NCube
+      NCubeInterior
+      NormalVector
+      PeriodicDomain
+      PlusNormalVector
+      Square
+      TangentVector
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.mapping.rst b/_sources/documentation/_autosummary/sympde.topology.mapping.rst
new file mode 100644
index 0000000..17f3c62
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.mapping.rst
@@ -0,0 +1,51 @@
+sympde.topology.mapping
+=======================
+
+.. automodule:: sympde.topology.mapping
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      cancel
+      get_logical_test_function
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BasicCallableMapping
+      Contravariant
+      Covariant
+      InterfaceMapping
+      InverseMapping
+      Jacobian
+      JacobianInverseSymbol
+      JacobianSymbol
+      LogicalExpr
+      MappedDomain
+      Mapping
+      MappingApplication
+      MultiPatchMapping
+      PullBack
+      SymbolicExpr
+      SymbolicWeightedVolume
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.measure.rst b/_sources/documentation/_autosummary/sympde.topology.measure.rst
new file mode 100644
index 0000000..6e57fb0
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.measure.rst
@@ -0,0 +1,32 @@
+sympde.topology.measure
+=======================
+
+.. automodule:: sympde.topology.measure
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BasicMeasure
+      CanonicalMeasure
+      CartesianMeasure
+      Measure
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.rst b/_sources/documentation/_autosummary/sympde.topology.rst
new file mode 100644
index 0000000..b4c4618
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.rst
@@ -0,0 +1,40 @@
+sympde.topology
+===============
+
+.. automodule:: sympde.topology
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.topology.analytical_mapping
+   sympde.topology.basic
+   sympde.topology.callable_mapping
+   sympde.topology.datatype
+   sympde.topology.derivatives
+   sympde.topology.domain
+   sympde.topology.mapping
+   sympde.topology.measure
+   sympde.topology.space
+   sympde.topology.tests
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.space.rst b/_sources/documentation/_autosummary/sympde.topology.space.rst
new file mode 100644
index 0000000..551b405
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.space.rst
@@ -0,0 +1,49 @@
+sympde.topology.space
+=====================
+
+.. automodule:: sympde.topology.space
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      TestFunction
+      element_of
+      elements_of
+      trace_0
+      trace_1
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BasicFunctionSpace
+      Derham
+      IndexedVectorFunction
+      ProductSpace
+      Projection
+      Projector
+      ScalarFunction
+      ScalarFunctionSpace
+      Trace
+      VectorFunction
+      VectorFunctionSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.tests.rst b/_sources/documentation/_autosummary/sympde.topology.tests.rst
new file mode 100644
index 0000000..50d9cfd
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.tests.rst
@@ -0,0 +1,39 @@
+sympde.topology.tests
+=====================
+
+.. automodule:: sympde.topology.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.topology.tests.test_callable_mapping
+   sympde.topology.tests.test_derivatives
+   sympde.topology.tests.test_field_2d
+   sympde.topology.tests.test_gallery
+   sympde.topology.tests.test_logical_expr
+   sympde.topology.tests.test_mapping
+   sympde.topology.tests.test_space
+   sympde.topology.tests.test_symbolic_expr
+   sympde.topology.tests.test_topology
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.tests.test_callable_mapping.rst b/_sources/documentation/_autosummary/sympde.topology.tests.test_callable_mapping.rst
new file mode 100644
index 0000000..fa4971b
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.tests.test_callable_mapping.rst
@@ -0,0 +1,43 @@
+sympde.topology.tests.test\_callable\_mapping
+=============================================
+
+.. automodule:: sympde.topology.tests.test_callable_mapping
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_affine_mapping_1d
+      test_affine_mapping_2d
+      test_affine_mapping_3d
+      test_affine_mapping_array_1d
+      test_affine_mapping_array_2d
+      test_affine_mapping_array_3d
+      test_identity_mapping_1d
+      test_identity_mapping_2d
+      test_identity_mapping_3d
+      test_identity_mapping_array_1d
+      test_identity_mapping_array_2d
+      test_identity_mapping_array_3d
+      test_polar_mapping
+      test_polar_mapping_array
+      test_user_defined_callable_mapping
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.tests.test_derivatives.rst b/_sources/documentation/_autosummary/sympde.topology.tests.test_derivatives.rst
new file mode 100644
index 0000000..3bba647
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.tests.test_derivatives.rst
@@ -0,0 +1,33 @@
+sympde.topology.tests.test\_derivatives
+=======================================
+
+.. automodule:: sympde.topology.tests.test_derivatives
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      indices_as_str
+      teardown_function
+      teardown_module
+      test_partial_derivatives_1
+      test_partial_derivatives_2
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.tests.test_field_2d.rst b/_sources/documentation/_autosummary/sympde.topology.tests.test_field_2d.rst
new file mode 100644
index 0000000..de021b0
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.tests.test_field_2d.rst
@@ -0,0 +1,31 @@
+sympde.topology.tests.test\_field\_2d
+=====================================
+
+.. automodule:: sympde.topology.tests.test_field_2d
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_field_2d_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.tests.test_gallery.rst b/_sources/documentation/_autosummary/sympde.topology.tests.test_gallery.rst
new file mode 100644
index 0000000..1b14fe6
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.tests.test_gallery.rst
@@ -0,0 +1,37 @@
+sympde.topology.tests.test\_gallery
+===================================
+
+.. automodule:: sympde.topology.tests.test_gallery
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_generic_line
+      test_interval
+      test_orthogonal_hexahedron
+      test_rectangle
+      test_unit_cube
+      test_unit_line
+      test_unit_square
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.tests.test_logical_expr.rst b/_sources/documentation/_autosummary/sympde.topology.tests.test_logical_expr.rst
new file mode 100644
index 0000000..3b223d1
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.tests.test_logical_expr.rst
@@ -0,0 +1,50 @@
+sympde.topology.tests.test\_logical\_expr
+=========================================
+
+.. automodule:: sympde.topology.tests.test_logical_expr
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_collela_mapping_2d_1
+      test_czarny_mapping_2d_1
+      test_identity_mapping_2d_1
+      test_identity_mapping_2d_2
+      test_logical_expr_1d_1
+      test_logical_expr_2d_1
+      test_logical_expr_2d_2
+      test_logical_expr_2d_3
+      test_logical_expr_3d_1
+      test_logical_expr_3d_2
+      test_logical_expr_3d_3
+      test_logical_expr_3d_4
+      test_logical_expr_3d_5
+      test_polar_mapping_2d_1
+      test_symbolic_expr_1d_1
+      test_symbolic_expr_2d_1
+      test_symbolic_expr_3d_1
+      test_target_mapping_2d_1
+      test_torus_mapping_3d_1
+      test_twisted_target_mapping_3d_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.tests.test_mapping.rst b/_sources/documentation/_autosummary/sympde.topology.tests.test_mapping.rst
new file mode 100644
index 0000000..d79b591
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.tests.test_mapping.rst
@@ -0,0 +1,34 @@
+sympde.topology.tests.test\_mapping
+===================================
+
+.. automodule:: sympde.topology.tests.test_mapping
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_mapping_1d
+      test_mapping_2d
+      test_mapping_2d_2
+      test_mapping_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.tests.test_space.rst b/_sources/documentation/_autosummary/sympde.topology.tests.test_space.rst
new file mode 100644
index 0000000..068ffe9
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.tests.test_space.rst
@@ -0,0 +1,35 @@
+sympde.topology.tests.test\_space
+=================================
+
+.. automodule:: sympde.topology.tests.test_space
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_projector_2d_1
+      test_space_1d_1
+      test_space_2d_1
+      test_space_2d_2
+      test_space_3d_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.rst b/_sources/documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.rst
new file mode 100644
index 0000000..fb9037b
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.rst
@@ -0,0 +1,32 @@
+sympde.topology.tests.test\_symbolic\_expr
+==========================================
+
+.. automodule:: sympde.topology.tests.test_symbolic_expr
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_derivatives_2d_with_mapping
+      test_derivatives_2d_without_mapping
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.topology.tests.test_topology.rst b/_sources/documentation/_autosummary/sympde.topology.tests.test_topology.rst
new file mode 100644
index 0000000..2216bce
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.topology.tests.test_topology.rst
@@ -0,0 +1,43 @@
+sympde.topology.tests.test\_topology
+====================================
+
+.. automodule:: sympde.topology.tests.test_topology
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      teardown_function
+      teardown_module
+      test_2d_domain_without_bnd
+      test_3d_domain_without_bnd
+      test_boundary_1
+      test_boundary_2
+      test_boundary_3
+      test_domain_1
+      test_domain_join_line
+      test_domain_join_square
+      test_element
+      test_get_subdomain
+      test_hash
+      test_interior_domain
+      test_topology_1
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.utilities.rst b/_sources/documentation/_autosummary/sympde.utilities.rst
new file mode 100644
index 0000000..56f23fa
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.utilities.rst
@@ -0,0 +1,32 @@
+sympde.utilities
+================
+
+.. automodule:: sympde.utilities
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.utilities.tests
+   sympde.utilities.utils
+
diff --git a/_sources/documentation/_autosummary/sympde.utilities.tests.rst b/_sources/documentation/_autosummary/sympde.utilities.tests.rst
new file mode 100644
index 0000000..ad48f44
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.utilities.tests.rst
@@ -0,0 +1,31 @@
+sympde.utilities.tests
+======================
+
+.. automodule:: sympde.utilities.tests
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
+.. rubric:: Modules
+
+.. autosummary::
+   :toctree:
+   :recursive:
+
+   sympde.utilities.tests.test_utilities
+
diff --git a/_sources/documentation/_autosummary/sympde.utilities.tests.test_utilities.rst b/_sources/documentation/_autosummary/sympde.utilities.tests.test_utilities.rst
new file mode 100644
index 0000000..fdee215
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.utilities.tests.test_utilities.rst
@@ -0,0 +1,30 @@
+sympde.utilities.tests.test\_utilities
+======================================
+
+.. automodule:: sympde.utilities.tests.test_utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      test_lambdify_sympde_1d
+      test_lambdify_sympde_2d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/_autosummary/sympde.utilities.utils.rst b/_sources/documentation/_autosummary/sympde.utilities.utils.rst
new file mode 100644
index 0000000..9373d71
--- /dev/null
+++ b/_sources/documentation/_autosummary/sympde.utilities.utils.rst
@@ -0,0 +1,34 @@
+sympde.utilities.utils
+======================
+
+.. automodule:: sympde.utilities.utils
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      lambdify_sympde
+      plot_2d
+      plot_2d_single_patch
+      plot_3d
+      plot_3d_single_patch
+      plot_domain
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/documentation/psydac.rst b/_sources/documentation/psydac.rst
new file mode 100644
index 0000000..dd90723
--- /dev/null
+++ b/_sources/documentation/psydac.rst
@@ -0,0 +1,19 @@
+Psydac API Reference
+====================
+
+.. autosummary::
+   :toctree: _autosummary
+   :recursive:
+
+   psydac.api       
+   psydac.cad
+   psydac.cmd
+   psydac.core
+   psydac.ddm
+   psydac.feec
+   psydac.fem
+   psydac.linalg
+   psydac.mapping
+   psydac.polar
+   psydac.pyccel
+   psydac.utilities
diff --git a/_sources/documentation/sympde.rst b/_sources/documentation/sympde.rst
new file mode 100644
index 0000000..3fc651f
--- /dev/null
+++ b/_sources/documentation/sympde.rst
@@ -0,0 +1,14 @@
+SymPDE API Reference
+====================
+
+.. autosummary::
+   :toctree: _autosummary
+   :recursive:
+
+   sympde.core
+   sympde.calculus 
+   sympde.topology
+   sympde.expr
+   sympde.printing
+   sympde.exterior
+   sympde.utilities
diff --git a/_sources/intro.md b/_sources/intro.md
new file mode 100644
index 0000000..f24f164
--- /dev/null
+++ b/_sources/intro.md
@@ -0,0 +1,16 @@
+# IGA-Python 
+
+Welcome to IGA-Python, a tutorial for Isogeometric Analysis using Python.
+
+## TODO
+
+- Subdomains
+- Geometry and Meshes
+- More models
+- Visualization
+- Documentation using docstrings from sympde and psydac
+- GLT 
+- DOcumentation using docstrings from gelato
+
+```{tableofcontents}
+```
diff --git a/_sources/psydac/README.md b/_sources/psydac/README.md
new file mode 100644
index 0000000..8a37e8d
--- /dev/null
+++ b/_sources/psydac/README.md
@@ -0,0 +1,163 @@
+# Welcome to PSYDAC
+
+[![devel_tests](https://github.com/pyccel/psydac/actions/workflows/continuous-integration.yml/badge.svg)](https://github.com/pyccel/psydac/actions/workflows/continuous-integration.yml) [![docs](https://github.com/pyccel/psydac/actions/workflows/documentation.yml/badge.svg)](https://github.com/pyccel/psydac/actions/workflows/documentation.yml)
+
+**PSYDAC** is a Python 3 Library for isogeometric analysis.
+
+## Table of contents
+
+-   [Requirements](#requirements)
+-   [Python setup and project download](#python-setup-and-project-download)
+-   [Installing the library](#installing-the-library)
+-   [Uninstall](#uninstall)
+-   [Running tests](#running-tests)
+-   [Speeding up Psydac's core](#speeding-up-psydacs-core)
+-   [User Documentation](#user-documentation)
+-   [Code Documentation](#code-documentation)
+
+## Requirements
+
+Psydac requires a certain number of components to be installed on the machine:
+
+-   Fortran and C compilers with OpenMP support
+-   OpenMP library
+-   BLAS and LAPACK libraries
+-   MPI library
+-   HDF5 library with MPI support
+
+The installations instructions depend on the operating system and on the packaging manager used.
+
+### Linux Debian-Ubuntu-Mint
+
+To install all requirements on a Linux Ubuntu operating system, just use APT, the Advanced Packaging Tool:
+```sh
+sudo apt update
+sudo apt install python3 python3-dev python3-pip
+sudo apt install gcc gfortran
+sudo apt install libblas-dev liblapack-dev
+sudo apt install libopenmpi-dev openmpi-bin
+sudo apt install libomp-dev libomp5
+sudo apt install libhdf5-openmpi-dev
+```
+
+### macOS
+
+To install all the requirements on a macOS operating system we recommend using [Homebrew](https://brew.sh/):
+
+```eh
+brew update
+brew install gcc
+brew install openblas
+brew install lapack
+brew install open-mpi
+brew install libomp
+brew install hdf5-mpi
+```
+
+### Other operating systems
+
+Please see the [instructions for the pyccel library](https://github.com/pyccel/pyccel#Requirements) for further details.
+
+## Python setup and project download
+
+We recommend creating a clean Python virtual environment using [venv](https://packaging.python.org/en/latest/guides/installing-using-pip-and-virtual-environments/#creating-a-virtual-environment):
+```sh
+python3 -m venv <ENV-PATH>
+```
+where `<ENV-PATH>` is the location to create the virtual environment.
+(A new directory will be created at the required location.)
+
+In order to activate the environment from a new terminal session just run the command
+```sh
+source <ENV-PATH>/bin/activate
+```
+
+One can clone the Psydac repository at any location `<ROOT-PATH>` in the filesystem which does not require administrator privileges, using either
+```sh
+git clone https://github.com/pyccel/psydac.git
+```
+or
+```sh
+git clone git@github.com:pyccel/psydac.git
+```
+The latter command requires a GitHub account.
+
+## Installing the library
+
+Psydac depends on several Python packages, which should be installed in the newly created virtual environment.
+These dependencies can be installed from the cloned directory `<ROOT-PATH>/psydac` using
+```sh
+export CC="mpicc"
+export HDF5_MPI="ON"
+export HDF5_DIR=/path/to/parallel/hdf5
+
+python3 -m pip install --upgrade pip
+python3 -m pip install -r requirements.txt
+python3 -m pip install -r requirements_extra.txt --no-build-isolation
+```
+where the `HDF5_DIR` environment variable should store the absolute path to the **installation point** of your parallel HDF5 library, which can be found with
+```sh
+h5pcc -showconfig
+```
+or (on macOS)
+```sh
+brew info hdf5-mpi
+```
+
+At this point the Psydac library may be installed in **standard mode**, which copies the relevant files to the correct locations of the virtual environment, or in **development mode**, which only installs symbolic links to the Psydac directory. The latter mode allows one to effect the behavior of Psydac by modifying the source files.
+
+-   **Standard mode**:
+    ```bash
+    python3 -m pip install .
+    ```
+
+-   **Development mode**:
+    ```bash
+    python3 -m pip install --editable .
+    ```
+
+## Uninstall
+
+-   **Whichever the install mode**:
+    ```bash
+    python3 -m pip uninstall psydac
+    ```
+
+## Running tests
+
+```bash
+export PSYDAC_MESH_DIR=/path/to/psydac/mesh/
+python3 -m pytest --pyargs psydac -m "not parallel"
+python3 /path/to/psydac/mpi_tester.py --pyargs psydac -m "parallel"
+```
+
+## Speeding up **Psydac**'s core
+
+Many of Psydac's low-level Python functions can be translated to a compiled language using the [Pyccel](https://github.com/pyccel/pyccel) transpiler. Currently, all of those functions are collected in modules which follow the name pattern `[module]_kernels.py`.
+
+The classical installation translates all kernel files to Fortran without user intervention. This does not happen in the case of an editable install, but the command `psydac-accelerate` is made available to the user instead. This command applies Pyccel to all the kernel files in the source directory, and the C language may be selected instead of Fortran (which is the default).
+
+-   **Only in development mode**:
+    ```bash
+    python3 /path/to/psydac/psydac_accelerate.py [--language LANGUAGE] [--openmp]
+    ```
+
+## User documentation
+
+-   [Output formats](./output.md)
+-   [Notebook examples](./examples/notebooks/)
+-   [Other examples](./examples/)
+
+## Code documentation
+
+Find our latest code documentation [here](https://pyccel.github.io/psydac/).
+
+## Mesh Generation
+
+After installation, a command `psydac-mesh` will be available.
+
+### Example of usage
+
+```bash
+psydac-mesh -n='16,16' -d='3,3' square mesh.h5
+```
diff --git a/_sources/psydac/TODO.rst b/_sources/psydac/TODO.rst
new file mode 100644
index 0000000..7119d57
--- /dev/null
+++ b/_sources/psydac/TODO.rst
@@ -0,0 +1,63 @@
+TODO
+====
+
+* mv psydac/feec/tests/todo_test_derivatives.py to psydac/feec/tests/test_derivatives.py once it is fixed
+
+* Add GMRES solver to 'psydac.linalg.solvers'
+
+* Add unit tests for Cart subcommunicators in 'psydac.ddm.cart'
+
+* Create parallel 'KroneckerLinearOperator' in 'psydac.linalg.kronecker' using Cart subcommunicators
+
+* Create object of type 'psydac.fem.psydacines.SplineSpace' with Cart
+
+* Extend functionality of 'SplineSpace' class:
+  . add (private?) methods 'init_fem' and 'init_collocation'
+  . call methods above when required
+  . add method 'compute_interpolant( self, values, field )'
+  . implement 'eval_field_gradient' method
+
+* Create 'SplineMapping' class in module 'psydac.mapping.discrete'
+
+* add a section in documentation about hdf5 installation (serial/parallel) + h5-tools (install+usage)
+
+
+Core
+****
+
+* interface to *psydac_eval_psydacines_ders*
+
+API
+***
+
+- reorganize code in api (no change to codgen)
+
+- additional tests and unit tests in api/tests
+
+- for the moment the codegen is not using the support from space:
+
+  * allow *fem_context* to be called with grid and degrees, and return a discrete space and a None for mapping.
+
+  * use support in codegen/interface
+
+- normal vector in 3d
+
+- periodic bc in codegen
+
+- init_fem is called everytime wa call discretize: maybe we should first do some checks (nderiv is ok norder etc)
+
+- hessian not yet implemented in sympde and api
+
+- add other solvers to the solver_driver (only cg is available now)
+
+- however, we need to clear the cache of sympy after 1d, 2d and 3d tests, otherwise pytest will crash.
+
+- add sympde and pyccel install procedure; maybe wget to download the requierements files as requirements_pyccel.txt etc then call pip3
+
+- api tests should be moved to examples
+
+- shall we add the Boundayr object as an attribut of SplineMapping? what to do if we don't use a mapping? we can also add the notion of a patch ... 
+
+- DiscreteModel is not up to date; changes need to be done because of the SumForm concept that was introduced lately. in fact, things will become even easier. a Model will be like a namespace or a Module.
+
+- add BoundaryDomain to define the whole boundary of a domain
diff --git a/_sources/psydac/doc/abstract.rst b/_sources/psydac/doc/abstract.rst
new file mode 100644
index 0000000..6392387
--- /dev/null
+++ b/_sources/psydac/doc/abstract.rst
@@ -0,0 +1,9 @@
+**PSYDAC** is a Python/Fortran 2003 library for B-Splines/NURBS and Computer Aided Design Algorithms.
+
+**PSYDAC** can be used in three different ways:
+
+1. *Fortran 90/95* subroutines through the file **fortran/src/bsplines/bspline.F90**
+
+2. *Fortran 2003* objects, mainly the **mapping** and **cad** objects
+
+3. The same objects as in *2.* but through *Python*
diff --git a/_sources/psydac/doc/api-fortran/spl.rst b/_sources/psydac/doc/api-fortran/spl.rst
new file mode 100644
index 0000000..1333ed7
--- /dev/null
+++ b/_sources/psydac/doc/api-fortran/spl.rst
@@ -0,0 +1 @@
+TODO
diff --git a/_sources/psydac/doc/api.rst b/_sources/psydac/doc/api.rst
new file mode 100644
index 0000000..6f80e78
--- /dev/null
+++ b/_sources/psydac/doc/api.rst
@@ -0,0 +1,19 @@
+API
+===
+
+You will find here both the *Fortran* **doxygen** documentation as well as the *Python-API*.
+
+Fortran API
+***********
+
+.. raw:: html
+
+  <a href="_static/html_doxygen/index.html" target="_blank">Doxygen</a>
+
+Python API
+**********
+
+.. toctree::
+   :maxdepth: 4
+
+   api-python/modules
diff --git a/_sources/psydac/doc/dive/index.rst b/_sources/psydac/doc/dive/index.rst
new file mode 100644
index 0000000..1bb1e35
--- /dev/null
+++ b/_sources/psydac/doc/dive/index.rst
@@ -0,0 +1,10 @@
+Dive into PSYDAC
+================
+
+Contents
+********
+
+.. toctree::
+
+  introduction
+  io
diff --git a/_sources/psydac/doc/dive/introduction.rst b/_sources/psydac/doc/dive/introduction.rst
new file mode 100644
index 0000000..8bb81aa
--- /dev/null
+++ b/_sources/psydac/doc/dive/introduction.rst
@@ -0,0 +1,4 @@
+Introduction
+************
+
+
diff --git a/_sources/psydac/doc/dive/io.rst b/_sources/psydac/doc/dive/io.rst
new file mode 100644
index 0000000..77d1a9c
--- /dev/null
+++ b/_sources/psydac/doc/dive/io.rst
@@ -0,0 +1,2 @@
+Input and Output
+****************
diff --git a/_sources/psydac/doc/index.rst b/_sources/psydac/doc/index.rst
new file mode 100644
index 0000000..39b85a4
--- /dev/null
+++ b/_sources/psydac/doc/index.rst
@@ -0,0 +1,27 @@
+.. spl documentation master file, created by
+   sphinx-quickstart on Fri Feb 16 16:21:17 2018.
+   You can adapt this file completely to your liking, but it should at least
+   contain the root `toctree` directive.
+
+Welcome to spl's documentation!
+===============================
+
+.. include:: abstract.rst
+
+.. toctree::
+   :maxdepth: 1
+   :caption: Contents:
+
+   tutorial
+   dive/index
+
+   math/index
+
+   api
+
+Indices and tables
+==================
+
+* :ref:`genindex`
+* :ref:`modindex`
+* :ref:`search`
diff --git a/_sources/psydac/doc/math/derham.rst b/_sources/psydac/doc/math/derham.rst
new file mode 100644
index 0000000..be138a7
--- /dev/null
+++ b/_sources/psydac/doc/math/derham.rst
@@ -0,0 +1,341 @@
+DeRham sequences
+****************
+
+.. sectionauthor:: A. Ratnani
+
+here without boundary conditions
+
+.. math::
+
+  \mathbb{R} \hookrightarrow \Hgrad  \xrightarrow{\quad \Grad \quad}  \Hcurl  \xrightarrow{\quad \Curl \quad}   \Hdiv  \xrightarrow{\quad \Div \quad}  \Ltwo  \xrightarrow{\quad} 0  
+
+Pullbacks
+^^^^^^^^^
+
+In the case where the physical domain :math:`\Omega := \mathcal{F}(\hat{\Omega})` is the *image* of a *logical* domain :math:`\hat{\Omega}` by a smooth mapping :math:`\mathcal{F}` (at least :math:`\mathcal{C}^1`), we have the following *parallel* diagrams 
+
+.. math::
+
+  \begin{array}{ccccccc}
+  \Hgrad & \xrightarrow{\quad \Grad \quad} & \Hcurl & \xrightarrow{\quad \Curl \quad} &  \Hdiv & \xrightarrow{\quad \Div \quad} & \Ltwo \\  
+  \igrad \Bigg\uparrow   &     & \icurl \Bigg\uparrow  &   & \idiv \Bigg\uparrow &  & \iltwo \Bigg\uparrow       \\
+  \HgradLogical & \xrightarrow{\quad \Grad \quad} & \HcurlLogical & \xrightarrow{\quad \Curl \quad} &  \HdivLogical & \xrightarrow{\quad \Div \quad} & \LtwoLogical \\  
+  %
+  \end{array}
+
+Where the *mappings* :math:`\igrad, \icurl, \idiv` and :math:`\iltwo` are called **pullbacks** and are given by
+
+.. math::
+
+  \phi (x) :=& \igrad \hat{\phi} (\hat{x}) = \hat{\phi}(\mathcal{F}^{-1}(x)) 
+  \\      
+  \Psi (x) :=& \icurl \hat{\Psi} (\hat{x}) = \left( D \mathcal{F} \right)^{-T} \hat{\Psi}(\mathcal{F}^{-1}(x)) 
+  \\     
+  \Phi (x) :=& \idiv \hat{\Phi} (\hat{x})  = \frac{1}{J} D \mathcal{F} \hat{\Phi}(\mathcal{F}^{-1}(x)) 
+  \\ 
+  \rho (x) :=& \iltwo \hat{\rho} (\hat{x}) = \hat{\rho}(\mathcal{F}^{-1}(x)) 
+
+where :math:`D \mathcal{F}` is the **jacobian matrix** of the mapping :math:`\mathcal{F}`. 
+
+.. note:: The *pullbacks* :math:`\igrad, \icurl, \idiv` and :math:`\iltwo` are **isomorphisms** between the corresponding spaces. 
+
+Discrete Spaces
+^^^^^^^^^^^^^^^
+
+Let us suppose that we have a sequence of finite subspaces for each of the spaces involved in the DeRham sequence. The discrete DeRham sequence stands for the following commutative diagram between continuous and discrete spaces
+
+.. math::
+
+  \begin{array}{ccccccc}
+  \Hgrad & \xrightarrow{\quad \Grad \quad} & \Hcurl & \xrightarrow{\quad \Curl \quad} &  \Hdiv & \xrightarrow{\quad \Div \quad} & \Ltwo \\  
+  \Pigrad \Bigg\downarrow   &     & \Picurl \Bigg\downarrow  &   & \Pidiv \Bigg\downarrow &  & \Piltwo \Bigg\downarrow       \\
+  \Vgrad & \xrightarrow{\quad \Grad \quad} & \Vcurl & \xrightarrow{\quad \Curl \quad}  & \Vdiv & \xrightarrow{\quad \Div \quad} & \Vltwo    \\
+  %
+  \end{array}
+
+When using a Finite Elements methods, we often deal with a reference element, and thus we need also to apply the *pullbacks* on the discrete spaces. In fact, we have again the following parallel diagram
+
+.. math::
+
+  \begin{array}{ccccccc}
+  \Vgrad & \xrightarrow{\quad \Grad \quad} & \Vcurl & \xrightarrow{\quad \Curl \quad} &  \Vdiv & \xrightarrow{\quad \Div \quad} & \Vltwo \\  
+  \igrad \Bigg\uparrow   &     & \icurl \Bigg\uparrow  &   & \idiv \Bigg\uparrow &  & \iltwo \Bigg\uparrow       \\
+  \VgradLogical & \xrightarrow{\quad \Grad \quad} & \VcurlLogical & \xrightarrow{\quad \Curl \quad} &  \VdivLogical & \xrightarrow{\quad \Div \quad} & \VltwoLogical \\  
+  %
+  \end{array}
+
+Since, the *pullbacks* are **isomorphisms** in the previous diagram, we can define a **one-to-one** correspondance 
+
+.. math::
+
+  \phi :=& \igrad \hat{\phi}, \quad \phi \in \Vgrad, \hat{\phi} \in \VgradLogical 
+  \\
+  \Psi :=& \icurl \hat{\Psi}, \quad \Psi \in \Vcurl, \hat{\Psi} \in \VcurlLogical 
+  \\
+  \Phi :=& \idiv \hat{\Phi}, \quad \Phi \in \Vdiv, \hat{\Phi} \in \VdivLogical 
+  \\
+  \rho :=& \iltwo \hat{\rho}, \quad \rho \in \Vltwo, \hat{\rho} \in \VltwoLogical 
+
+We have then, the following results
+
+.. math::
+
+  \Grad \phi =& \icurl \Grad \hat{\phi} , \quad \phi \in \Vgrad
+  \\
+  \Curl \Psi =& \idiv \Curl \hat{\Psi} , \quad \Psi \in \Vcurl
+  \\
+  \Div \Phi =& \iltwo \Div \hat{\Phi} , \quad \Phi \in \Vdiv
+
+
+
+Projectors
+^^^^^^^^^^
+
+In some cases, one may need to define projectors on *smooth* functions
+
+.. math::
+
+  \begin{array}{ccccccc}
+  \Cinfinity & \xrightarrow{\quad \Grad \quad} & \Cinfinity & \xrightarrow{\quad \Curl \quad}  & \Cinfinity & \xrightarrow{\quad \Div \quad} & \Cinfinity    \\
+  \Pigrad \Bigg\downarrow   &     & \Picurl \Bigg\downarrow  &   & \Pidiv \Bigg\downarrow &  & \Piltwo \Bigg\downarrow       \\
+  \Vgrad & \xrightarrow{\quad \Grad \quad} & \Vcurl & \xrightarrow{\quad \Curl \quad}  & \Vdiv & \xrightarrow{\quad \Div \quad} & \Vltwo    \\
+  \end{array}
+
+
+Discrete DeRham sequence for B-Splines
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+
+Buffa et al :cite:`Buffa2009a` show the construction of a discrete DeRham sequence using B-Splines, (here without boundary conditions)
+
+.. math::
+
+  \begin{array}{ccccccc}
+  \Hgrad & \xrightarrow{\quad \Grad \quad} & \Hcurl & \xrightarrow{\quad \Curl \quad}  & \Hdiv & \xrightarrow{\quad \Div \quad} & \Ltwo    \\
+  \Pigrad \Bigg\downarrow   &     & \Picurl \Bigg\downarrow  &   & \Pidiv \Bigg\downarrow &  & \Piltwo \Bigg\downarrow       \\
+  \Vgradspline &  \xrightarrow{\quad \Grad \quad}&   \Vcurlspline &  \xrightarrow{\quad \Curl \quad}  &  \Vdivspline&   \xrightarrow{\quad \Div \quad} &  \Vltwospline \\ 
+  \end{array}
+
+1d case
+_______
+
+1. DeRham sequence is reduced to
+
+.. math::
+
+      \mathbb{R} \hookrightarrow 
+      \underbrace{\mathcal{S}^{p}}_{\VgradLogical}  \xrightarrow{\quad \Grad \quad}  
+      \underbrace{\mathcal{S}^{p-1}}_{\VltwoLogical}  \xrightarrow{\quad} 0  
+
+2. The recursion formula for derivative writes 
+
+.. math::
+
+      {N_i^p}'(t)=D_i^{p}(t)-D_{i+1}^{p}(t)
+      \quad \mbox{where} \quad
+      D_{i}^{p}(t) = \frac{p}{t_{i+p+1}-t_i}N_i^{p-1}(t) 
+
+3. we have :math:`\mathcal{S}^{p-1} = \mathbf{span}\{ N_i^{p-1}, 1 \leq i \leq n-1 \} = \mathbf{span}\{ D_i^p, 1 \leq i \leq n-1 \}` which is a change of basis as a diagonal matrix
+
+4. Now if :math:`u \in S^p`, with and expansion :math:`u = \sum_i u_i N_i^p`, we have
+
+.. math::
+
+    \begin{align*}
+    u^{\prime} = \sum_i u_i \left( N_i^p \right)^{\prime} = \sum_i (-u_{i-1} + u_i) D_i^p 
+  %  \label{}
+    \end{align*}
+
+5. If we introduce the B-Splines coefficients vector :math:`\mathbf{u} := \left( u_i \right)_{1 \leq i \leq n}` (and :math:`\mathbf{u}^{\star}` for the derivative), we have
+
+.. math::
+
+  \mathbf{u}^{\star} = D \mathbf{u}    
+
+where :math:`D` is the incidence matrix (of entries :math:`-1` and :math:`+1`)
+
+.. topic:: Discrete derivatives:
+
+  .. math::
+
+        \mathcal{G} = D 
+
+.. .. math::
+.. 
+..   \begin{array}{ccccccc}
+..   \Hgrad & \xrightarrow{\quad \Grad \quad} & \Hcurl & \xrightarrow{\quad \Curl \quad} &  \Hdiv & \xrightarrow{\quad \Div \quad} & \Ltwo \\  
+..   \Pigrad \Bigg\downarrow   &     & \Picurl \Bigg\downarrow  &   & \Pidiv \Bigg\downarrow &  & \Piltwo \Bigg\downarrow       \\
+..     \Vgrad & \xrightleftharpoons[\quad \mathcal{G}^{T} \quad]{\quad \mathcal{G} \quad} & \Vcurl & \xrightleftharpoons[\quad \mathcal{C}^{T} \quad]{\quad \mathcal{C} \quad}  & \Vdiv & \xrightleftharpoons[\quad \mathcal{D}^{T} \quad]{\quad \mathcal{D} \quad} & \Vltwo    \\
+..   %
+..   \end{array}
+
+2d case
+_______
+
+In *2d*, the are two De-Rham complexes:
+
+.. math::
+
+  \begin{array}{ccccc}
+  \Hgrad & \xrightarrow{\quad \Grad \quad} & \Hcurl & \xrightarrow{\quad \Rots \quad} & \Ltwo \\
+  \Pigrad \Bigg\downarrow   &     & \Picurl \Bigg\downarrow  &   & \Piltwo \Bigg\downarrow   \\
+  \Vgrad & \xrightarrow{\quad \Grad \quad} & \Vcurl & \xrightarrow{\quad \Rots \quad} & \Vltwo \\
+  \end{array}
+
+and
+
+.. math::
+
+  \begin{array}{ccccc}
+  \Hgrad & \xrightarrow{\quad \Curl \quad} & \Hdiv & \xrightarrow{\quad \Div \quad} & \Ltwo \\
+  \Pigrad \Bigg\downarrow   &     & \Pidiv \Bigg\downarrow  &   & \Piltwo \Bigg\downarrow   \\
+  \Vgrad & \xrightarrow{\quad \Grad \quad} & \Vdiv & \xrightarrow{\quad \Div \quad} & \Vltwo \\
+  \end{array}
+
+.. where
+.. 
+.. .. math::
+.. 
+..   \Vgrad = \Vgradspline2d, \quad \Vcurl = \Vcurlspline2d, \quad \Vdiv = \Vdivspline2d, \quad  \mbox{and} \quad \Vltwo = \Vltwospline2d.
+
+Let :math:`I` be the identity matrix, we have
+
+.. topic:: Discrete derivatives:
+
+  .. math::
+
+        \mathcal{G} = 
+        \begin{pmatrix}
+          D \otimes I
+          \\
+          I \otimes D
+        \end{pmatrix}
+
+  .. math::
+
+        \mathcal{C} = 
+        \begin{pmatrix}
+          I \otimes D
+          \\
+        - D \otimes I
+        \end{pmatrix} 
+        \quad \mbox{[scalar curl],} \quad 
+        \mathcal{C} = 
+        \begin{pmatrix}
+        - I \otimes D
+         & 
+          D \otimes I
+        \end{pmatrix} 
+        \quad \mbox{[vectorial curl]}
+
+  .. math::
+
+        \mathcal{D} = 
+        \begin{pmatrix}
+          D \otimes I
+         & 
+          I \otimes D
+        \end{pmatrix}
+
+3d case
+_______
+
+.. topic:: Discrete derivatives:
+
+  .. math::
+
+        \mathcal{G} = 
+        \begin{pmatrix}
+          D \otimes I \otimes I
+          \\
+          I \otimes D \otimes I 
+          \\
+          I \otimes I \otimes D
+        \end{pmatrix}
+
+  .. math::
+
+        \mathcal{C} = 
+        \begin{pmatrix}
+          0    &    - I \otimes I \otimes D     &     I \otimes D \otimes I 
+          \\
+          I \otimes I \otimes D   &    0   &   - D \otimes I \otimes I 
+          \\
+          - I \otimes D \otimes I  & D \otimes I \otimes I & 0 
+        \end{pmatrix} 
+
+  .. math::
+
+        \mathcal{D} = 
+        \begin{pmatrix}
+          D \otimes I \otimes I
+          & 
+          I \otimes D \otimes I 
+          &
+          I \otimes I \otimes D
+        \end{pmatrix}
+
+.. note:: From now on, we will denote the discrete derivative by :math:`\mathbb{D}_k` for the one going from :math:`V_k` to :math:`V_{k+1}`.
+
+Algebraic identities
+^^^^^^^^^^^^^^^^^^^^
+
+Let us consider the discretization of the exterior derivative
+
+.. math::
+
+  \omega^{k+1} = \diff  \omega^k
+
+multiplying by a test function :math:`\eta^{k+1}` and integrating over the whole computation domain, we get
+
+.. math::
+
+  \left( \eta^{k+1}, \omega^{k+1} \right)_{k+1} = \left( \eta^{k+1}, \diff \omega^{k} \right)_{k+1}
+
+let :math:`E^{k+1}`, :math:`W^{k}` and :math:`W^{k+1}` be the vector representation of :math:`\eta^{k+1}`, :math:`\omega^{k}` and  :math:`\omega^{k+1}`. We get
+
+.. math::
+
+  {E^{k+1}}^T M_{k+1} W^{k+1} = {E^{k+1}}^T D_{k+1,k} W^{k}  
+
+where 
+
+.. math::
+
+  D_{k+1,k} = \left( \left( \eta^{k+1}_i, \diff \omega^{k}_j \right)_{k+1} \right)_{i,j}
+
+On the other hand, using the coderivative, we get 
+
+.. math::
+
+  \left( \eta^{k+1}, \omega^{k+1} \right)_{k+1} = \left( \delta \eta^{k+1}, \omega^{k} \right)_{k} + BC
+
+Let us now introduce the following matrix
+
+.. math::
+
+  D_{k,k+1} = \left( \left( \delta \eta^{k+1}_i, \omega^{k}_j \right)_{k} \right)_{i,j}
+
+hence,
+
+.. math::
+
+  {E^{k+1}}^T D_{k,k+1} W^{k} = \left( \mathbb{D}^{\star}_{k+1} E^{k+1} \right)^T M_{k} W^{k} 
+
+
+Therefor, we have the following important result
+
+.. topic:: Proposition:
+
+  * :math:`D_{k+1,k} = D_{k,k+1} + BC`
+
+  * :math:`D_{k+1,k} = M_{k+1} \mathbb{D}^T_k`
+
+  * :math:`D_{k,k+1} = {\mathbb{D}^{\star}_{k+1}}^T M_{k}`
+
+
+
+.. rubric:: References
+
+.. bibliography:: refs_derham.bib
+   :cited:
diff --git a/_sources/psydac/doc/math/feec.rst b/_sources/psydac/doc/math/feec.rst
new file mode 100644
index 0000000..61d01fb
--- /dev/null
+++ b/_sources/psydac/doc/math/feec.rst
@@ -0,0 +1,689 @@
+Exterior Algebra 
+****************
+
+.. sectionauthor:: A. Ratnani
+
+Let :math:`V` be a real vector space of dimension :math:`n`. 
+
+.. topic:: Definition, Alternating algebraic forms:
+
+  For each :math:`k`, we define :math:`\Alt^k V` as the space of alternating :math:`k`-linear maps :math:`V \times \cdots \times V \rightarrow \mathbb{R}`.
+
+.. note:: * :math:`\Alt^0 = \mathbb{R}`,
+          * :math:`\Alt^1 = V^{*}` is the dual space of :math:`V` (the space of covectors)
+
+
+.. topic:: Definition, Exterior product:
+
+  For :math:`\omega \in \Alt^j` and :math:`\eta \in \Alt^k`, their exterior (wedge) product is given by:
+
+  .. math::
+
+    (\omega \wedge \eta ) (v_1, \cdots, v_{j+k}) = \sum_{\sigma} (\mathrm{sign}~ \sigma) 
+    \omega (v_{\sigma(1)}, \cdots, v_{\sigma(j)}) 
+    \eta (v_{\sigma(j+1)}, \cdots, v_{\sigma(j+k)})
+
+  for all :math:`v_i \in V`. Where the sum is over all permutations :math:`\sigma` of :math:`\{ 1,\cdots,j+k \}`, 
+  for which :math:`\sigma(1)< \cdots <\sigma(j)` and :math:`\sigma(j+1)< \cdots <\sigma(j+k)`.
+
+.. note:: * The exterior product is **bilinear**, **associative**,
+          * **anti-commutative**: :math:`\eta \wedge \omega = (-1)^{jk} \omega \wedge \eta` for all :math:`\omega \in \Alt^j` and :math:`\eta \in \Alt^k`.
+
+
+.. topic:: Definition, Grassmann Algebra:
+
+  Grassmann Algebra is defined by:
+
+  .. math::
+
+    \Alt V := \bigoplus_k \Alt^k V
+
+  This is a **anti-commutative graded algebra**. Also called **Exterior Algebra** of :math:`V^{*}`
+
+In the case of :math:`V=\mathbb{R}^n`, we have:
+
+* :math:`\Alt V^0 \sim \mathbb{R}`,
+* :math:`\Alt V^1 \sim \mathbb{R}^n`,
+* :math:`\Alt V^{n-1} \sim \mathbb{R}^n`, using Riesz representation theorem,
+* :math:`\Alt V^n \sim \mathbb{R}`, using the map :math:`v \longmapsto \det(v,v_1,\cdots,v_{n-1})`.
+
+
+Basis
+^^^^^
+
+Let :math:`v_1,\cdots,v_n` be a basis of :math:`V` and :math:`\mu_1,\cdots,\mu_n` the associated dual basis for :math:`V^*` (:math:`\mu_i(v_j) = \delta_{ij}`). 
+
+For any increasing permutations :math:`\sigma, \rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}`, we have:
+
+.. math::
+
+  \mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)} (v_{\rho(1)}, \cdots, v_{\rho(k)}) = \chi_{\sigma,\rho}
+
+thus the :math:`\binom {n}{k}` algebraic :math:`k`-forms :math:`\mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)}`, 
+form a basis for :math:`\Alt^k V` and :math:`\dim \Alt^k V = \binom {n}{k}`.
+
+
+.. topic:: Definition, Interior product:
+
+  Let :math:`\omega` be a :math:`k`-form, and :math:`v \in V`. The **interior product** of :math:`\omega` and :math:`v` is the :math:`(k-1)`-form  :math:`\omega \lrcorner v` defined by:
+
+  .. math::
+    \omega \lrcorner v (v_1,\cdots,v_{k-1}) = \omega (v,v_1,\cdots,v_{k-1})
+
+
+* We have for :math:`\omega \in \Alt^k V`, :math:`\eta \in \Alt^l V` and :math:`v \in V`: 
+
+.. math::
+
+  (\omega \wedge \eta) \lrcorner v = (\omega \lrcorner v)\wedge \eta + (-1)^k \omega \wedge (\eta \lrcorner v)
+
+
+.. topic:: Definition, Inner product:
+
+  If :math:`V` is has an inner product, then :math:`\Alt^k V` is endowed with an inner product given by:
+
+  .. math::
+
+    (\omega , \eta) = \sum_{\rho} \omega (v_{\rho(1)}, \cdots, v_{\rho(k)}) \eta (v_{\rho(1)}, \cdots, v_{\rho(k)}), ~~~\forall \omega, \eta \in \Alt^k V. 
+
+  where the sum is over increasing sequences :math:`\rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}`, and :math:`v_1, \cdots,v_n` is any orthonormal basis.
+
+
+Orientation and Volume form
+^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+.. todo:: add Orientation and Volume form 
+
+
+
+.. topic:: Definition, Pullback:
+
+  A linear transformation of vector spaces :math:`L: V \rightarrow W` induces a transformation 
+  :math:`L^{*}: \Alt W \rightarrow \Alt V`, called the **pullback**, and given by:
+
+  .. math::
+
+    L^{*} \omega (v_{1}, \cdots, v_{k}) = \omega (L v_{1}, \cdots, L v_{k}),~~~~~ \forall \omega \in \Alt^k W,~~~ v_{1}, \cdots, v_{k} \in V
+
+
+* The pullback acts **contravariantly**: if :math:`U \xrightarrow{~K~} V \xrightarrow{~L~} W` then,
+
+  .. math::
+
+    \Alt W \xrightarrow{~K^{*}~} \Alt V \xrightarrow{~L^{*}~} \Alt U 
+
+* :math:`L^{*} (\omega \wedge \eta) = L^{*} \omega \wedge L^{*} \eta`
+
+
+Let V be a subspace of W. For the inclusion :math:`\imath_V : V \longrightarrow W`, we can define its pullback :math:`\imath_V^{*}`: 
+this is a **surjection** of :math:`\Alt W` onto :math:`\Alt V`.
+
+If W has an inner product and :math:`\pi_V : W \longrightarrow V` is the orthogonal projection. We can define its pullback :math:`\pi_V^{*}` : 
+this an **injection** of :math:`\Alt V` onto :math:`\Alt W`.
+
+Let us consider the composition :  :math:`W` \shortstack{:math:`\pi_V` \\ :math:`\longrightarrow`} :math:`V` \shortstack{:math:`\imath_V` \\ :math:`\longrightarrow`} :math:`W`, and its pullback :math:`\pi_V^* \imath_V^*`.
+
+
+.. topic:: Definition, The tangential and normal parts:
+
+  * :math:`\pi_V^* \imath_V^*` associates for each :math:`\omega \in \Alt^k` its **tangential** part :math:`\omega_{\parallel}` with respect to :math:`V` :
+
+  .. math::
+
+    (\pi_V^* \imath_V^* \omega) (v_1,\cdots,v_k) = \omega (\pi_V v_1, \cdots, \pi_V v_k), ~~~~~\forall v_1,\cdots,v_k \in W.
+
+  * :math:`\omega - \pi_V^* \imath_V^* \omega` associates for each :math:`\omega \in \Alt^k` its **normal** part :math:`\omega_{\perp}` with respect to :math:`V`.
+
+
+The **tangential part** of :math:`\omega` vanishes if and only if the image of :math:`\omega` in :math:`\Alt^k V` vanishes.
+
+Let :math:`V` be an oriented inner product space, with volume form :math:`\mbox{vol}`. Let :math:`\omega \in \Alt^k V`. 
+We can define a new linear map :math:`L_{\omega}` as the composition of :math:`\Alt^{n-k} V \longrightarrow \Alt^n V` such as:
+
+.. math::
+
+  \mu \longmapsto \omega \wedge \mu
+
+and the canonical isomorphism of :math:`\Alt^n V` onto :math:`\mathbb{R}`, and using the Riesz representation theorem, 
+there exists an element :math:`\star \omega \in \Alt^{n-k} V` such that : :math:`L_{\omega} (\mu) = (\star \omega , \mu)`, *i.e.*:
+
+.. math::
+
+  \omega \wedge \mu = (\star \omega , \mu) \mbox{vol}, ~~~\omega \in \Alt^{k}, ~\mu \in \Alt^{n-k}
+
+
+.. topic:: Definition, The Hodge star operation:
+
+  The linear map which maps :math:`\Alt^k V` onto :math:`\Alt^{n-k} V` :math:`\omega \longmapsto \star \omega` is called the **Hodge star** operator.
+
+
+* If :math:`e_1,\cdots,e_n` is any positively oriented orthonormal basis, and :math:`\sigma` a permutation, we have
+
+.. math::
+
+  \omega(e_{\sigma(1)}, \cdots, e_{\sigma(k)}) = (\mathrm{sign} \sigma) \star \omega(e_{\sigma(k+1)}, \cdots, e_{\sigma(n)})
+
+* :math:`\star \star \omega = (-1)^{k(n-k)} \omega, ~~~\forall \omega \in \Alt^k V`, thus the Hodge star is an **isometry**.
+* :math:`(\star \omega)_{\parallel} = \star (\omega_{\perp})` and :math:`(\star \omega)_{\perp} = \star (\omega_{\parallel})`
+* the image of :math:`\star \omega` in :math:`\Alt^k V` vanishes if and only if :math:`\omega_{\perp}` vanishes.
+
+
+.. math::
+
+  \begin{tabular}{|c|l|}
+    \hline
+   $\Alt^0 \mathbb{R}^3 \cong \mathbb{R}$ &  $c \leftrightarrow c$ \\
+  %   \hline
+   $\Alt^1 \mathbb{R}^3 \cong \mathbb{R}^3$ & $u_1 \diff x_1 + u_2 \diff x_2 + u_3 \diff x_3 \leftrightarrow u$ \\
+  %    \hline
+   $\Alt^2 \mathbb{R}^3 \cong \mathbb{R}^3$ & $u_3 \diff x_1 \wedge \diff x_2 - u_2 \diff x_1 \wedge \diff x_3 + u_1 \diff x_2 \wedge \diff x_3 +  \leftrightarrow u$ \\
+  %   \hline
+   $\Alt^3 \mathbb{R}^3 \cong \mathbb{R}$ & $c \diff x_1 \wedge \diff x_2 \wedge \diff x_3 \leftrightarrow c$  \\
+    \hline
+  \end{tabular}
+..   \caption{Correspondence}
+
+.. math::
+
+  \begin{tabular}{|c|l|}
+    \hline
+   $ \wedge : \Alt^1 \mathbb{R}^3 \times \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3$ 
+  &  $\times : \mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}^3$ 
+  \\
+   $ \wedge : \Alt^1 \mathbb{R}^3 \times \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3$ 
+  &  $\cdot : \mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}$  
+  \\
+    \hline
+  \end{tabular}
+..   \caption{Exterior product}
+
+.. math::
+
+  \begin{tabular}{|c|l|}
+    \hline
+   $ L^* : \Alt^0 \mathbb{R}^3 \longrightarrow \Alt^0 \mathbb{R}^3 $ & $\id : \mathbb{R} \longrightarrow \mathbb{R}$
+  \\
+   $ L^* : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^1 \mathbb{R}^3 $ & $L^T : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$
+  \\
+   $ L^* : \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $(\det L )L^{-1} : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$
+  \\
+   $ L^* : \Alt^3 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3 $ & $(\det L) : \mathbb{R} \longrightarrow \mathbb{R}$ ~~~($c \longmapsto c \det L$)
+  \\
+    \hline
+  \end{tabular}
+..   \caption{Pullback by a linear map L :$\mathbb{R}^3 \longrightarrow \mathbb{R}^3$}
+
+.. math::
+
+  \begin{tabular}{|c|l|}
+    \hline
+   $ \lrcorner v : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^0 \mathbb{R}^3 $ & $v \cdot : \mathbb{R}^3 \longrightarrow \mathbb{R}$
+  \\
+   $ \lrcorner v : \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^1 \mathbb{R}^3 $ & $v \times : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$
+  \\
+   $ \lrcorner v : \Alt^3 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $v : \mathbb{R} \longrightarrow \mathbb{R}^3$ ~~~($c \longmapsto c v$)
+  \\
+    \hline
+  \end{tabular}
+..   \caption{Interior product with a vector $v \in \mathbb{R}^3$}
+
+.. math::
+
+  \begin{tabular}{|c|l|}
+    \hline
+   inner product on $\Alt^k \mathbb{R}^3$ induced  & dot product on $\mathbb{R}$ and $\mathbb{R}^3$
+  \\
+   by dot product on $\mathbb{R}^3$ & 
+  \\
+   $\volume = \diff x_1 \wedge \diff x_2 \wedge \diff x_3$ & $(v_1,v_2,v_3) \longmapsto \det(v_1|v_2|v_3)$
+  \\
+    \hline
+  \end{tabular}
+..   \caption{Inner product and volume form}
+
+.. math::
+
+  \begin{tabular}{|c|l|}
+    \hline
+   $ \star : \Alt^0 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3 $ & $\id : \mathbb{R} \longrightarrow \mathbb{R}$
+  \\
+   $ \star : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ & $\id : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$
+  \\
+    \hline
+  \end{tabular}
+..   \caption{Hodge star}
+
+
+
+Exterior Calculus on manifolds and Differential forms
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Let :math:`\Omega` be a smooth manifold, of dimension :math:`n`.
+
+*  :math:`\forall x \in \Omega` we denote by :math:`T_x \Omega` the tangent space. This is a vector space of dimension :math:`n`,
+*  tangent bundle :math:`\{ (x,v), ~~ x \in \Omega, v \in T_x \Omega \}`,
+*  Applying the exterior algebra to the tangent spaces, we obtain the exterior forms bundle, whose elements are pairs :math:`(x,\mu)` with :math:`x \in \Omega` and :math:`\mu \in \Alt^k T_x \Omega`.
+*  a **differential** :math:`k`-form :math:`\omega` is a section of this bundle. This is a map which associates to each :math:`x \in \Omega` an element :math:`\omega_x \in \Alt^k T_x \Omega`,
+*  if the map :math:`\mathcal{L}_{\omega}^k : x \longmapsto \omega_x (v_1(x), \cdots, v_k(x))` is smooth (whenever :math:`v_i` are smooth), we say that :math:`\omega` is a smooth differential :math:`k`-form,
+*  we define :math:`\Lambda^k(\Omega)` the space of all smooth :math:`k`-forms on :math:`\Omega`,
+*  :math:`\Lambda^0(\Omega) = \mathcal{C}^{\infty}(\Omega)`,
+*  if the map :math:`\mathcal{L}_{\omega}^k` is :math:`\mathcal{C}^{m}(\Omega)`, we define differential :math:`k`-forms with less smoothness :math:`\mathcal{C}^{m} \Lambda^k (\Omega)`.
+
+Let :math:`\Omega` be a smooth manifold, of dimension :math:`n`.
+
+.. topic:: Exterior product:
+
+  if :math:`\omega \in \Lambda^k(\Omega)` and :math:`\eta \in \Lambda^j(\Omega)`, we may define :math:`\omega \wedge \eta` as  :math:`(\omega \wedge \eta)_x = \omega_x \wedge \eta_x` and the Grassmann algebra :math:`\Lambda(\Omega) := \bigoplus_k \Lambda^k(\Omega)`
+
+Differential forms can be differentiated and integrated, without recourse to any additional structure, such as a metric or a measure.
+
+.. topic:: Exterior differentiation:
+
+   For each :math:`\omega \in \Lambda^k(\Omega)`, can define the :math:`(k+1)`-form :math:`\diff \omega \in \Lambda^{k+1}(\Omega)`, such as:
+
+  .. math::
+
+    \diff\omega_x(v_1,\cdots,v_{k+1}) = \sum_{j=1}^{k+1} (-1)^j \partial_{v_j} \omega_x(v_1,\cdots,\hat{v_j},\cdots,v_{k+1})
+
+  where the hat is used to indicated a suppressed argument.
+
+  This defines a graded linear operator of degree :math:`+1`, of :math:`\Lambda(\Omega)` onto :math:`\Lambda(\Omega)`.
+
+We have the following properties:
+
+*  :math:`\diff \circ \diff = 0`
+*  :math:`\diff (\omega \wedge \eta) = \diff \omega \wedge \eta + (-1)^k \omega \wedge \diff \eta, ~~\forall \omega \in \Lambda^k(\Omega), \eta \in \Lambda^j(\Omega)`,
+*  (Pullback) let :math:`\phi` be a smooth map of :math:`\Omega` onto :math:`\Omega^{\prime}`. Then :math:`\phi^*(\omega \wedge \eta) = \phi^*(\omega) \wedge \phi^*(\eta)` and :math:`\phi^* (\diff \omega) = \diff (\phi^* \omega)`,
+*  (Interior product) the interior product of a differential :math:`k`-form :math:`\omega` with a vector field :math:`v`, 
+*  we obtain a :math:`(k-1)`-form by : :math:`(\omega \lrcorner v)_x := \omega_x \lrcorner v_x`,
+*  (Trace operator) the pullback :math:`i_{\partial \Omega}^*` of :math:`i_{\partial \Omega}` is the trace operator :math:`\trace`
+
+
+.. topic:: Integration:
+
+  * If :math:`f` is an oriented, piecewise smooth :math:`k`-dimensional submanifold of :math:`\Omega`, and :math:`\omega` is a continuous :math:`k`-form, then th integral :math:`\int_f \omega` is well defined :
+
+    * [0-forms] can be evaluated at points,
+    * [1-forms] can be integrated over directed curves,
+    * [2-forms] can be integrated over directed surfaces,
+
+  *  (Inner product) The :math:`L^2`-inner product of two differential :math:`k`-forms on an oriented Riemannian manifold :math:`\Omega` is defined as :
+
+  .. math::
+
+    (\omega,\eta)_{L^2 \Lambda^k} = \int_{\Omega} (\omega_x,\eta_x) \volume = \int \omega \wedge \star \eta
+
+  The completion of :math:`\Lambda^k(\Omega)` in the corresponding norm defines the Hilbert space :math:`L^2 \Lambda^k(\Omega)`.
+
+We have the following results:
+
+* (Integration) if :math:`\phi` is an orientation-preserving diffeomorphism, then 
+
+.. math::
+
+  \int_{\Omega} \phi^* \omega = \int_{\Omega^{\prime}} \omega, ~~~ \forall \omega \in \Lambda^n(\Omega^{\prime})
+
+.. topic:: Theorem, Stokes theorem:
+
+  If :math:`\Omega` is an oriented :math:`n`-manifold with boundary :math:`\partial \Omega`, then
+
+  .. math::
+
+    \int_{\Omega} \diff \omega = \int_{\partial \Omega} \trace \omega, ~~~ \forall \omega \in \Lambda^{n-1}(\Omega)
+
+.. topic:: Theorem, Integration by parts:
+
+  If :math:`\Omega` is an oriented :math:`n`-manifold with boundary :math:`\partial \Omega`, then
+
+  .. math::
+
+    \int_{\Omega} \diff \omega \wedge \eta = (-1)^{k-1} \int_{\Omega} \omega \wedge \diff \eta + \int_{\partial \Omega} \trace \omega \wedge \trace \eta, ~~~ \forall \omega \in \Lambda^{k}(\Omega), \eta \in \Lambda^{n-k-1}(\Omega)
+
+
+
+Sobolev spaces of differential forms
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+As for the classical case, we can define the Sobolev spaces as:
+
+*  :math:`H^s \Lambda^k(\Omega)` is the space of differential :math:`k`-forms such that :math:`\mathcal{L}_{\omega}^k \in H^s(\Omega)`.
+*  :math:`H \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \diff \omega \in L^2 \Lambda^{k+1}(\Omega) \}`. The associated norm is :
+
+  .. math::
+
+    \| \omega \|_{H \Lambda^k}^2 = \| \omega \|_{H \Lambda}^2 := \| \omega \|_{L^2 \Lambda^k}^2 + \| \diff \omega \|_{L^2 \Lambda^{k+1}}^2
+
+*  :math:`H \Lambda^{0}(\Omega)` coincides with :math:`H^1 \Lambda^{0}(\Omega)`,
+*  :math:`H \Lambda^{n}(\Omega)` coincides with :math:`L^2 \Lambda^{n}(\Omega)`,
+*  for :math:`0 < k < n`, we have :math:`H^1 \Lambda^k(\Omega) \subset H \Lambda^k(\Omega) \subset L^2 \Lambda^k(\Omega)`, strictly.
+
+.. math::
+
+  \begin{tabular}{|c|c c c c c|}
+    \hline
+   $k$ & $\Lambda^k$ & $H \Lambda^k$ & $\diff \omega$ & $\int_f \omega$ & $\kappa \omega$ 
+  \\
+   \hline
+  & & & & & \\
+   0 & $\mathcal{C}^{\infty}$ & $H^1$ & $\nabla \omega$ & $\omega(f)$ & $0$
+  \\
+   1 & $\mathcal{C}^{\infty}(\mathbb{R}^3)$ & $H(\rots,\mathbb{R}^3)$ & $\rots \omega$ & $\int_f \omega \cdot t \diff \mathcal{H}_1$ & $x \longmapsto x \cdot \omega(x)$
+  \\
+   2 & $\mathcal{C}^{\infty}(\mathbb{R}^3)$ & $H(\divs, \mathbb{R}^3)$ & $\divs \omega$ & $\int_f \omega \cdot n \diff \mathcal{H}_2$ & $x \longmapsto x \times \omega(x)$
+  \\
+   3 & $\mathcal{C}^{\infty}$ & $L^2$ & $0$ & $\int_f \omega \diff \mathcal{H}_3$ & $x \longmapsto x \omega(x)$
+  \\
+  & & & & & \\
+    \hline
+  \end{tabular}
+..   \caption{Correspondences between differential forms in $3$D, and scalar/vector fields.}
+
+Cohomology and De Rham Complex
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+The De Rham complex is the sequence of spaces and mappings
+
+.. math::
+
+  0 \xrightarrow{\quad} \Lambda^0(\Omega)  \xrightarrow{~\diff~}  \Lambda^1(\Omega)  \xrightarrow{~\diff~}   \cdots  \xrightarrow{~\diff~}  \Lambda^n(\Omega)  \xrightarrow{\quad} 0  
+
+Since, :math:`\diff \circ \diff = 0`, we have 
+
+.. math::
+
+  \mathcal{R}(\diff : \Lambda^{k-1}(\Omega) \longrightarrow \Lambda^k(\Omega)) \subset \mathcal{N}(\diff : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k+1}(\Omega))
+
+If :math:`\Omega` is an oriented Riemannian manifold, we have the following cohomology:
+
+.. math::
+
+  0 \xrightarrow{\quad} H \Lambda^0(\Omega)  \xrightarrow{~\diff~}  H \Lambda^1(\Omega)  \xrightarrow{~\diff~}   \cdots  \xrightarrow{~\diff~}  H \Lambda^n(\Omega)  \xrightarrow{\quad} 0  
+
+The *coderivative operator* :math:`\delta : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k-1}(\Omega)` is defined as:
+
+.. math::
+
+  \star \delta \omega = (-1)^k \diff \star \omega,~~~ \omega \in \Lambda^k(\Omega)
+
+*  we have 
+
+  .. math::
+
+    (\diff \omega , \eta ) = (\omega , \delta \eta )  + \int_{\partial \Omega} \trace \omega \wedge \trace \eta,  ~~~ \forall \omega \in \Lambda^{k}(\Omega), \eta \in \Lambda^{k+1}(\Omega),
+
+*  :math:`\delta` is a graded linear operator of degree :math:`-1`.
+*  :math:`\delta` is the formal adjoint of :math:`\diff` whenever :math:`\omega` or :math:`\eta` vanishes near the boundary.
+*  we define the spaces 
+
+  .. math::
+
+    H^* \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \delta \omega \in L^2 \Lambda^{k-1}(\Omega) \}.
+
+  we have :math:`H^* \Lambda^k(\Omega) = \star H \Lambda^{n-k}(\Omega)`.
+
+*  we obtain the dual complex
+
+  .. math::
+
+    0 \xleftarrow{\quad} H^* \Lambda^0(\Omega)  \xleftarrow{~\delta~}  H^* \Lambda^1(\Omega)  \xleftarrow{~\delta~}   \cdots  \xleftarrow{~\delta~}  H^* \Lambda^n(\Omega)  \xleftarrow{\quad} 0  
+
+
+Cohomology with boundary conditions
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Let :math:`\Lambda_0^k(\Omega)` be the subspace of :math:`\Lambda^k(\Omega)` of smooth :math:`k`-forms with compact support. We have :math:`\diff \Lambda_0^k \subset \Lambda_0^{k+1}`.
+
+The De Rham complex with the compact support is 
+
+.. math::
+
+  0 \xrightarrow{\quad} \Lambda^0_0(\Omega)  \xrightarrow{~\diff~}  \Lambda^1_0(\Omega)  \xrightarrow{~\diff~}   \cdots  \xrightarrow{~\diff~}  \Lambda^n_0(\Omega)  \xrightarrow{\quad} 0  
+
+Recall that the closure of :math:`\Lambda_0^k(\Omega)` in :math:`H \Lambda^k(\Omega)` is 
+
+.. math::
+
+  H_0 \Lambda^k(\Omega) = \{ \omega \in H \Lambda^k(\Omega),~~ \trace \omega =0\}.
+
+The :math:`L^2` version of the last complex is 
+
+.. math::
+
+  0 \xrightarrow{\quad} H_0 \Lambda^0(\Omega)  \xrightarrow{~\diff~}  H_0 \Lambda^1(\Omega)  \xrightarrow{~\diff~}   \cdots  \xrightarrow{~\diff~}  H_0 \Lambda^n(\Omega)  \xrightarrow{\quad} 0  
+
+
+.. topic:: Definition, Harmonic forms:
+
+  The harmonic :math:`k`-forms are the differential :math:`k`-forms that verify the differential equations 
+
+  .. math::
+
+      \left\{
+          \begin{aligned}
+            \diff \omega &=& 0,\\
+            \delta \omega &=& 0,\\
+            \trace \star \omega &=& 0.\\
+          \end{aligned}
+        \right.
+
+  this defines the following space,
+
+  .. math::
+
+    \mathfrak{H}^k (\Omega) = \{ \omega \in H \Lambda^k(\Omega) \cap H_0^* \Lambda^k(\Omega),~~\diff \omega = 0, \delta \omega = 0 \}
+
+We can also define the following space,
+
+.. math::
+
+  \mathfrak{H}_0^k (\Omega) = \{ \omega \in H_0 \Lambda^k(\Omega) \cap H^* \Lambda^k(\Omega),~~\diff \omega = 0, \delta \omega = 0 \}
+
+As we can see, :math:`\star \mathfrak{H}^k (\Omega) = \mathfrak{H}_0^{n-k} (\Omega)`.
+
+.. topic:: Proposition, Poincaré duality:
+
+  There is an isomorphism between the :math:`k` th De Rham cohomology space and the :math:`(n-k)` th cohomology space with boundary conditions.
+
+Homological Algebra and Hilbert complexes
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Homological Algebra
+___________________
+
+*  A cochain complex is a sequence of vector spaces and linear maps
+
+.. TODO
+.. .. math::
+.. 
+..   \cdots \longrightarrow V_{k-1}~\mbox{\shortstack{:math:`\diff_{k-1}` \\ :math:`\longrightarrow`}}~V_{k} \mbox{\shortstack{:math:`\diff_k` \\ :math:`\longrightarrow`}}~V_{k+1}~\longrightarrow~\cdots,~~~~\mbox{with}~ \diff_{k+1} \circ \diff_k = 0.
+.. 
+
+*  :math:`k`-cocycles :math:`\mathfrak{Z}^k := \mathcal{N}(d_k)`,
+
+*  :math:`k`-coboundaries :math:`\mathfrak{B}^k := \mathcal{R}(d_{k-1})`,
+
+*  :math:`k`-cohomology :math:`\mathcal{H}^k(V) := \mathfrak{Z}^k / \mathfrak{B}^k`,
+
+*  we say that the sequence is **exact**, if the **cohomology vanishes** (*i.e.* :math:`\forall~k,~~ \mathcal{H}^k(V) = \{0\}`),
+
+*  Given two cochain complexes :math:`V,V^{\prime}`, a **cochain map** :math:`f =(f_k)` (such as :math:`\diff^{\prime}_k f_k = f_{k+1} \diff_k`)
+
+  .. math::
+
+    \begin{array}{ccccccccc}
+    \cdots & \longrightarrow & V_{k-1} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k} & \mbox{\shortstack{$\diff_k$ \\ $\longrightarrow$}} & V_{k+1} & \longrightarrow~\cdots \\
+     & & \downarrow f_{k-1} & & \downarrow f_{k} &  & \downarrow f_{k+1} & & \\
+    \cdots & \longrightarrow & V_{k-1}^{\prime} & \mbox{\shortstack{$\diff_{k-1}^{\prime}$ \\ $\longrightarrow$}} & V_{k}^{\prime} & \mbox{\shortstack{$\diff_k^{\prime}$ \\ $\longrightarrow$}} & V_{k+1}^{\prime} & \longrightarrow~\cdots 
+    \end{array}
+ 
+*  :math:`f_k` maps :math:`k`-cochains to :math:`k`-cochains and :math:`k`-coboundaries to :math:`k`-coboundaries, thus induces a map :math:`\mathcal{H}^k(f) : \mathcal{H}^k(V) \longrightarrow \mathcal{H}^k(V^{\prime})`.
+
+Let :math:`V^{\prime} \subset V` be two cochain complexes,
+
+* The inclusion :math:`\imath_V` is a cochain map and thus induces a map of cohomology :math:`\mathcal{H}^k(V^{\prime}) \longrightarrow \mathcal{H}^k(V)`,
+
+* If there exists a cochain projection of :math:`V` onto :math:`V^{\prime}`, (this leads to :math:`\pi \circ \imath = \id_{V^{\prime}}`) so :math:`\mathcal{H}^k(\pi) \circ \mathcal{H}^k(\imath) = \id_{\mathcal{H}^k(V^{\prime})}`.
+
+  .. math::
+
+    \begin{array}{ccccccc}
+    \cdots & \longrightarrow & V_{k-1} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k} & \longrightarrow~\cdots \\
+     & & \pi_{k-1} \downarrow \uparrow \imath & & \pi_{k} \downarrow \uparrow \imath & & \\
+    \cdots & \longrightarrow & V_{k-1}^{\prime} & \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} & V_{k}^{\prime} & \longrightarrow~\cdots 
+    \end{array}
+
+Thus, :math:`\mathcal{H}^k(\imath)` is **injective** and :math:`\mathcal{H}^k(\pi)` is **surjective**. Hence, if one of the cohomology spaces :math:`\mathcal{H}^k(V)` vanishes, 
+then so does :math:`\mathcal{H}^k(V^{\prime})`
+
+Cycles and boundaries of the De Rham complex
+____________________________________________
+
+* :math:`k`-cocycles 
+
+.. math::
+
+  \mathfrak{Z}^k = \{ \omega \in H\Lambda^k(\Omega),~~ \diff \omega = 0 \}, ~~~ \mathfrak{Z}^{*k} = \{ \omega \in H^*\Lambda^k(\Omega),~~ \delta \omega = 0 \},
+
+.. math::
+
+  \mathfrak{Z}_0^k = \{ \omega \in H_0\Lambda^k(\Omega),~~ \diff \omega = 0 \}, ~~~ \mathfrak{Z}_0^{*k} = \{ \omega \in H_0^*\Lambda^k(\Omega),~~ \delta \omega = 0 \},
+
+.. math::
+
+* :math:`k`-coboundaries
+
+.. math::
+
+  \mathfrak{B}^k = \diff H\Lambda^{k-1}(\Omega), ~~~ \mathfrak{B}^{* k} = \delta \Lambda^{k+1}(\Omega),
+
+.. math::
+
+  \mathfrak{B}_0^k = \diff H_0\Lambda^{k-1}(\Omega), ~~~ \mathfrak{B}_0^{* k} = \delta \Lambda_0^{k+1}(\Omega),
+
+* each of the spaces of cycles is closed in :math:`\mathcal{H} \Lambda^k(\Omega)` (:math:`\mathcal{H}^* \Lambda^k(\Omega)`), as well in :math:`L^2 \Lambda^k(\Omega)`.
+
+* each of the spaces of boundaries is closed in :math:`L^2 \Lambda^k(\Omega)`.
+
+* let :math:`\perp` denotes the orthogonal complement in :math:`L^2 \Lambda^k(\Omega)`,
+
+.. math::
+
+  \mathfrak{Z}^{k \perp} \subset \mathfrak{B}^{k \perp} = \mathfrak{Z}_0^{* k} , ~~~ \mathfrak{Z}^{* k \perp} \subset \mathfrak{B}^{* k \perp} = \mathfrak{Z}_0^{k}
+
+.. math::
+
+  \mathfrak{Z}_0^{k \perp} \subset \mathfrak{B}_0^{k \perp} = \mathfrak{Z}^{* k} , ~~~ \mathfrak{Z}_0^{* k \perp} \subset \mathfrak{B}_0^{* k \perp} = \mathfrak{Z}^{k}
+
+
+The Hodge decomposition
+_______________________
+
+There are two Hodge decompositions, with different boundary conditions,
+
+1.
+
+  .. math::
+
+    L^2 \Lambda^k(\Omega) 
+    = 
+    \underbrace{\mathfrak{B}^{k}}_{\mathfrak{Z}_0^{* k\perp}} 
+    \oplus 
+    \underbrace{\mathfrak{H}^{k} 
+    \oplus 
+    \mathfrak{B}_0^{* k}}_{\mathfrak{Z}_0^{* k}=\mathfrak{B}^{k\perp}}
+    = 
+    \overbrace{\mathfrak{B}^{k}
+    \oplus 
+    \mathfrak{H}^{k}}^{\mathfrak{Z}^{k}=\mathfrak{B}_0^{* k\perp}}  
+    \oplus 
+    \overbrace{\mathfrak{B}_0^{* k}}^{\mathfrak{Z}^{k\perp}}
+
+2.
+
+  .. math::
+
+    L^2 \Lambda^k(\Omega) 
+    = 
+    \underbrace{\mathfrak{B}_0^{k}}_{\mathfrak{Z}^{* k\perp}} 
+    \oplus 
+    \underbrace{\mathfrak{H}_0^{k} 
+    \oplus 
+    \mathfrak{B}^{* k}}_{\mathfrak{Z}^{* k}=\mathfrak{B}_0^{k\perp}}
+    = 
+    \overbrace{\mathfrak{B}_0^{k}
+    \oplus 
+    \mathfrak{H}_0^{k}}^{\mathfrak{Z}_0^{k}=\mathfrak{B}^{* k\perp}}  
+    \oplus 
+    \overbrace{\mathfrak{B}^{* k}}^{\mathfrak{Z}_0^{k\perp}}
+
+Summary
+^^^^^^^
+
+.. math::
+
+  \begin{tabular}{|c||c|c|c|c|}
+   \hline
+   $\omega^k \in \Lambda^k(\Omega)$             & $k=0$ 
+                                                & $k=1$ 
+                                                & $k=2$ 
+                                                & $k=3$ 
+   \\
+   \hline
+   $\diff \omega^k$                             & $\Grad u$ 
+                                                & $\Curl \uu$ 
+                                                & $\Div \uu$ 
+                                                & $-$
+   \\
+   $\delta \omega^k$                            & $-$ 
+                                                & $-\Div \uu$ 
+                                                & $\Curl \uu$ 
+                                                & $-\Grad u$
+   \\
+   $\mathfrak{i}_{\boldsymbol{\beta}} \omega^k$ & $-$ 
+                                                & $\boldsymbol{\beta} \cdot \uu$ 
+                                                & $\uu \times \boldsymbol{\beta}$ 
+                                                & $u \boldsymbol{\beta}$
+   \\
+   $\mathfrak{j}_{\boldsymbol{\beta}} \omega^k$ & $u \boldsymbol{\beta}$ 
+                                                & $-\uu \times \boldsymbol{\beta}$ 
+                                                & $\boldsymbol{\beta} \cdot \uu$ 
+                                                & $-$
+   \\
+   $L_{\boldsymbol{\beta}} \omega^k$            & $\boldsymbol{\beta} \cdot \Grad u$ 
+                                                & $\Grad \left(\boldsymbol{\beta} \cdot \uu \right)  + \left(\Curl \uu \right) \times \boldsymbol{\beta}$ 
+                                                & $\Curl \left(\uu \times \boldsymbol{\beta} \right) + \boldsymbol{\beta} \Div \uu$ 
+                                                & $\Div \left( u \boldsymbol{\beta} \right)$
+   \\
+   $\mathcal{L}_{\boldsymbol{\beta}} \omega^k$  & $-\Div \left( u \boldsymbol{\beta} \right)$ 
+                                                & $-\Curl \left(\uu \times \boldsymbol{\beta} \right) - \boldsymbol{\beta} \Div \uu$ 
+                                                & $-\Grad \left(\boldsymbol{\beta} \cdot \uu \right)  - \left(\Curl \uu \right) \times \boldsymbol{\beta}$ 
+                                                & $-\boldsymbol{\beta} \cdot \Grad u$
+   \\
+   \hline
+   $\tr \omega^k$                               & $u(\xx)$ 
+                                                & $\uu(\xx) \times \nn(\xx)$ 
+                                                & $\uu(\xx) \cdot  \nn(\xx)$ 
+                                                & $-$
+   \\
+   \hline
+   \hline
+   $H \Lambda^k(\Omega)$                        & $\Hgrad$ 
+                                                & $\Hcurl$ 
+                                                & $\Hdiv$ 
+                                                & $\Ltwo$
+   \\
+   $V_k$                                        & $\Vgrad$ 
+                                                & $\Vcurl$ 
+                                                & $\Vdiv$ 
+                                                & $\Vltwo$
+   \\
+   \hline
+  \end{tabular}
+..   \caption{Correspondences between differential forms in $3$D, and scalar/vector fields.}
+
+
+
+.. rubric:: References
+
+.. bibliography:: refs_feec.bib
+   :cited:
diff --git a/_sources/psydac/doc/math/glt.rst b/_sources/psydac/doc/math/glt.rst
new file mode 100644
index 0000000..5d274ea
--- /dev/null
+++ b/_sources/psydac/doc/math/glt.rst
@@ -0,0 +1,230 @@
+GLT
+***
+
+Where do the GLTs come from?
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+The main aim of this paragraph is to present a crucial example that highlights the importance of the GLT algebra when dealing with linear systems coming from the discretization of PDEs. Let us start with some preliminaries. In detail, we will recall the notion of symbol of a matrix-sequence and the basic idea behind the GLT theory.
+
+
+Spectral preliminaries
+______________________
+
+.. %\subsubsection{Symbol}
+
+The following one is a rather informal definition of symbol of a matrix-sequence.
+
+.. .. math:: 
+.. 
+..   \begin{definition}\label{def}Let
+.. 
+..   \begin{itemize}
+..           \item $\{A_n\}_n$ be a matrix-sequence, ${\rm dim}(A_n)=d_n\uparrow\infty$,
+..           \item $f:D\subset\mathbb R^d\to\mathbb C$,\quad $0<{\rm measure}(D)<\infty$.
+..   \end{itemize}
+..   We say that $\{A_n\}_n$ is \emph{spectrally distributed} as $f$ (in symbols $\{A_n\}_n\sim_\lambda f$) if the eigenvalues of $A_n$ are approximately a uniform sampling of $f$ over $D$. In turn, the function $f$ is called \emph{spectral symbol} of $\{A_n\}_n$.
+..   \end{definition}
+
+**example**:
+
+When :math:`d_n=n`, :math:`d=1`, :math:`D=[0,\pi]`, :math:`\{A_n\}_n\sim_\lambda f` means
+
+.. .. math::
+.. 
+..   \begin{equation*}
+..   \lambda_j(A_n)\approx f\left(\frac{j\pi}{n}\right), \quad j=0, \ldots, n-1.
+..   \end{equation*}
+
+.. \begin{remark} This definition can also be given is the singular values sense (replacing $f\rightarrow |f|$). Notation: $\{A_n\}_n\sim_\sigma f$.
+.. \end{remark}
+.. 
+.. In what follows, we briefly recall the basics of the GLT theory. For the sake of simplicity, we focus on the 1D setting.
+.. %\subsubsection{GLT theory}
+.. \begin{definition}\label{glt}
+.. A \emph{GLT sequence} is a matrix-sequence $\{A_n\}_n$, with ${\rm dim}(A_n)=d_n\uparrow\infty$ obtained as \emph{linear combination, product, inversion or conjugation} of
+.. \begin{itemize}
+.. \item $\{T_n(f)\}_n$, \emph{Toeplitz sequences} with $T_n(f)$ $f: \left[-\pi,\pi \right] \rightarrow \mathbb{C}$, $f\in L^1([-\pi,\pi])$ defined as follows
+.. \[ T_n(f) = \left[\begin{array}{cccc}
+.. f_0 & f_{-1} & \cdots & f_{-(n-1)}\\
+.. f_1 & \ddots & \ddots &  \vdots\\
+.. %\vdots & \ddots & \ddots & \ddots & \vdots\\
+.. \vdots & \ddots & \ddots & f_{-1}\\
+.. f_{n-1} & \cdots  & f_1 & f_0
+.. \end{array}\right],\]
+.. where
+.. \begin{equation*}
+.. f_j = \frac1{2\pi}\int_{-\pi}^\pi f(\theta)\textup{e}^{-\textup{i}j\theta}d\theta,\qquad j\in\mathbb Z,
+.. \end{equation*}
+.. \item $\{D_n(a)\}_n$, \emph{diagonal sampling sequences}, with $D_n(a)$, $a: \left[0,1\right] \rightarrow \mathbb{C}$ Riemann integrable function, s.t.
+.. \begin{equation} \label{dsm} D_n(a) = \left[\begin{array}{cccc}
+.. a(\frac1n) & & & \\
+.. & a(\frac2n) & & \\
+.. & & \ddots & \\
+.. & & & a(1)
+.. \end{array}\right], \end{equation}
+.. \item $\{Z_n\}_n$, \emph{low-rank+small-norm sequences}, with
+.. $$Z_n=R_n+S_n,$$
+.. and
+.. \begin{itemize}
+.. \item $R_n$: $\lim_{n\to\infty}\frac{\textup{rank}(R_n)}{d_n}=0$,
+.. \item $S_n$: $\lim_{n\to\infty}\|S_n\|=0$.
+.. \end{itemize}
+.. Here and throughout these notes, $\|A_n\|:=\sigma_{\rm max}(A_n)$.
+.. \end{itemize}
+.. \end{definition}
+.. \begin{remark}\label{rem}
+.. Let us observe that according to Definition \ref{glt}
+.. \begin{enumerate}
+.. \item the sequences $\{T_n(f)\}_n$, $\{D_n(a)\}_n$, and $\{Z_n\}_n$ are GLT sequences themselves;
+.. \item the set of the GLT sequences form a $*$-algebra, i.e., it is closed under linear combinations, products, inversion, conjugation: hence, the sequence obtained via algebraic operations on a finite set of input GLT sequences is still a GLT sequence.
+.. \end{enumerate}
+.. \end{remark}
+.. 
+.. What follows provides a link between the notions of symbol and of GLT sequence.
+.. \begin{theorem}\label{szego} Let $f: \left[-\pi,\pi \right] \rightarrow \mathbb{C}$, with $f\in L^1([-\pi,\pi])$. Then
+.. \begin{equation*}
+.. \{T_n(f)\}_n\sim_\sigma f.
+.. \end{equation*}
+.. \end{theorem}
+.. \begin{remark}\label{diag} Let $\{D_n(a)\}_n$ be a diagonal sampling sequence defined as in \eqref{dsm}. Then, recalling Definition \ref{def}, it yields
+.. \[\{D_n(a)\}_n\sim_{\sigma,\lambda} a.\]
+.. \end{remark}
+.. \begin{theorem}\label{zero}
+.. Let $\{Z_n\}_n$ be a matrix-sequence. Then the following conditions are equivalent
+.. \begin{itemize}
+.. \item $\{Z_n\}_n\sim_\sigma 0$;
+.. \item For any $n$ we have $Z_n=R_n+S_n$, where $\lim_{n\to\infty}\frac{\textup{rank}(R_n)}{N_n}=\lim_{n\to\infty}\|S_n\|=0$.
+.. \end{itemize}
+.. \end{theorem}
+.. Because of the equivalence in Theorem \ref{zero} the low-rank+small-norm matrix-sequences are also known as \emph{zero-distributed} matrix-sequences.
+.. 
+.. 
+.. Summarizing, thanks to Theorems \ref{szego},\ref{zero} and Remark \ref{diag}, all the basic GLT sequences (Toeplitz, diagonal sampling, zero-distributed) are equipped with a symbol and then an asymptotical description of their spectrum is available. More in general, as stated in the following proposition, each GLT sequence is provided with a symbol.
+.. \begin{proposition}
+.. Each GLT sequence $\left\{A_n\right\}_n$ is equipped with a symbol in the singular value sense, i.e. there exists a function $\chi: [0,1]\times[-\pi,\pi]\rightarrow\mathbb{C}$ such that $$\left\{A_n\right\}_n \sim_{\sigma} \chi.$$
+.. \end{proposition}
+.. 
+.. At this point a question arises: how to compute the symbol of a GLT sequence? The following proposition provides the answer.
+.. \begin{proposition}\label{sym}
+.. Let $\{ A_{n} \}_{n}$ and $\{ B_{n} \}_{n}$ be two GLT sequences such that $\{ A_{n} \}_{n}\sim_\sigma \kappa_1$ and $\{ B_{n} \}_{n}\sim_\sigma \kappa_2$. Then
+..         \begin{itemize}
+.. 
+..         \item $\{\alpha A_{ n} + \beta B_{ n}\}_{n} \sim_\sigma \alpha\kappa_1+\beta \kappa_2, \quad \alpha, \beta \in \mathbb{C};$
+..         \item $\{A_{n}B_{n}\}_{n} \sim_\sigma\kappa_1 \kappa_2;$
+..         \item if $\kappa_1$ vanishes, at most, in a set of zero Lebesgue measure, then $\{A^{-1}_{n}\}_{n} \sim_\sigma \kappa_1^{-1};$
+..         \item $\{ A_{n}^{H} \}_{n} \sim_\sigma\bar{\kappa_1}.$
+..       \end{itemize}
+.. \end{proposition}
+.. 
+.. Let us summarize the previous results in the following three properties:
+.. \begin{itemize}
+.. \item[\textbf{GLT1}] $\left\{A_n\right\}_n$ a GLT sequence $\Rightarrow$ $\left\{A_n\right\}_n \sim_{\sigma} \chi$ with $\chi: [0,1]\times[-\pi,\pi]\rightarrow\mathbb{C}$.
+.. \item[\textbf{GLT2}] The GLT sequences form a $*$-algebra and the symbol of a GLT sequence is computed according to Proposition \ref{sym}.
+.. \item[\textbf{GLT3}]
+.. \begin{itemize}
+.. \item if $f\in L^1([-\pi,\pi])$, then $\{T_n(f)\}_n$ is a GLT sequence and $\{T_n(f)\}_n\sim_\sigma f$;
+.. \item if $a:[0,1]\rightarrow\mathbb{C}$ is Riemann integrable, then $\{D_n(a)\}_n$ is a GLT sequence and $\{D_n(a)\}_n\sim_{\sigma,\lambda} a$;
+.. \item if $\{Z_n\}_n$ is such that for any $n$ $Z_n=R_n+S_n$, where $\lim_{n\to\infty}\frac{\textup{rank}(R_n)}{N_n}=\lim_{n\to\infty}\|S_n\|=0$, then $\{Z_n\}_n$ is a GLT sequence and $\{Z_n\}_n\sim_\sigma0$.
+.. \end{itemize}
+.. \end{itemize}
+.. 
+.. \subsection*{GLTs come into life: an FD discretization of a diffusion PDE}
+.. Let us consider the following diffusion problem
+.. \begin{eqnarray}\label{pro} \left\{\begin{array}{l}
+.. -(a(x)u'(x))'=g(x),\qquad x\in(0,1),\vspace{5pt}\\
+.. u(0)=u(1)=0,
+.. \end{array}\right.
+.. \end{eqnarray}
+.. where $a,g\in C([0,1])$.
+.. Let us discretize it with a classical second-order central FD scheme. Let $n\in\mathbb{N}$ be the discretization parameter, and set $h=\frac{1}{n+1}$ and $x_j=jh$ for all $j\in[0,n+1]$. For $j=1,\ldots,n$
+.. \begin{align*} \left.-(a(x)u'(x))'\right|_{x=x_j}&\approx-\frac{a(x_{j+\frac12})u'(x_{j+\frac12})-a(x_{j-\frac12})u'(x_{j-\frac12})}{h}\\[5pt]
+.. &\approx-\frac{a(x_{j+\frac12})\dfrac{u(x_{j+1})-u(x_j)}{h}-a(x_{j-\frac12})\dfrac{u(x_j)-u(x_{j-1})}{h}}{h}\\
+.. &=\frac{-a(x_{j+\frac12})u(x_{j+1})+\bigl(a(x_{j+\frac12})+a(x_{j-\frac12})\bigr)u(x_j)-a(x_{j-\frac12})u(x_{j-1})}{h^2}.
+.. \end{align*}
+.. Let us define $u_j\approx u(x_j)$, $j=1,\ldots,n$ and $u_0=u_{n+1}=0$. Then problem \eqref{pro} can be reformulated in a matrix form as follows: find $\mathbf u_n=(u_1,\ldots,u_n)^T$ s.t.
+.. \[ A_n\mathbf u_n=\mathbf g_n\]
+.. where $\mathbf g_n=h^2(g(x_1),\ldots,g(x_n))^T$ and
+.. \begin{eqnarray}\label{matr}
+.. A_n =
+.. \begin{bmatrix}
+.. a(x_{\frac12})+a(x_{\frac32}) & -a(x_{\frac32}) & & & \\[12pt]
+.. -a(x_{\frac32}) & a(x_{\frac32})+a(x_{\frac52}) & -a(x_{\frac52}) & & \\[12pt]
+.. & \ddots & \ddots & \ddots & \\[12pt]
+.. & & -a(x_{n-\frac32}) & a(x_{n-\frac32})+a(x_{n-\frac12}) & -a(x_{n-\frac12}) \\[12pt]
+.. & & & -a(x_{n-\frac12}) & a(x_{n-\frac12})+a(x_{n+\frac12})
+.. \end{bmatrix}.
+.. \end{eqnarray}
+.. \begin{remark}
+.. Note that when $a(x)\equiv1$, then $A_n$ is the Laplacian matrix
+.. \begin{eqnarray*}
+.. A_n = \left[\begin{array}{ccccc}
+.. 2 & -1 & 0 & \cdots & 0\\
+.. -1 & 2 & -1 & \ddots & \vdots\\
+.. 0 & \ddots & \ddots & \ddots & 0\\
+.. \vdots &\ddots & -1 & 2 & -1\\
+.. 0 & \cdots & 0 & -1 & 2
+.. \end{array}\right],
+.. \end{eqnarray*}
+.. or, in other words, the Toeplitz matrix associated to the function $2-2\cos\theta$, i.e., $A_n=T_n(2-2\cos\theta)$. Using Theorem \ref{szego} it holds that $$\{A_n\}\sim_\sigma 2-2\cos\theta.$$
+.. \end{remark}
+.. Our aim is to recognize in the matrix-sequence $\{A_n\}_n$, with $A_n$ defined as in \eqref{matr} a GLT sequence. Consider the following matrix
+.. \begin{align*}
+.. B_n=D_n(a)T_n(2-2\cos\theta)&= \left[\begin{array}{cccc}
+.. a(\frac1n) & & & \\
+.. & a(\frac2n) & & \\
+.. & & \ddots & \\
+.. & & & a(1)
+.. \end{array}\right]
+.. \left[\begin{array}{cccc}
+.. 2 & -1 & & \\
+.. -1 & \ddots & \ddots & \\
+..  & \ddots & \ddots & -1\\
+..  & & -1 & 2
+.. \end{array}\right]
+.. \end{align*}
+.. \begin{align*}
+.. =\begin{bmatrix}
+.. 2a(\frac1n)\, & -a(\frac1n) & & & \\[5pt]
+.. -a(\frac2n) & 2\,a(\frac2n) & -a(\frac2n) & & \\[5pt]
+.. & \ddots & \ddots & \ddots & \\[5pt]
+.. & & -a(\frac{n-1}n) & 2\,a(\frac{n-1}n) & -a(\frac{n-1}n) \\[5pt]
+.. & & & -a(1) & 2\,a(1)
+.. \end{bmatrix}.
+.. \end{align*}
+.. Let us define $Z_n=A_n-B_n$. In the view of the inequalities
+.. \begin{equation*}
+.. \left|x_j-\frac{j}{n}\right|=\left|\frac{j}{n+1}-\frac{j}{n}\right|\le h, \quad j=1,\ldots,n,
+.. \end{equation*}
+.. and thanks to the continuity of $a$, the off-diagonal entries of $Z_n$ satisfy
+.. \begin{align*}
+.. \left|a\left(\frac{j}{n}\right)-a(x_{j+\frac12})\right|&\le \omega_a(3h/2),\\
+.. \left|a\left(\frac{j}{n}\right)-a(x_{j-\frac12})\right|&\le \omega_a(3h/2),
+.. \end{align*}
+.. where $\omega_a$ is the modulus of continuity of $a$. As a consequence, the modulus of each diagonal entry of $Z_n$ is bounded by $2\omega_a(3h/2)$. Therefore, it holds that
+.. \begin{align*}
+.. \|Z_n\|_1:=\max_{j=1,\ldots,n}\sum_{i=1}^n|z_{ij}|&\le 4 \omega_a(3h/2),\\
+.. \|Z_n\|_\infty:=\max_{i=1,\ldots,n}\sum_{j=1}^n|z_{ij}|&\le 4 \omega_a(3h/2),
+.. \end{align*}
+.. and then
+.. \begin{equation*}
+.. \|Z_n\|\le\sqrt{\|Z_n\|_1\|Z_n\|_\infty}\le 4 \omega_a(3h/2)\underset{h\rightarrow 0}{\longrightarrow} 0
+.. \end{equation*}
+.. (recall that when $a$ is a continuous function, then $\omega_a(h)\rightarrow0$ as $h\rightarrow 0$). Therefore, $\{Z_n\}_n$ is a small-norm sequence. This, together with Remark \ref{rem}, let us to conclude that $\{A_n\}_n$ with
+.. $$A_n=B_n+Z_n=D_n(a)T_n(2-2\cos\theta)+Z_n.$$
+.. is a GLT sequence.
+.. 
+.. Now, it is easy to compute the symbol of $\{A_n\}_n$. By \textbf{GLT3} we have that
+.. \begin{itemize}
+.. \item $\{T_n(2-2\cos\theta)\}_n \sim_\sigma 2-2\cos\theta$;
+.. \item $\{D_n(a)\}_n\sim_\sigma a$;
+.. \item $\{Z_n\}_n\sim_\sigma 0$;
+.. \end{itemize}
+.. and then by \textbf{GLT2} it holds
+.. $$\{A_n\}_n\sim_\sigma a(x)\cdot (2-2\cos\theta).$$
+
+
+.. rubric:: References
+
+.. bibliography:: refs_glt.bib
+   :cited:
+
diff --git a/_sources/psydac/doc/math/index.rst b/_sources/psydac/doc/math/index.rst
new file mode 100644
index 0000000..dec0d0d
--- /dev/null
+++ b/_sources/psydac/doc/math/index.rst
@@ -0,0 +1,18 @@
+.. role:: envvar(literal)
+.. role:: command(literal)
+.. role:: file(literal)
+.. _feec:
+
+.. toctree::
+   :maxdepth: 1
+   :caption: Contents:
+
+   splines
+   glt 
+   feec
+   derham
+
+.. Local Variables:
+.. mode: rst
+.. End:
+
diff --git a/_sources/psydac/doc/math/splines.rst b/_sources/psydac/doc/math/splines.rst
new file mode 100644
index 0000000..e626c67
--- /dev/null
+++ b/_sources/psydac/doc/math/splines.rst
@@ -0,0 +1,324 @@
+B-Splines and NURBS
+*******************
+
+We start this section by recalling some basic properies about B-splines curves and surfaces. We also recall some fundamental algorithms (knot insertion and degree elevation). 
+
+For a basic introduction to the subject, we refer to the books :cite:`piegl` and :cite:`Farin_Book2002`.  
+
+A B-Splines family, :math:`(N_i)_{ 1 \leqslant i \leqslant n}` of order :math:`k`, can be generated using a non-decreasing sequence **of knots** :math:`T=(t_i)_{1\leqslant i \leqslant n + k}`.
+
+B-Splines series
+^^^^^^^^^^^^^^^^
+
+The j-th B-Spline of order :math:`k` is defined by the recurrence relation:
+
+.. math::
+  
+   N_j^k = w_j^k N_j^{k-1} + ( 1 - w_{j+1}^k ) N_{j+1}^{k-1}
+
+where,
+
+.. math::
+
+   w_j^k (x) = \frac{x-t_j}{t_{j+k-1}-t_{j}} \hspace{2cm} N_j^1(x) = \chi_{ \left[ t_j, t_{j+1} \right[ }(x)
+
+for :math:`k \geq 1` and :math:`1 \leq j \leq n`.
+
+We note some important properties of a B-splines basis:
+
+* B-splines are piecewise polynomial of degree :math:`p=k-1`,
+
+* Compact support; the support of :math:`N_j^k` is contained in :math:`\left[ t_j, t_{j+k} \right]` ,
+
+* If :math:`x \in~ ] t_j,t_{j+1} [`, then only the *B-splines* :math:`\{ N_{j-k+1}^k,\cdots,N_{j}^k \}` are non vanishing at :math:`x`,
+
+* Positivity: :math:`\forall j \in \{1,\cdots,n \}~~N_j(x) >0, ~~\forall x \in ] t_j, t_{j+k} [`,
+
+* Partition of unity  :math:`\sum_{i=1}^n N_i^{k}(x) = 1, \forall x \in \mathbb{R}`,
+
+* Local linear independence,
+
+* If a knot :math:`t_i` has a multiplicity :math:`m_i` then the B-spline is :math:`\mathcal{C}^{(p-m_i)}` at :math:`t_i`.
+
+Knots vector families
+^^^^^^^^^^^^^^^^^^^^^
+
+There are two kind of **knots vectors**, called **clamped** and **unclamped**. Both families contains **uniform** and **non-uniform** sequences. 
+
+The following are examples of such knots vectors
+
+1. **Clamped knots** (open knots vector)
+
+* uniform
+
+.. math::
+
+  T_1 &= \{0, 0, 0, 1, 2, 3, 4, 5, 5, 5 \}
+  \\
+  T_2 &= \{-0.2, -0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 0.8 \}
+
+.. image:: ../include/splines/bsplines_t1_p2.png
+   :width: 8cm
+   :height: 8cm
+
+.. image:: ../include/splines/bsplines_t2_p2.png
+   :width: 8cm
+   :height: 8cm
+
+* non-uniform 
+
+.. math::
+
+  T_3 &= \{0, 0, 0, 1, 3, 4, 5, 5, 5 \}
+  \\
+  T_4 &= \{-0.2, -0.2, 0.4, 0.6, 0.8, 0.8 \}
+
+.. image:: ../include/splines/bsplines_t3_p2.png
+   :width: 8cm
+   :height: 8cm
+
+.. image:: ../include/splines/bsplines_t4_p2.png
+   :width: 8cm
+   :height: 8cm
+
+2. **Unclamped knots**
+
+* uniform
+
+.. math::
+
+  T_5 &= \{0, 1, 2, 3, 4, 5, 6, 7 \}
+  \\
+  T_6 &= \{-0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 \}
+
+.. image:: ../include/splines/bsplines_t5_p2.png
+   :width: 8cm
+   :height: 8cm
+
+.. image:: ../include/splines/bsplines_t6_p2.png
+   :width: 8cm
+   :height: 8cm
+
+* non-uniform 
+
+.. math::
+
+  T_7 &= \{0, 0, 3, 4, 7, 8, 9 \}
+  \\
+  T_8 &= \{-0.2, 0.2, 0.4, 0.6, 1.0, 2.0, 2.5 \}
+
+.. image:: ../include/splines/bsplines_t7_p2.png
+   :width: 8cm
+   :height: 8cm
+
+.. image:: ../include/splines/bsplines_t8_p2.png
+   :width: 8cm
+   :height: 8cm
+
+B-Spline curve
+^^^^^^^^^^^^^^
+
+The B-spline curve in :math:`\mathbb{R}^d` associated to knots vector :math:`T=(t_i)_{1\leqslant i \leqslant n + k}` and the control polygon :math:`(\mathbf{P}_i)_{ 1 \leqslant i \leqslant n}` is defined by :
+
+.. math::
+
+   \mathcal{C}(t) = \sum_{i=1}^n N_i^k(t) \textbf{P}_i
+
+
+In (Fig. \ref{figBSplineCurve}), we give an example of a quadratic B-Spline curve, and its corresponding knot vector and control points.
+
+
+.. image:: ../include/splines/courbe_bsplines.png
+   :width: 8cm
+   :height: 8cm
+
+.. image:: ../include/splines/basis_fct_p2_N5.png
+   :width: 8cm
+   :height: 8cm
+
+.. \caption{(left) A quadratic B-Spline curve and its control points using the knot vector :math:`T = \{ 000~  \frac{1}{2}~ \frac{3}{4} \frac{3}{4}~ 1 1 1 \}`, (right) the corresponding B-Splines.}
+
+We have the following properties for a *B-spline* curve:
+
+* If :math:`n=k`, then :math:`\mathcal{C}` is just a B\'ezier-curve,
+
+* :math:`\mathcal{C}` is a piecewise polynomial curve,
+
+* The curve interpolates its extremas if the associated multiplicity of the first and the last knot are maximum (*i.e.* equal to :math:`k`), *i.e.* open knot vector,
+
+* Invariance with respect to affine transformations,
+
+* Strong convex-hull property:
+
+if :math:`t_i \leq t \leq t_{i+1}`, then :math:`\mathcal{C}(t)` is inside the convex-hull associated to the control points :math:`\mathbf{P}_{i-p},\cdots,\mathbf{P}_{i}`,
+
+* Local modification : moving the :math:`i^{th}` control point :math:`\mathbf{P}_{i}` affects :math:`\mathcal{C}(t)`, only in the interval :math:`[t_i,t_{i+k}]`,
+* The control polygon approaches the behavior of the curve.
+
+.. note::
+
+   In order to model a singular curve, we can use multiple control points : :math:`\mathbf{P}_{i}=\mathbf{P}_{i+1}`.
+
+Multivariate tensor product splines
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Let us consider :math:`d` knot vectors :math:`\mathcal{T} = \{T^1,T^2,\cdots,T^d\}`. For simplicity, we consider that these knot vectors are open, which means that :math:`k` knots on each side are duplicated so that the spline is interpolating on the boundary, and of bounds :math:`0` and :math:`1`. In the sequel we will use the notation :math:`I=[0,1]`.
+Each knot vector :math:`T^i`, will generate a basis for a Schoenberg space, :math:`\mathcal{S}_{k_{i}}(T^i,I)`. The tensor product of all these spaces is also a Schoenberg space, namely :math:`\mathcal{S}_{\mathbf{k}}(\mathcal{T})`, where :math:`\mathbf{k}=\{k_1,\cdots,k_d\}`. The cube :math:`\mathcal{P}=I^d=[0,1]^d`, will be referred to as a patch.
+
+The basis for :math:`\mathcal{S}_{\mathbf{k}}(\mathcal{T})` is defined by a tensor product :
+
+.. math::
+
+   N_{\mathbf{i}}^{\mathbf{k}} := N_{i_1}^{k_1} \otimes N_{i_2}^{k_2} \otimes \cdots \otimes N_{i_d}^{k_d}
+
+where, :math:`\mathbf{i}=\{i_1,\cdots , i_d \}`.
+
+A typical cell from :math:`\mathcal{P}` is a cube of the form : :math:`Q_{\mathbf{i}}=[\xi_{i_1}, \xi_{i_1+1}] \otimes \cdots \otimes [\xi_{i_d}, \xi_{i_d+1}]`. 
+
+Deriving a B-spline curve
+^^^^^^^^^^^^^^^^^^^^^^^^^
+
+The derivative of a B-spline curve is obtained as:
+
+.. math::
+
+   \mathcal{C}^{\prime}(t) = \sum_{i=1}^{n} {N_{i}^{k}}^{\prime}(t) \mathbf{P}_i = \sum_{i=1}^{n} \left(\frac{p}{t_{i+p}-t_{i}}N_{i}^{k-1}(t) \mathbf{P}_i - \frac{p}{t_{i+1+p}-t_{i+1}}N_{i+1}^{k-1}(t) \mathbf{P}_i \right)
+  = \sum_{i=1}^{n-1} {N_{i}^{k-1}}^{\ast}(t) \mathbf{Q}_i
+
+where :math:`\mathbf{Q}_i = p \frac{\mathbf{P}_{i+1} - \mathbf{P}_i}{t_{i+1+p}-t_{i+1}}`, and :math:`\{{N_{i}^{k-1}}^{\ast},~~1 \leq i \leq n-1\}` are generated using the knot vector :math:`T^{\ast}`, which is obtained from :math:`T` by reducing by one the multiplicity of the first and the last knot (in the case of open knot vector), *i.e.* by removing the first and the last knot.
+
+More generally, by introducing the B-splines family :math:`\{ {N_{i}^{k-j}}^{\ast}, 1 \leq i \leq n-j \}` generated by the knots vector :math:`T^{j^{\ast}}` obtained from :math:`T` by removing the first and the last knot :math:`j` times, we have the following result:
+
+proposition
+___________
+
+The :math:`j^{th}` derivative of the curve :math:`\mathcal{C}` is given by
+
+.. math::
+
+  \mathcal{C}^{(j)}(t) = \sum_{i=1}^{n-j} {N_{i}^{k-j}}^{\ast}(t) \mathbf{P}_i^{(j)}`
+   
+where, for :math:`j>0`
+
+.. math::
+
+  \mathbf{P}_i^{(j)} = \frac{p-j+1}{t_{i+p+1}-t_{i+j}} \left( \mathbf{P}_{i+1}^{(j-1)} - \mathbf{P}_i^{(j-1)} \right)
+  \\
+  \mbox{and} ~ ~ ~ \mathbf{P}_i^{(0)} = \mathbf{P}_i.
+
+By denoting :math:`\mathcal{C}^{\prime}` and :math:`\mathcal{C}^{\prime\prime}` the first and second derivative of the B-spline curve :math:`\mathcal{C}`, it is easy to show that:
+
+We have,
+
+* :math:`\mathcal{C}^{\prime}(0) = \frac{p}{t_{p+2}} \left(\mathbf{P}_{2} - \mathbf{P}_1\right)`,
+
+* :math:`\mathcal{C}^{\prime}(1) = \frac{p}{1-t_{n}} \left(\mathbf{P}_{n} - \mathbf{P}_{n-1}\right)`,
+
+* :math:`\mathcal{C}^{\prime\prime}(0) = \frac{p(p-1)}{t_{p+2}} \left( \frac{1}{t_{p+2}}\mathbf{P}_{1} - \{ \frac{1}{t_{p+2}} + \frac{1}{t_{p+3}} \} \mathbf{P}_2 + \frac{1}{t_{p+3}}\mathbf{P}_{3} \right)`,
+
+* :math:`\mathcal{C}^{\prime\prime}(1) = \frac{p(p-1)}{1-t_{n}} \left( \frac{1}{1-t_{n}}\mathbf{P}_{n} - \{ \frac{1}{1-t_{n}} + \frac{1}{1-t_{n-1}} \} \mathbf{P}_{n-1} + \frac{1}{1-t_{n-1}}\mathbf{P}_{n-2} \right)`.
+
+
+Example
+_______
+
+Let us consider the quadratic B-spline curve associated to the knots vector :math:`T=\{000~\frac{2}{5}~\frac{3}{5}~111 \}` and the control points :math:`\{ P_i, 1 \leq i \leq 5 \}`:
+
+.. math::
+
+   \mathcal{C}(t) = \sum_{i=1}^{5} {N_{i}^{3}}^{\prime}(t) \mathbf{P}_i 
+
+we have, 
+
+.. math::
+
+   \mathcal{C}^{\prime}(t) = \sum_{i=1}^{4} {N_{i}^{2}}^{\ast}(t) \mathbf{Q}_i
+
+where 
+
+.. math::
+
+   \mathbf{Q}_1 = 5 \{\mathbf{P}_{2} - \mathbf{P}_1\}, ~~~~\mathbf{Q}_2 = \frac{10}{3} \{ \mathbf{P}_{3} - \mathbf{P}_2\},
+   \\
+   \mathbf{Q}_3 = \frac{10}{3} \{ \mathbf{P}_{4} - \mathbf{P}_3\},~~~~\mathbf{Q}_4 = 5 \{\mathbf{P}_{5} - \mathbf{P}_4\}.
+
+The *B-splines* :math:`\{ {N_{i}^{2}}^{\ast},~~1 \leq i \leq 4\}` are associated to the knot vector :math:`T^{\ast}=\{00~\frac{2}{5}~\frac{3}{5}~11 \}`. 
+
+
+.. rubric:: Fundamental geometric operations
+
+By inserting new knots into the knot vector, we add new control points without changing the shape of the B-Spline curve. This can be done using the DeBoor algorithm :cite:`DeBoor_Book2001`. We can also elevate the degree of the B-Spline family and keep unchanged the curve :cite:`qi`. In (Fig. \ref{refinement_curve_B_Spline}), we apply these algorithms on a quadratic B-Spline curve and we show the position of the new control points.  
+
+
+Knot insertion
+^^^^^^^^^^^^^^
+
+
+After modification, we denote by :math:`\widetilde{n}, \widetilde{k}, \widetilde{T}` the new parameters. :math:`(\textbf{Q}_i)` are the new control points.
+
+One can insert a new knot :math:`t`, where :math:`t_j \leqslant t < t_{j+1}`. For this purpose we use the DeBoor algorithm :cite:`DeBoor_Book2001`:
+
+.. math::
+
+   \widetilde{n} = n+1
+   \\
+   \widetilde{k} = k
+   \\
+   \widetilde{T} = \{ t_1,.., t_j, t, t_{j+1},.., t_{n+k}\}
+   \\
+   \alpha_i = \left\{\begin{array}{cc}1 & 1 \leqslant i \leqslant j-k+1 \\\frac{t-t_i}{t_{i+k-1}-t_i} & j-k+2 \leqslant i \leqslant j \\0 & j+1 \leqslant i \end{array}\right.
+   \\
+   \textbf{Q}_i = \alpha_i \textbf{P}_i + (1-\alpha_i) \textbf{P}_{i-1}
+
+Many other algorithms exist, like blossoming for fast insertion algorithm. For more details about this topic, we refer to :cite:`goldman_lyche_book`. 
+
+Order elevation
+^^^^^^^^^^^^^^^
+
+We can elevate the order of the basis, without changing the curve. Several algorithms exist for this purpose. We used the one by Huang et al. :cite:`prautzsch`, :cite:`qi`.
+
+A quadratic B-spline curve and its control points. The knot vector is :math:`T = \{ 000, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 1 1 \}`.
+
+.. image:: ../include/splines/curve.png 
+   :width: 8cm
+   :height: 8cm
+
+The curve after a h-refinement by inserting the knots :math:`\{ 0.15, 0.35\}` while the degree is kept equal to :math:`2`.
+
+.. image:: ../include/splines/curve_p0_n9.png 
+   :width: 8cm
+   :height: 8cm
+
+The curve after a p-refinement, the degree was raised by :math:`1` (using cubic B-splines).
+
+.. image:: ../include/splines/curve_p2_n0.png 
+   :width: 8cm
+   :height: 8cm
+
+The curve after duplicating the multiplicity of the internal knots :math:`\{ \frac{1}{4}, \frac{1}{2}, \frac{3}{4} \}`, 
+this leads to a B\'ezier description. We can then, split the curve into :math:`4` pieces (sub-domains), each one will corresponds to a quadratic B\'ezier curve.
+
+.. image:: ../include/splines/curve_p0_n3_bezier.png 
+   :width: 8cm
+   :height: 8cm
+
+Translation
+^^^^^^^^^^^
+
+Rotation
+^^^^^^^^
+
+.. todo:: not yet available
+
+Scaling
+^^^^^^^
+
+.. todo:: not yet available
+
+
+
+.. rubric:: References
+
+.. bibliography:: refs_spline.bib
+   :cited:
+
diff --git a/_sources/psydac/doc/tutorial.rst b/_sources/psydac/doc/tutorial.rst
new file mode 100644
index 0000000..e55b93d
--- /dev/null
+++ b/_sources/psydac/doc/tutorial.rst
@@ -0,0 +1,30 @@
+.. highlight:: rst
+
+First Steps with PSYDAC
+=======================
+
+This document is meant to give a tutorial-like overview of PSYDAC.
+
+The green arrows designate "more info" links leading to advanced sections about
+the described task.
+
+By reading this tutorial, you'll be able to:
+
+* compile a simple *PSYDAC* file
+
+* get familiar with parallel programing paradigms 
+
+* create, modify and build a *PSYDAC* project.
+
+Install PSYDAC
+**************
+
+Examples
+********
+
+In this section, we describe some features of *PSYDAC* on simple examples.
+
+.. literalinclude:: ../../tests/mapping/1d/test_1d.py 
+  :language: python
+
+See :download:`script <../../tests/mapping/1d/test_1d.py>`.
diff --git a/_sources/psydac/docs/source/_templates/autosummary/module.rst b/_sources/psydac/docs/source/_templates/autosummary/module.rst
new file mode 100644
index 0000000..7e06df5
--- /dev/null
+++ b/_sources/psydac/docs/source/_templates/autosummary/module.rst
@@ -0,0 +1,77 @@
+{{ fullname | smart_fullname | escape | underline}}
+
+.. automodule:: {{ fullname }}
+   :no-index:
+
+   {% block attributes %}
+   {% if attributes %}
+   Module Attributes
+   -----------------
+
+   .. autosummary::
+   {% for item in attributes %}
+      {{ item }}
+   {%- endfor %}
+   {% endif %}
+   {% endblock %}
+
+   {% block functions %}
+   {% if functions %}
+   Functions
+   ---------
+
+   .. autosummary::
+   {% for item in functions %}
+      {{ item }}
+   {%- endfor %}
+   {% endif %}
+   {% endblock %}
+
+   {% block classes %}
+   {% if classes %}
+   Classes
+   -------
+   .. inheritance-diagram:: {{ fullname }}
+      :parts: 1
+      :top-classes: psydac.linalg.basic.LinearOperator, psydac.linalg.basic.Vector, psydac.linalg.basic.VectorSpace, psydac.linalg.basic.LinearSolver
+   .. autosummary::
+   {% for item in classes %}
+      {{ item }}
+   {%- endfor %}      
+   {% endif %}
+   {% endblock %}
+
+   {% block exceptions %}
+   {% if exceptions %}
+   Exceptions
+   ----------
+
+   .. autosummary::
+   {% for item in exceptions %}
+      {{ item }}
+   {%- endfor %}
+   {% endif %}
+   {% endblock %}
+
+{% block modules %}
+{% if modules %}
+Modules
+-------
+
+.. autosummary::
+   :toctree:
+   :recursive:
+{% for item in modules %}
+   {{ item }}
+{%- endfor %}
+{% endif %}
+{% endblock %}
+
+Details
+-------
+
+.. automodule:: {{ fullname }}
+   :members:
+   :undoc-members:
+   :exclude-members:
+   :show-inheritance:
diff --git a/_sources/psydac/docs/source/examples.rst b/_sources/psydac/docs/source/examples.rst
new file mode 100644
index 0000000..d05be7e
--- /dev/null
+++ b/_sources/psydac/docs/source/examples.rst
@@ -0,0 +1,6 @@
+Examples
+========
+
++------------------------------------------------------------------------------------------------------------------------+
+| Here you will find examples of how to use PSYDAC and explanations thereof as well as links to notebooks in the future. |
++------------------------------------------------------------------------------------------------------------------------+
diff --git a/_sources/psydac/docs/source/index.rst b/_sources/psydac/docs/source/index.rst
new file mode 100644
index 0000000..a5bca81
--- /dev/null
+++ b/_sources/psydac/docs/source/index.rst
@@ -0,0 +1,41 @@
+.. PSYDAC documentation master file, created by
+   sphinx-quickstart on Fri Apr 21 11:24:31 2023.
+
+   You can adapt this file completely to your liking, but it should at least
+   contain the root `toctree` directive.
+
+Welcome to PSYDAC's documentation!
+
+Contents
+--------
+.. toctree::
+   :maxdepth: 1
+   
+   modules
+   examples
+
+.. _note:
+.. note::
+
+   This documentation is still under construction.
+   For the time being, its purpose is to assist the developers.
+   Find our GitHub repository `here <https://github.com/pyccel/psydac/tree/devel/>`_.
+
+Psydac is a Python 3 library for isogeometric analysis (IGA). It uses high-order tensor-product splines, multi-patch mapped
+domains, and hybrid MPI-OpenMP parallelization.
+
+In order to use Psydac, the user provides a geometry analytically or through an input file, and then defines the model equations
+in symbolic form (weak formulation) using `SymPDE <https://github.com/pyccel/sympde>`_, which provides the mathematical
+expressions and checks the semantic validity of the model.
+
+Once a finite element discretization has been chosen, Psydac maps the abstract concepts to concrete objects, the basic building
+blocks being MPI-distributed vectors and matrices.
+For all the computationally intensive operations (assembly of matrices, vectors, and norms, etc.), Psydac generates ad-hoc
+Python code which is then accelerated to Fortran speed using `Pyccel <https://github.com/pyccel/pyccel>`_.
+
+Indices and tables
+------------------
+
+* :ref:`genindex`
+* :ref:`modindex`
+* :ref:`search`
diff --git a/_sources/psydac/docs/source/modules.rst b/_sources/psydac/docs/source/modules.rst
new file mode 100644
index 0000000..3b64639
--- /dev/null
+++ b/_sources/psydac/docs/source/modules.rst
@@ -0,0 +1,20 @@
+Modules
+=======
+
+Find here every class and every function of every module in PSYDAC.
+Relying solely on docstrings.
+
+.. toctree::
+    :maxdepth: 1
+
+    modules/api
+    modules/cad
+    modules/cmd
+    modules/core
+    modules/ddm
+    modules/feec
+    modules/fem
+    modules/linalg
+    modules/mapping
+    modules/polar
+    modules/utilities
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.basic.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.basic.rst
new file mode 100644
index 0000000..fe57f37
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.basic.rst
@@ -0,0 +1,29 @@
+psydac.api.ast.basic
+====================
+
+.. automodule:: psydac.api.ast.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      SplBasic
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.evaluation.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.evaluation.rst
new file mode 100644
index 0000000..e0f3ee4
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.evaluation.rst
@@ -0,0 +1,30 @@
+psydac.api.ast.evaluation
+=========================
+
+.. automodule:: psydac.api.ast.evaluation
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      EvalArrayField
+      EvalArrayMapping
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.expr.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.expr.rst
new file mode 100644
index 0000000..815be46
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.expr.rst
@@ -0,0 +1,38 @@
+psydac.api.ast.expr
+===================
+
+.. automodule:: psydac.api.ast.expr
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_atoms_expr
+      is_scalar_field
+      is_vector_field
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      ExprInterface
+      ExprKernel
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.fem.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.fem.rst
new file mode 100644
index 0000000..097f3e9
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.fem.rst
@@ -0,0 +1,44 @@
+psydac.api.ast.fem
+==================
+
+.. automodule:: psydac.api.ast.fem
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_diag_len
+      convert
+      expand
+      expand_hdiv_hcurl
+      get_degrees
+      get_multiplicity
+      get_nquads
+      regroup
+      toInteger
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      AST
+      DefNode
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.glt.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.glt.rst
new file mode 100644
index 0000000..e562170
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.glt.rst
@@ -0,0 +1,30 @@
+psydac.api.ast.glt
+==================
+
+.. automodule:: psydac.api.ast.glt
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      GltInterface
+      GltKernel
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.linalg.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.linalg.rst
new file mode 100644
index 0000000..4beb2fe
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.linalg.rst
@@ -0,0 +1,39 @@
+psydac.api.ast.linalg
+=====================
+
+.. automodule:: psydac.api.ast.linalg
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      Mod
+      compute_diag_len
+      toInteger
+      variable_to_sympy
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      LinearOperatorDot
+      VectorDot
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.nodes.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.nodes.rst
new file mode 100644
index 0000000..d759b28
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.nodes.rst
@@ -0,0 +1,189 @@
+psydac.api.ast.nodes
+====================
+
+.. automodule:: psydac.api.ast.nodes
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      construct_itergener
+      construct_logical_expressions
+      get_number_derivatives
+      random_string
+      toInteger
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      AddNode
+      Allocate
+      AndNode
+      ArityType
+      Array
+      ArrayNode
+      AtomicNode
+      BaseNode
+      BasisAtom
+      BilinearArity
+      Block
+      BlockLinearOperatorNode
+      BlockScalarLocalBasis
+      BlockStencilMatrixGlobalBasis
+      BlockStencilMatrixLocalBasis
+      BlockStencilVectorGlobalBasis
+      BlockStencilVectorLocalBasis
+      CoefficientBasis
+      ComputeKernelExpr
+      ComputeLogical
+      ComputeLogicalBasis
+      ComputeNode
+      ComputePhysical
+      ComputePhysicalBasis
+      Element
+      ElementOf
+      EqNode
+      EvalField
+      EvalMapping
+      Evaluation
+      ExprNode
+      Expression
+      FieldEvaluation
+      FloorDiv
+      FunctionalArity
+      GeneratorBase
+      GeometryAtom
+      GeometryExpr
+      GeometryExpressions
+      GlobalElementBasis
+      GlobalSpanArray
+      GlobalTensorQuadratureBasis
+      GlobalTensorQuadratureGrid
+      GlobalTensorQuadratureTestBasis
+      GlobalTensorQuadratureTrialBasis
+      GlobalThreadEnds
+      GlobalThreadSizes
+      GlobalThreadSpanArray
+      GlobalThreadStarts
+      Grid
+      IfNode
+      IndexDerivative
+      IndexDof
+      IndexDofTest
+      IndexDofTrial
+      IndexElement
+      IndexInnerDofTest
+      IndexNode
+      IndexOuterDofTest
+      IndexQuadrature
+      IntDivNode
+      IteratorBase
+      LengthDof
+      LengthDofTest
+      LengthDofTrial
+      LengthElement
+      LengthInnerDofTest
+      LengthNode
+      LengthOuterDofTest
+      LengthQuadrature
+      LinearArity
+      LocalElementBasis
+      LocalIndexElement
+      LocalSpanArray
+      LocalTensorQuadratureBasis
+      LocalTensorQuadratureGrid
+      LocalTensorQuadratureTestBasis
+      LocalTensorQuadratureTrialBasis
+      LocalThreadEnds
+      LocalThreadStarts
+      LogicalBasisValue
+      LogicalGeometryValue
+      LogicalValueNode
+      Loop
+      MappingEvaluation
+      Mask
+      MatrixCoordsFromRank
+      MatrixGlobalBasis
+      MatrixLocalBasis
+      MatrixNode
+      MatrixQuadrature
+      MatrixRankFromCoords
+      Max
+      Min
+      MulNode
+      NeighbourThreadCoordinates
+      NotNode
+      NumThreads
+      Pads
+      ParallelBlock
+      Pattern
+      PhysicalBasisValue
+      PhysicalGeometryValue
+      PhysicalValueNode
+      PlusGlobalTensorQuadratureGrid
+      PlusLocalTensorQuadratureGrid
+      PlusTensorQuadrature
+      ProductGenerator
+      ProductIteration
+      ProductIterator
+      RAT
+      RankDimension
+      RankNode
+      Reduce
+      Reduction
+      Reset
+      ScalarLocalBasis
+      ScalarNode
+      Span
+      SpanArray
+      SplitArray
+      StencilMatrixGlobalBasis
+      StencilMatrixLocalBasis
+      StencilVectorGlobalBasis
+      StencilVectorLocalBasis
+      StrictLessThanNode
+      TensorAdd
+      TensorAssignExpr
+      TensorBasis
+      TensorExpression
+      TensorGenerator
+      TensorIntDiv
+      TensorInteger
+      TensorIteration
+      TensorIterator
+      TensorMax
+      TensorMul
+      TensorQuadrature
+      TensorQuadratureBasis
+      TensorQuadratureTestBasis
+      TensorQuadratureTrialBasis
+      TensorTestBasis
+      TensorTrialBasis
+      ThreadCoordinates
+      ThreadId
+      ValueNode
+      VectorAssign
+      WeightedVolumeQuadrature
+      WhileLoop
+      Zeros
+      ZerosLike
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.parser.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.parser.rst
new file mode 100644
index 0000000..28ca790
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.parser.rst
@@ -0,0 +1,38 @@
+psydac.api.ast.parser
+=====================
+
+.. automodule:: psydac.api.ast.parser
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      is_scalar_array
+      parse
+      random_string
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Parser
+      Shape
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.utilities.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.utilities.rst
new file mode 100644
index 0000000..c22fb84
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.ast.utilities.rst
@@ -0,0 +1,47 @@
+psydac.api.ast.utilities
+========================
+
+.. automodule:: psydac.api.ast.utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      build_pyccel_type_annotations
+      build_pythran_types_header
+      compute_atoms_expr
+      compute_atoms_expr_field
+      compute_atoms_expr_mapping
+      compute_boundary_jacobian
+      compute_normal_vector
+      compute_tangent_vector
+      filter_loops
+      filter_product
+      fusion_loops
+      get_name
+      is_mapping
+      logical2physical
+      math_atoms_as_str
+      random_string
+      rationalize_eval_mapping
+      select_loops
+      variables
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.basic.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.basic.rst
new file mode 100644
index 0000000..a7d8e00
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.basic.rst
@@ -0,0 +1,30 @@
+psydac.api.basic
+================
+
+.. automodule:: psydac.api.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BasicCodeGen
+      BasicDiscrete
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.discretization.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.discretization.rst
new file mode 100644
index 0000000..1528391
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.discretization.rst
@@ -0,0 +1,34 @@
+psydac.api.discretization
+=========================
+
+.. automodule:: psydac.api.discretization
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      change_dtype
+      discretize
+      discretize_derham
+      discretize_domain
+      discretize_space
+      reduce_space_degrees
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.equation.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.equation.rst
new file mode 100644
index 0000000..bb54853
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.equation.rst
@@ -0,0 +1,36 @@
+psydac.api.equation
+===================
+
+.. automodule:: psydac.api.equation
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      l2_boundary_projection
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteEquation
+      LinearSystem
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.essential_bc.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.essential_bc.rst
new file mode 100644
index 0000000..77917f3
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.essential_bc.rst
@@ -0,0 +1,34 @@
+psydac.api.essential\_bc
+========================
+
+.. automodule:: psydac.api.essential_bc
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      apply_essential_bc
+      apply_essential_bc_BlockLinearOperator
+      apply_essential_bc_BlockVector
+      apply_essential_bc_kronecker_dense_matrix
+      apply_essential_bc_stencil
+      check_boundary_type
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.expr.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.expr.rst
new file mode 100644
index 0000000..d3b0daf
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.expr.rst
@@ -0,0 +1,29 @@
+psydac.api.expr
+===============
+
+.. automodule:: psydac.api.expr
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteExpr
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.feec.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.feec.rst
new file mode 100644
index 0000000..1ecb32a
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.feec.rst
@@ -0,0 +1,29 @@
+psydac.api.feec
+===============
+
+.. automodule:: psydac.api.feec
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteDerham
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.fem.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.fem.rst
new file mode 100644
index 0000000..4ae7043
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.fem.rst
@@ -0,0 +1,47 @@
+psydac.api.fem
+==============
+
+.. automodule:: psydac.api.fem
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      collect_spaces
+      compute_diag_len
+      construct_quad_grids_arguments
+      construct_test_space_arguments
+      construct_trial_space_arguments
+      do_nothing
+      extract_stencil_mats
+      get_nquads
+      reset_arrays
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteBilinearForm
+      DiscreteFunctional
+      DiscreteLinearForm
+      DiscreteSesquilinearForm
+      DiscreteSumForm
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.glt.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.glt.rst
new file mode 100644
index 0000000..a45c350
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.glt.rst
@@ -0,0 +1,30 @@
+psydac.api.glt
+==============
+
+.. automodule:: psydac.api.glt
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteGltExpr
+      GltBasicCodeGen
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.grid.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.grid.rst
new file mode 100644
index 0000000..136f1b8
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.grid.rst
@@ -0,0 +1,38 @@
+psydac.api.grid
+===============
+
+.. automodule:: psydac.api.grid
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      create_collocation_basis
+      get_points_weights
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BasisValues
+      CollocationBasisValues
+      QuadratureGrid
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.postprocessing.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.postprocessing.rst
new file mode 100644
index 0000000..6bee388
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.postprocessing.rst
@@ -0,0 +1,36 @@
+psydac.api.postprocessing
+=========================
+
+.. automodule:: psydac.api.postprocessing
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_grid_lines_2d
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      OutputManager
+      PostProcessManager
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.printing.pycode.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.printing.pycode.rst
new file mode 100644
index 0000000..3233191
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.api.printing.pycode.rst
@@ -0,0 +1,35 @@
+psydac.api.printing.pycode
+==========================
+
+.. automodule:: psydac.api.printing.pycode
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      pycode
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      PythonCodePrinter
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.cad.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.cad.rst
new file mode 100644
index 0000000..632aa39
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.cad.rst
@@ -0,0 +1,32 @@
+psydac.cad.cad
+==============
+
+.. automodule:: psydac.cad.cad
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      elevate
+      random_string
+      refine
+      translate
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.gallery.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.gallery.rst
new file mode 100644
index 0000000..149f076
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.gallery.rst
@@ -0,0 +1,31 @@
+psydac.cad.gallery
+==================
+
+.. automodule:: psydac.cad.gallery
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      annulus
+      circle
+      quart_circle
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.geometry.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.geometry.rst
new file mode 100644
index 0000000..259c2b4
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.geometry.rst
@@ -0,0 +1,38 @@
+psydac.cad.geometry
+===================
+
+.. automodule:: psydac.cad.geometry
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      export_nurbs_to_hdf5
+      import_geopdes_to_nurbs
+      refine_knots
+      refine_nurbs
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Geometry
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.multipatch.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.multipatch.rst
new file mode 100644
index 0000000..c3672c2
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.multipatch.rst
@@ -0,0 +1,29 @@
+psydac.cad.multipatch
+=====================
+
+.. automodule:: psydac.cad.multipatch
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      export_multipatch_nurbs_to_hdf5
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.utils.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.utils.rst
new file mode 100644
index 0000000..0a1b8b0
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cad.utils.rst
@@ -0,0 +1,29 @@
+psydac.cad.utils
+================
+
+.. automodule:: psydac.cad.utils
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      plot_mapping
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.cmd.mesh.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cmd.mesh.rst
new file mode 100644
index 0000000..864825e
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.cmd.mesh.rst
@@ -0,0 +1,31 @@
+psydac.cmd.mesh
+===============
+
+.. automodule:: psydac.cmd.mesh
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      export_analytical_mapping
+      export_caid_geometry
+      main
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.core.bsplines.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.core.bsplines.rst
new file mode 100644
index 0000000..288eaaa
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.core.bsplines.rst
@@ -0,0 +1,47 @@
+psydac.core.bsplines
+====================
+
+.. automodule:: psydac.core.bsplines
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      basis_ders_on_irregular_grid
+      basis_ders_on_quad_grid
+      basis_funs
+      basis_funs_1st_der
+      basis_funs_all_ders
+      basis_funs_array
+      basis_integrals
+      breakpoints
+      cell_index
+      collocation_matrix
+      elements_spans
+      elevate_knots
+      find_span
+      find_spans
+      greville
+      histopolation_matrix
+      hrefinement_matrix
+      make_knots
+      quadrature_grid
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.core.bsplines_kernels.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.core.bsplines_kernels.rst
new file mode 100644
index 0000000..03f8724
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.core.bsplines_kernels.rst
@@ -0,0 +1,47 @@
+psydac.core.bsplines\_kernels
+=============================
+
+.. automodule:: psydac.core.bsplines_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      basis_ders_on_irregular_grid_p
+      basis_ders_on_quad_grid_p
+      basis_funs_1st_der_p
+      basis_funs_all_ders_p
+      basis_funs_array_p
+      basis_funs_p
+      basis_integrals_p
+      breakpoints_p
+      cell_index_p
+      collocation_matrix_p
+      elements_spans_p
+      elevate_knots_p
+      find_span_p
+      find_spans_p
+      greville_p
+      histopolation_matrix_p
+      make_knots_p
+      merge_sort
+      quadrature_grid_p
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.core.field_evaluation_kernels.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.core.field_evaluation_kernels.rst
new file mode 100644
index 0000000..5674096
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.core.field_evaluation_kernels.rst
@@ -0,0 +1,66 @@
+psydac.core.field\_evaluation\_kernels
+======================================
+
+.. automodule:: psydac.core.field_evaluation_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      eval_fields_2d_irregular_no_weights
+      eval_fields_2d_irregular_weighted
+      eval_fields_2d_no_weights
+      eval_fields_2d_weighted
+      eval_fields_3d_irregular_no_weights
+      eval_fields_3d_irregular_weighted
+      eval_fields_3d_no_weights
+      eval_fields_3d_weighted
+      eval_jac_det_2d
+      eval_jac_det_2d_weights
+      eval_jac_det_3d
+      eval_jac_det_3d_weights
+      eval_jac_det_irregular_2d
+      eval_jac_det_irregular_2d_weights
+      eval_jac_det_irregular_3d
+      eval_jac_det_irregular_3d_weights
+      eval_jacobians_2d
+      eval_jacobians_2d_weights
+      eval_jacobians_3d
+      eval_jacobians_3d_weights
+      eval_jacobians_inv_2d
+      eval_jacobians_inv_2d_weights
+      eval_jacobians_inv_3d
+      eval_jacobians_inv_3d_weights
+      eval_jacobians_inv_irregular_2d
+      eval_jacobians_inv_irregular_2d_weights
+      eval_jacobians_inv_irregular_3d
+      eval_jacobians_inv_irregular_3d_weights
+      eval_jacobians_irregular_2d
+      eval_jacobians_irregular_2d_weights
+      eval_jacobians_irregular_3d
+      eval_jacobians_irregular_3d_weights
+      pushforward_2d_hcurl
+      pushforward_2d_hdiv
+      pushforward_2d_l2
+      pushforward_3d_hcurl
+      pushforward_3d_hdiv
+      pushforward_3d_l2
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.basic.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.basic.rst
new file mode 100644
index 0000000..c6e0ced
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.basic.rst
@@ -0,0 +1,29 @@
+psydac.ddm.basic
+================
+
+.. automodule:: psydac.ddm.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      CartDataExchanger
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.blocking_data_exchanger.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.blocking_data_exchanger.rst
new file mode 100644
index 0000000..7619bcb
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.blocking_data_exchanger.rst
@@ -0,0 +1,29 @@
+psydac.ddm.blocking\_data\_exchanger
+====================================
+
+.. automodule:: psydac.ddm.blocking_data_exchanger
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BlockingCartDataExchanger
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.cart.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.cart.rst
new file mode 100644
index 0000000..2bda703
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.cart.rst
@@ -0,0 +1,39 @@
+psydac.ddm.cart
+===============
+
+.. automodule:: psydac.ddm.cart
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      create_interfaces_cart
+      find_mpi_type
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      CartDecomposition
+      DomainDecomposition
+      InterfaceCartDecomposition
+      MultiPatchDomainDecomposition
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.interface_data_exchanger.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.interface_data_exchanger.rst
new file mode 100644
index 0000000..d3c1e29
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.interface_data_exchanger.rst
@@ -0,0 +1,29 @@
+psydac.ddm.interface\_data\_exchanger
+=====================================
+
+.. automodule:: psydac.ddm.interface_data_exchanger
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      InterfaceCartDataExchanger
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.nonblocking_data_exchanger.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.nonblocking_data_exchanger.rst
new file mode 100644
index 0000000..57b1a50
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.nonblocking_data_exchanger.rst
@@ -0,0 +1,29 @@
+psydac.ddm.nonblocking\_data\_exchanger
+=======================================
+
+.. automodule:: psydac.ddm.nonblocking_data_exchanger
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      NonBlockingCartDataExchanger
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.partition.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.partition.rst
new file mode 100644
index 0000000..26936be
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.partition.rst
@@ -0,0 +1,32 @@
+psydac.ddm.partition
+====================
+
+.. automodule:: psydac.ddm.partition
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_dims
+      compute_dims_general
+      compute_dims_uniform
+      partition_procs_per_patch
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.petsc.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.petsc.rst
new file mode 100644
index 0000000..e1b3dea
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.petsc.rst
@@ -0,0 +1,29 @@
+psydac.ddm.petsc
+================
+
+.. automodule:: psydac.ddm.petsc
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      PetscCart
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.utilities.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.utilities.rst
new file mode 100644
index 0000000..b2842b8
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.ddm.utilities.rst
@@ -0,0 +1,29 @@
+psydac.ddm.utilities
+====================
+
+.. automodule:: psydac.ddm.utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      get_data_exchanger
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.derivatives.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.derivatives.rst
new file mode 100644
index 0000000..5324c41
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.derivatives.rst
@@ -0,0 +1,44 @@
+psydac.feec.derivatives
+=======================
+
+.. automodule:: psydac.feec.derivatives
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      block_tostencil
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Curl_3D
+      Derivative_1D
+      DiffOperator
+      DirectionalDerivativeOperator
+      Divergence_2D
+      Divergence_3D
+      Gradient_2D
+      Gradient_3D
+      ScalarCurl_2D
+      VectorCurl_2D
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.global_projectors.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.global_projectors.rst
new file mode 100644
index 0000000..265e03d
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.global_projectors.rst
@@ -0,0 +1,51 @@
+psydac.feec.global\_projectors
+==============================
+
+.. automodule:: psydac.feec.global_projectors
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      evaluate_dofs_1d_0form
+      evaluate_dofs_1d_1form
+      evaluate_dofs_2d_0form
+      evaluate_dofs_2d_1form_hcurl
+      evaluate_dofs_2d_1form_hdiv
+      evaluate_dofs_2d_2form
+      evaluate_dofs_2d_vec
+      evaluate_dofs_3d_0form
+      evaluate_dofs_3d_1form
+      evaluate_dofs_3d_2form
+      evaluate_dofs_3d_3form
+      evaluate_dofs_3d_vec
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      GlobalProjector
+      Projector_H1
+      Projector_H1vec
+      Projector_Hcurl
+      Projector_Hdiv
+      Projector_L2
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.api.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.api.rst
new file mode 100644
index 0000000..96cc3d1
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.api.rst
@@ -0,0 +1,36 @@
+psydac.feec.multipatch.api
+==========================
+
+.. automodule:: psydac.feec.multipatch.api
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      discretize
+      discretize_derham_multipatch
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DiscreteDerhamMultipatch
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.fem_linear_operators.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.fem_linear_operators.rst
new file mode 100644
index 0000000..4eba7ad
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.fem_linear_operators.rst
@@ -0,0 +1,33 @@
+psydac.feec.multipatch.fem\_linear\_operators
+=============================================
+
+.. automodule:: psydac.feec.multipatch.fem_linear_operators
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      ComposedLinearOperator
+      FemLinearOperator
+      IdLinearOperator
+      MultLinearOperator
+      SumLinearOperator
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.operators.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.operators.rst
new file mode 100644
index 0000000..9d5faf6
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.operators.rst
@@ -0,0 +1,50 @@
+psydac.feec.multipatch.operators
+================================
+
+.. automodule:: psydac.feec.multipatch.operators
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      allocate_interface_matrix
+      get_K0_and_K0_inv
+      get_K1_and_K1_inv
+      get_interface_from_corners
+      get_patch_index_from_face
+      get_row_col_index
+      ortho_proj_Hcurl
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BrokenGradient_2D
+      BrokenScalarCurl_2D
+      BrokenTransposedGradient_2D
+      BrokenTransposedScalarCurl_2D
+      ConformingProjection_V0
+      ConformingProjection_V1
+      HodgeOperator
+      Multipatch_Projector_H1
+      Multipatch_Projector_Hcurl
+      Multipatch_Projector_L2
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.utilities.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.utilities.rst
new file mode 100644
index 0000000..1dd3621
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.utilities.rst
@@ -0,0 +1,35 @@
+psydac.feec.multipatch.utilities
+================================
+
+.. automodule:: psydac.feec.multipatch.utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      error_fn
+      get_fem_name
+      get_load_dir
+      get_method_name
+      sol_ref_fn
+      source_name
+      time_count
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.pull_push.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.pull_push.rst
new file mode 100644
index 0000000..1255923
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.pull_push.rst
@@ -0,0 +1,50 @@
+psydac.feec.pull\_push
+======================
+
+.. automodule:: psydac.feec.pull_push
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      pull_1d_h1
+      pull_1d_l2
+      pull_2d_h1
+      pull_2d_h1vec
+      pull_2d_hcurl
+      pull_2d_hdiv
+      pull_2d_l2
+      pull_3d_h1
+      pull_3d_h1vec
+      pull_3d_hcurl
+      pull_3d_hdiv
+      pull_3d_l2
+      push_1d_h1
+      push_1d_l2
+      push_2d_h1
+      push_2d_hcurl
+      push_2d_hdiv
+      push_2d_l2
+      push_3d_h1
+      push_3d_hcurl
+      push_3d_hdiv
+      push_3d_l2
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.pushforward.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.pushforward.rst
new file mode 100644
index 0000000..aaf49ce
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.feec.pushforward.rst
@@ -0,0 +1,29 @@
+psydac.feec.pushforward
+=======================
+
+.. automodule:: psydac.feec.pushforward
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Pushforward
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.basic.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.basic.rst
new file mode 100644
index 0000000..c921083
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.basic.rst
@@ -0,0 +1,30 @@
+psydac.fem.basic
+================
+
+.. automodule:: psydac.fem.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      FemField
+      FemSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.context.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.context.rst
new file mode 100644
index 0000000..a0e0fad
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.context.rst
@@ -0,0 +1,29 @@
+psydac.fem.context
+==================
+
+.. automodule:: psydac.fem.context
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      fem_context
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.grid.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.grid.rst
new file mode 100644
index 0000000..7d8e571
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.grid.rst
@@ -0,0 +1,29 @@
+psydac.fem.grid
+===============
+
+.. automodule:: psydac.fem.grid
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      FemAssemblyGrid
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.partitioning.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.partitioning.rst
new file mode 100644
index 0000000..7b29130
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.partitioning.rst
@@ -0,0 +1,35 @@
+psydac.fem.partitioning
+=======================
+
+.. automodule:: psydac.fem.partitioning
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      construct_connectivity
+      construct_interface_spaces
+      construct_reduced_interface_spaces
+      create_cart
+      get_minus_starts_ends
+      get_plus_starts_ends
+      partition_coefficients
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.projectors.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.projectors.rst
new file mode 100644
index 0000000..d2c989d
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.projectors.rst
@@ -0,0 +1,30 @@
+psydac.fem.projectors
+=====================
+
+.. automodule:: psydac.fem.projectors
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      knot_insertion_projection_operator
+      knots_to_insert
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.splines.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.splines.rst
new file mode 100644
index 0000000..8ffa457
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.splines.rst
@@ -0,0 +1,29 @@
+psydac.fem.splines
+==================
+
+.. automodule:: psydac.fem.splines
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      SplineSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.tensor.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.tensor.rst
new file mode 100644
index 0000000..07390b9
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.tensor.rst
@@ -0,0 +1,29 @@
+psydac.fem.tensor
+=================
+
+.. automodule:: psydac.fem.tensor
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      TensorFemSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.vector.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.vector.rst
new file mode 100644
index 0000000..7606208
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.fem.vector.rst
@@ -0,0 +1,30 @@
+psydac.fem.vector
+=================
+
+.. automodule:: psydac.fem.vector
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      ProductFemSpace
+      VectorFemSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.basic.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.basic.rst
new file mode 100644
index 0000000..142146a
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.basic.rst
@@ -0,0 +1,39 @@
+psydac.linalg.basic
+===================
+
+.. automodule:: psydac.linalg.basic
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      ComposedLinearOperator
+      IdentityOperator
+      InverseLinearOperator
+      LinearOperator
+      LinearSolver
+      PowerLinearOperator
+      ScaledLinearOperator
+      SumLinearOperator
+      Vector
+      VectorSpace
+      ZeroOperator
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.block.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.block.rst
new file mode 100644
index 0000000..4e18355
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.block.rst
@@ -0,0 +1,31 @@
+psydac.linalg.block
+===================
+
+.. automodule:: psydac.linalg.block
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BlockLinearOperator
+      BlockVector
+      BlockVectorSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.direct_solvers.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.direct_solvers.rst
new file mode 100644
index 0000000..23e193e
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.direct_solvers.rst
@@ -0,0 +1,30 @@
+psydac.linalg.direct\_solvers
+=============================
+
+.. automodule:: psydac.linalg.direct_solvers
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BandedSolver
+      SparseSolver
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.fft.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.fft.rst
new file mode 100644
index 0000000..6d61c01
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.fft.rst
@@ -0,0 +1,35 @@
+psydac.linalg.fft
+=================
+
+.. automodule:: psydac.linalg.fft
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DistributedDCT
+      DistributedDST
+      DistributedFFT
+      DistributedFFTBase
+      DistributedIDCT
+      DistributedIDST
+      DistributedIFFT
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.axpy_kernels.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.axpy_kernels.rst
new file mode 100644
index 0000000..442f180
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.axpy_kernels.rst
@@ -0,0 +1,31 @@
+psydac.linalg.kernels.axpy\_kernels
+===================================
+
+.. automodule:: psydac.linalg.kernels.axpy_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      axpy_1d
+      axpy_2d
+      axpy_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.inner_kernels.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.inner_kernels.rst
new file mode 100644
index 0000000..759dabc
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.inner_kernels.rst
@@ -0,0 +1,31 @@
+psydac.linalg.kernels.inner\_kernels
+====================================
+
+.. automodule:: psydac.linalg.kernels.inner_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      inner_1d
+      inner_2d
+      inner_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.matvec_kernels.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.matvec_kernels.rst
new file mode 100644
index 0000000..995e06f
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.matvec_kernels.rst
@@ -0,0 +1,31 @@
+psydac.linalg.kernels.matvec\_kernels
+=====================================
+
+.. automodule:: psydac.linalg.kernels.matvec_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      matvec_1d
+      matvec_2d
+      matvec_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.rst
new file mode 100644
index 0000000..0bc1c55
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.rst
@@ -0,0 +1,23 @@
+psydac.linalg.kernels
+=====================
+
+.. automodule:: psydac.linalg.kernels
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.stencil2coo_kernels.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.stencil2coo_kernels.rst
new file mode 100644
index 0000000..5f337b4
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.stencil2coo_kernels.rst
@@ -0,0 +1,34 @@
+psydac.linalg.kernels.stencil2coo\_kernels
+==========================================
+
+.. automodule:: psydac.linalg.kernels.stencil2coo_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      stencil2coo_1d_C
+      stencil2coo_1d_F
+      stencil2coo_2d_C
+      stencil2coo_2d_F
+      stencil2coo_3d_C
+      stencil2coo_3d_F
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.transpose_kernels.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.transpose_kernels.rst
new file mode 100644
index 0000000..4e33ba7
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.transpose_kernels.rst
@@ -0,0 +1,34 @@
+psydac.linalg.kernels.transpose\_kernels
+========================================
+
+.. automodule:: psydac.linalg.kernels.transpose_kernels
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      interface_transpose_1d
+      interface_transpose_2d
+      interface_transpose_3d
+      transpose_1d
+      transpose_2d
+      transpose_3d
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kron.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kron.rst
new file mode 100644
index 0000000..2d0bf36
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.kron.rst
@@ -0,0 +1,37 @@
+psydac.linalg.kron
+==================
+
+.. automodule:: psydac.linalg.kron
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      kronecker_solve
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      KroneckerDenseMatrix
+      KroneckerLinearSolver
+      KroneckerStencilMatrix
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.solvers.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.solvers.rst
new file mode 100644
index 0000000..adee6bd
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.solvers.rst
@@ -0,0 +1,42 @@
+psydac.linalg.solvers
+=====================
+
+.. automodule:: psydac.linalg.solvers
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      inverse
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      BiConjugateGradient
+      BiConjugateGradientStabilized
+      ConjugateGradient
+      GMRES
+      LSMR
+      MinimumResidual
+      PBiConjugateGradientStabilized
+      PConjugateGradient
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.stencil.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.stencil.rst
new file mode 100644
index 0000000..0e6cdb1
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.stencil.rst
@@ -0,0 +1,40 @@
+psydac.linalg.stencil
+=====================
+
+.. automodule:: psydac.linalg.stencil
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      compute_diag_len
+      flip_axis
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      StencilDiagonalMatrix
+      StencilInterfaceMatrix
+      StencilMatrix
+      StencilVector
+      StencilVectorSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.topetsc.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.topetsc.rst
new file mode 100644
index 0000000..b10cade
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.topetsc.rst
@@ -0,0 +1,31 @@
+psydac.linalg.topetsc
+=====================
+
+.. automodule:: psydac.linalg.topetsc
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      flatten_vec
+      mat_topetsc
+      vec_topetsc
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.utilities.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.utilities.rst
new file mode 100644
index 0000000..8938315
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.linalg.utilities.rst
@@ -0,0 +1,30 @@
+psydac.linalg.utilities
+=======================
+
+.. automodule:: psydac.linalg.utilities
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      array_to_psydac
+      petsc_to_psydac
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.mapping.discrete.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.mapping.discrete.rst
new file mode 100644
index 0000000..d6ef398
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.mapping.discrete.rst
@@ -0,0 +1,36 @@
+psydac.mapping.discrete
+=======================
+
+.. automodule:: psydac.mapping.discrete
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      random_string
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      NurbsMapping
+      SplineMapping
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.mapping.discrete_gallery.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.mapping.discrete_gallery.rst
new file mode 100644
index 0000000..3cc8a67
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.mapping.discrete_gallery.rst
@@ -0,0 +1,35 @@
+psydac.mapping.discrete\_gallery
+================================
+
+.. automodule:: psydac.mapping.discrete_gallery
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      discrete_mapping
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      Collela3D
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_cart.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_cart.rst
new file mode 100644
index 0000000..c9f3816
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_cart.rst
@@ -0,0 +1,29 @@
+psydac.polar.c1\_cart
+=====================
+
+.. automodule:: psydac.polar.c1_cart
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      C1_Cart
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_linops.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_linops.rst
new file mode 100644
index 0000000..cfdae86
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_linops.rst
@@ -0,0 +1,30 @@
+psydac.polar.c1\_linops
+=======================
+
+.. automodule:: psydac.polar.c1_linops
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      LinearOperator_DenseToStencil
+      LinearOperator_StencilToDense
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_projections.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_projections.rst
new file mode 100644
index 0000000..e3b59ae
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_projections.rst
@@ -0,0 +1,29 @@
+psydac.polar.c1\_projections
+============================
+
+.. automodule:: psydac.polar.c1_projections
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      C1Projector
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_spaces.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_spaces.rst
new file mode 100644
index 0000000..cd75bcf
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.c1_spaces.rst
@@ -0,0 +1,29 @@
+psydac.polar.c1\_spaces
+=======================
+
+.. automodule:: psydac.polar.c1_spaces
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      new_c1_vector_space
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.dense.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.dense.rst
new file mode 100644
index 0000000..24cfe89
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.polar.dense.rst
@@ -0,0 +1,31 @@
+psydac.polar.dense
+==================
+
+.. automodule:: psydac.polar.dense
+
+   
+   
+   
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Classes
+
+   .. autosummary::
+   
+      DenseMatrix
+      DenseVector
+      DenseVectorSpace
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.utilities.quadratures.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.utilities.quadratures.rst
new file mode 100644
index 0000000..ae0c8c0
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.utilities.quadratures.rst
@@ -0,0 +1,31 @@
+psydac.utilities.quadratures
+============================
+
+.. automodule:: psydac.utilities.quadratures
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      gauss_legendre
+      gauss_lobatto
+      quadrature
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.utilities.utils.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.utilities.utils.rst
new file mode 100644
index 0000000..839e403
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.utilities.utils.rst
@@ -0,0 +1,35 @@
+psydac.utilities.utils
+======================
+
+.. automodule:: psydac.utilities.utils
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      animate_field
+      is_real
+      refine_array_1d
+      roll_edges
+      split_field
+      split_space
+      unroll_edges
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/STUBDIR/psydac.utilities.vtk.rst b/_sources/psydac/docs/source/modules/STUBDIR/psydac.utilities.vtk.rst
new file mode 100644
index 0000000..ade25f1
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/STUBDIR/psydac.utilities.vtk.rst
@@ -0,0 +1,29 @@
+psydac.utilities.vtk
+====================
+
+.. automodule:: psydac.utilities.vtk
+
+   
+   
+   
+
+   
+   
+   .. rubric:: Functions
+
+   .. autosummary::
+   
+      writeParallelVTKUnstructuredGrid
+   
+   
+
+   
+   
+   
+
+   
+   
+   
+
+
+
diff --git a/_sources/psydac/docs/source/modules/api.ast.rst b/_sources/psydac/docs/source/modules/api.ast.rst
new file mode 100644
index 0000000..98dd0c3
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/api.ast.rst
@@ -0,0 +1,18 @@
+api.ast
+=======
+
+.. currentmodule:: psydac.api
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    ast.basic
+    ast.evaluation
+    ast.expr
+    ast.fem
+    ast.glt
+    ast.linalg
+    ast.nodes
+    ast.parser
+    ast.utilities
diff --git a/_sources/psydac/docs/source/modules/api.printing.rst b/_sources/psydac/docs/source/modules/api.printing.rst
new file mode 100644
index 0000000..00313cc
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/api.printing.rst
@@ -0,0 +1,10 @@
+api.printing
+============
+
+.. currentmodule:: psydac.api
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    printing.pycode
diff --git a/_sources/psydac/docs/source/modules/api.rst b/_sources/psydac/docs/source/modules/api.rst
new file mode 100644
index 0000000..a105fe7
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/api.rst
@@ -0,0 +1,27 @@
+api
+===
+
+.. currentmodule:: psydac
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    api.basic
+    api.discretization
+    api.equation
+    api.essential_bc
+    api.expr
+    api.feec
+    api.fem
+    api.glt
+    api.grid
+    api.postprocessing
+
+api submodules
+--------------
+.. toctree::
+    :maxdepth: 1
+
+    api.ast
+    api.printing
diff --git a/_sources/psydac/docs/source/modules/cad.rst b/_sources/psydac/docs/source/modules/cad.rst
new file mode 100644
index 0000000..7891875
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/cad.rst
@@ -0,0 +1,14 @@
+cad
+===
+
+.. currentmodule:: psydac
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    cad.cad
+    cad.gallery
+    cad.geometry
+    cad.multipatch
+    cad.utils
diff --git a/_sources/psydac/docs/source/modules/cmd.rst b/_sources/psydac/docs/source/modules/cmd.rst
new file mode 100644
index 0000000..c1f33d9
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/cmd.rst
@@ -0,0 +1,10 @@
+cmd
+===
+
+.. currentmodule:: psydac
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    cmd.mesh
diff --git a/_sources/psydac/docs/source/modules/core.rst b/_sources/psydac/docs/source/modules/core.rst
new file mode 100644
index 0000000..bfb0ed7
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/core.rst
@@ -0,0 +1,12 @@
+core
+====
+
+.. currentmodule:: psydac
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    core.bsplines_kernels
+    core.bsplines
+    core.field_evaluation_kernels
diff --git a/_sources/psydac/docs/source/modules/ddm.rst b/_sources/psydac/docs/source/modules/ddm.rst
new file mode 100644
index 0000000..888772d
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/ddm.rst
@@ -0,0 +1,17 @@
+ddm
+===
+
+.. currentmodule:: psydac
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    ddm.basic
+    ddm.blocking_data_exchanger
+    ddm.cart
+    ddm.interface_data_exchanger
+    ddm.nonblocking_data_exchanger
+    ddm.partition
+    ddm.petsc
+    ddm.utilities
diff --git a/_sources/psydac/docs/source/modules/feec.multipatch.rst b/_sources/psydac/docs/source/modules/feec.multipatch.rst
new file mode 100644
index 0000000..03daf1d
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/feec.multipatch.rst
@@ -0,0 +1,13 @@
+feec.multipatch
+===============
+
+.. currentmodule:: psydac.feec
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    multipatch.api
+    multipatch.fem_linear_operators
+    multipatch.operators
+    multipatch.utilities
diff --git a/_sources/psydac/docs/source/modules/feec.rst b/_sources/psydac/docs/source/modules/feec.rst
new file mode 100644
index 0000000..5d424f3
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/feec.rst
@@ -0,0 +1,20 @@
+feec
+====
+
+.. currentmodule:: psydac
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    feec.derivatives
+    feec.global_projectors
+    feec.pull_push
+    feec.pushforward
+
+feec submodules
+---------------
+.. toctree::
+    :maxdepth: 1
+
+    feec.multipatch
diff --git a/_sources/psydac/docs/source/modules/fem.rst b/_sources/psydac/docs/source/modules/fem.rst
new file mode 100644
index 0000000..7d01e64
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/fem.rst
@@ -0,0 +1,17 @@
+fem
+===
+
+.. currentmodule:: psydac
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    fem.basic
+    fem.context
+    fem.grid
+    fem.partitioning
+    fem.projectors
+    fem.splines
+    fem.tensor
+    fem.vector
diff --git a/_sources/psydac/docs/source/modules/linalg.kernels.rst b/_sources/psydac/docs/source/modules/linalg.kernels.rst
new file mode 100644
index 0000000..7a99f04
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/linalg.kernels.rst
@@ -0,0 +1,14 @@
+linalg.kernels
+==============
+
+.. currentmodule:: psydac.linalg
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    kernels.axpy_kernels
+    kernels.inner_kernels
+    kernels.matvec_kernels
+    kernels.stencil2coo_kernels
+    kernels.transpose_kernels
diff --git a/_sources/psydac/docs/source/modules/linalg.rst b/_sources/psydac/docs/source/modules/linalg.rst
new file mode 100644
index 0000000..7ccbbf4
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/linalg.rst
@@ -0,0 +1,26 @@
+linalg
+======
+
+.. currentmodule:: psydac
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    linalg.basic
+    linalg.block
+    linalg.direct_solvers
+    linalg.fft
+    linalg.kernels
+    linalg.kron
+    linalg.solvers
+    linalg.stencil
+    linalg.topetsc
+    linalg.utilities
+
+linalg submodules
+-----------------
+.. toctree::
+    :maxdepth: 1
+
+    linalg.kernels
diff --git a/_sources/psydac/docs/source/modules/mapping.rst b/_sources/psydac/docs/source/modules/mapping.rst
new file mode 100644
index 0000000..029d595
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/mapping.rst
@@ -0,0 +1,11 @@
+mapping
+=======
+
+.. currentmodule:: psydac
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    mapping.discrete
+    mapping.discrete_gallery
diff --git a/_sources/psydac/docs/source/modules/polar.rst b/_sources/psydac/docs/source/modules/polar.rst
new file mode 100644
index 0000000..b2b3edc
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/polar.rst
@@ -0,0 +1,14 @@
+polar
+=====
+
+.. currentmodule:: psydac
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    polar.c1_cart
+    polar.c1_linops
+    polar.c1_projections
+    polar.c1_spaces
+    polar.dense
diff --git a/_sources/psydac/docs/source/modules/utilities.rst b/_sources/psydac/docs/source/modules/utilities.rst
new file mode 100644
index 0000000..1f51877
--- /dev/null
+++ b/_sources/psydac/docs/source/modules/utilities.rst
@@ -0,0 +1,12 @@
+utilities
+=========
+
+.. currentmodule:: psydac
+.. autosummary::
+    :nosignatures:
+    :toctree: STUBDIR
+    :template: autosummary/module.rst
+
+    utilities.quadratures
+    utilities.utils
+    utilities.vtk
diff --git a/_sources/psydac/examples/notebooks/Helmholtz_non_periodic.ipynb b/_sources/psydac/examples/notebooks/Helmholtz_non_periodic.ipynb
new file mode 100644
index 0000000..8d59495
--- /dev/null
+++ b/_sources/psydac/examples/notebooks/Helmholtz_non_periodic.ipynb
@@ -0,0 +1,312 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Solving Helmholtz's equation on non-periodic topological domains\n",
+    "In this notebook, we will show how to solve Helmholtz's equation on single patch and multipatch domains. As we are running this in a notebook, everything will be run in serial and hence we are limiting ourselves to a fairly coarse discretization to avoid taking too much time. However, Psydac allows for hybrid MPI + OpenMP parallelization with barely any changed to the code. The lines that are impacted by that will be preceeded by their MPI equivalent\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 1 : Building the domain\n",
+    "Psydac uses the powerful topological tools of SymPDE to build a large variety of domains. Here we show how to create a Square."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "from sympde.topology import Square\n",
+    "Omega=Square()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 2: Defining the Abstract PDE model using SymPDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.calculus import grad, dot\n",
+    "from sympde.calculus import minus, plus\n",
+    "\n",
+    "from sympde.expr.expr          import LinearForm, BilinearForm\n",
+    "from sympde.expr.expr          import integral              \n",
+    "from sympde.expr.expr          import Norm                       \n",
+    "from sympde.expr               import find, EssentialBC\n",
+    "\n",
+    "from sympde.topology import ScalarFunctionSpace\n",
+    "from sympde.topology import elements_of\n",
+    "from sympde.topology import NormalVector\n",
+    "from sympde.topology import Union\n",
+    "from sympy import pi, cos, sin, symbols, conjugate, exp, sqrt\n",
+    "\n",
+    "# Define the abstract model to solve Helmholtz's equation using the manufactured solution method\n",
+    "\n",
+    "x, y = Omega.coordinates\n",
+    "kappa = 2*pi\n",
+    "solution = exp(1j * kappa * x) * sin(kappa * y)\n",
+    "e_w_0 = sin(kappa * y) # value of incoming wave at x=0, forall y\n",
+    "dx_e_w_0 = 1j*kappa*sin(kappa * y) # derivative wrt. x of incoming wave at x=0, forall y\n",
+    "\n",
+    "V   = ScalarFunctionSpace('V', Omega, kind=None)\n",
+    "V.codomain_type='complex'\n",
+    "\n",
+    "u, v = elements_of(V, names='u, v')\n",
+    "\n",
+    "error  = u - solution\n",
+    "\n",
+    "expr   = dot(grad(u),grad(v)) - 2 * kappa ** 2 * u * v\n",
+    "\n",
+    "#we impose an incoming wave from the left and absorbing boundary conditions at the right.\n",
+    "boundary_expr = - 1j * kappa * u * v\n",
+    "x_boundary = Union(Omega.get_boundary(axis=0, ext=-1), Omega.get_boundary(axis=0, ext=1))\n",
+    "boundary_source_expr = - dx_e_w_0 * v - 1j * kappa * e_w_0 * v    \n",
+    "\n",
+    "a = BilinearForm((u, v), integral(Omega, expr) + integral(x_boundary, boundary_expr))\n",
+    "l = LinearForm(v, integral(Omega.get_boundary(axis=0, ext=-1), boundary_source_expr))\n",
+    "\n",
+    "equation = find(u, forall=v, lhs=a(u,v), rhs=l(v))\n",
+    "\n",
+    "l2norm = Norm(error, Omega, kind='l2')\n",
+    "h1norm = Norm(error, Omega, kind='h1')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 3: Discretizing the domain, spaces and equations"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Discretize the geometry and equation\n",
+    "from psydac.api.discretization import discretize\n",
+    "from psydac.api.settings import PSYDAC_BACKENDS\n",
+    "\n",
+    "# The backends are as follows and can all be run with OpenMP by setting backend['omp'] = True.\n",
+    "# For all but the Python backend, compilation flags can be accessed and changed using backend['flags'].\n",
+    "# - Python backend: Generate and runs python files. Accessed via PSYDAC_BACKENDS['python']\n",
+    "# - Pyccel GCC backend: Accessed via PSYDAC_BACKENDS['pyccel-gcc']\n",
+    "# - Pyccel Intel backend: Accessed via PSYDAC_BACKENDS['pyccel-intel']\n",
+    "# - Pyccel PGI backend: Accessed via PSYDAC_BACKENDS['pyccel-pgi']\n",
+    "# - Pyccel Numba backend: Accessed via PSYDAC_BACKENDS['numba']\n",
+    "\n",
+    "backend = PSYDAC_BACKENDS['python']\n",
+    "\n",
+    "# Uncomment to use OpenMp\n",
+    "# import os\n",
+    "# os.environ['OMP_NUM_THREADS'] = \"4\"\n",
+    "# backend['omp'] = True\n",
+    "\n",
+    "ncells = [30, 30]\n",
+    "degree = [3, 3]\n",
+    "periodic = [False, True]\n",
+    "\n",
+    "# MPI version\n",
+    "# from mpi4py import MPI\n",
+    "# comm = MPI.COMM_WORLD\n",
+    "# Omega_h = discretize(Omega, ncells=ncells, comm=comm)\n",
+    "Omega_h = discretize(Omega, ncells=ncells, periodic=periodic)\n",
+    "\n",
+    "Vh         = discretize(V, Omega_h, degree=degree)\n",
+    "equation_h = discretize(equation, Omega_h, [Vh, Vh], backend=backend)\n",
+    "l2norm_h   = discretize(l2norm, Omega_h, Vh, backend=backend)\n",
+    "h1norm_h   = discretize(h1norm, Omega_h, Vh, backend=backend)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 4: Solving the equation and computing the error norms"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "> CG info       ::  {'niter': 45, 'success': True, 'res_norm': 2.8926962059736328e-15}\n",
+      "> L2 error      :: 2.07e-07\n",
+      "> H1 error      :: 3.80e-05\n",
+      "> Solution time :: 8.96e+00s\n",
+      "> Evaluat. time :: 3.72e+00s \n"
+     ]
+    }
+   ],
+   "source": [
+    "# Set the solver parameters\n",
+    "# 'cbig' -> Biconjugate gradient method\n",
+    "equation_h.set_solver('bicg', info=True, tol=1e-14)\n",
+    "\n",
+    "import time\n",
+    "\n",
+    "t0_s     = time.time()\n",
+    "uh, info = equation_h.solve()\n",
+    "t1_s     = time.time()\n",
+    "\n",
+    "\n",
+    "t0_d     = time.time()\n",
+    "l2_error = l2norm_h.assemble(u=uh)\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "t1_d     = time.time()\n",
+    "\n",
+    "print( '> CG info       :: ',info )\n",
+    "print( '> L2 error      :: {:.2e}'.format(l2_error))\n",
+    "print( '> H1 error      :: {:.2e}'.format(h1_error))\n",
+    "print( '> Solution time :: {:.2e}s'.format(t1_s - t0_s))\n",
+    "print( '> Evaluat. time :: {:.2e}s '.format(t1_d - t0_d))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 5: Saving the results"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Save the results using OutputManager\n",
+    "from psydac.api.postprocessing import OutputManager\n",
+    "import os\n",
+    "\n",
+    "os.makedirs('results_Helmholtz', exist_ok=True)\n",
+    "\n",
+    "Om = OutputManager(\n",
+    "    f'results_Helmholtz/space_info_{Omega.name}',\n",
+    "    f'results_Helmholtz/field_info_{Omega.name}',\n",
+    "# MPI version\n",
+    "#   comm=comm,    \n",
+    ")\n",
+    "\n",
+    "Om.add_spaces(V=Vh)\n",
+    "Om.export_space_info()\n",
+    "\n",
+    "Om.set_static()\n",
+    "Om.export_fields(u=uh)\n",
+    "\n",
+    "Om.close()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 6: Exporting the results to VTK and visualizing them with Paraview"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Export the results to VTK using PostProcessManager\n",
+    "from psydac.api.postprocessing import PostProcessManager\n",
+    "from sympy import lambdify\n",
+    "\n",
+    "Pm = PostProcessManager(\n",
+    "    domain=Omega,\n",
+    "    space_file=f'results_Helmholtz/space_info_{Omega.name}.yaml',\n",
+    "    fields_file=f'results_Helmholtz/field_info_{Omega.name}.h5',\n",
+    "# MPI version\n",
+    "#   comm=comm,    \n",
+    ")\n",
+    "\n",
+    "# The complex fields will be exported in two field: a real and an imaginary field\n",
+    "Pm.export_to_vtk(\n",
+    "    f'results_Helmholtz/visu_{Omega.name}',\n",
+    "    grid=None,\n",
+    "    npts_per_cell=3,\n",
+    "    fields='u',\n",
+    "    additional_physical_functions={'exact_solution': lambdify(Omega.coordinates, solution, modules='numpy')}\n",
+    ")\n",
+    "\n",
+    "Pm.close()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Example: Results on the square domain\n",
+    "\n",
+    "<p float=\"left\">\n",
+    "<table><tr>\n",
+    "  <td><img src=\"./paraview_images/Helmholtz_Square_Real_Computed.png\" width=\"100%\" /> \n",
+    "  <td><img src=\"./paraview_images/Helmholtz_Square_Imag_Computed.png\" width=\"100%\" />\n",
+    "</tr></table>\n",
+    "</p>\n",
+    "\n",
+    "<p float=\"left\">\n",
+    "<table><tr>\n",
+    "  <td><img src=\"./paraview_images/Helmholtz_Square_Imag_exact.png\" width=\"90%\" /> \n",
+    "  <td><img src=\"./paraview_images/Helmholtz_Square_Real_exact.png\" width=\"100%\" /> \n",
+    "</tr></table> \n",
+    "</p>\n",
+    "      \n",
+    "<p float=\"left\">\n",
+    "<table><tr>\n",
+    "  <td><img src=\"./paraview_images/Helmholtz_Square_Real_error.png\" width=\"100%\" />\n",
+    "  <td><img src=\"./paraview_images/Helmholtz_Square_Imag_error.png\" width=\"100%\" />\n",
+    "</tr></table> \n",
+    "</p>"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "v_psydac",
+   "language": "python",
+   "name": "v_psydac"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.6"
+  },
+  "vscode": {
+   "interpreter": {
+    "hash": "e7370f93d1d0cde622a1f8e1c04877d8463912d04d973331ad4851f04de6915a"
+   }
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/psydac/examples/notebooks/Poisson_non_periodic.ipynb b/_sources/psydac/examples/notebooks/Poisson_non_periodic.ipynb
new file mode 100644
index 0000000..be49179
--- /dev/null
+++ b/_sources/psydac/examples/notebooks/Poisson_non_periodic.ipynb
@@ -0,0 +1,338 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Solving Poisson's equation on non-periodic topological domains\n",
+    "In this notebook, we will show how to solve poisson's equation on single patch and multipatch domains. As we are running this in a notebook, everything will be run in serial and hence we are limiting ourselves to a fairly coarse discretization to avoid taking too much time. However, Psydac allows for hybrid MPI + OpenMP parallelization with barely any changed to the code. The lines that are impacted by that will be preceeded by their MPI equivalent"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 1 : Building the domain\n",
+    "Psydac uses the powerful topological tools of SymPDE to build a large variety of domains. Here we show how to do a quarter annulus using a Square and a polar mapping and how to join two annulus to create a multipatch domain. Left commented is an example of a complex multipatch domain built using SymPDE."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "from sympde.topology import Square, PolarMapping\n",
+    "from sympde.utilities.utils import plot_domain\n",
+    "\n",
+    "from psydac.api.tests.build_domain import build_pretzel\n",
+    "\n",
+    "# Define the topological geometry for each patch\n",
+    "rmin, rmax = 0.3, 1.\n",
+    "\n",
+    "# First quarter annulus\n",
+    "domain_log_1 = Square('A_1', bounds1=(0., 1.), bounds2=(0., np.pi/2))\n",
+    "F_1 = PolarMapping('F_1', dim=2, c1=0., c2=0., rmin=rmin, rmax=rmax)\n",
+    "Omega_1 = F_1(domain_log_1)\n",
+    "\n",
+    "# Second quarter annulus\n",
+    "domain_log_2 = Square('A_2', bounds1=(0., 1.), bounds2=(np.pi, 3/2 * np.pi))\n",
+    "F_2 = PolarMapping('F_1', dim=2, c1=rmin+rmax, c2=0., rmin=rmin, rmax=rmax)\n",
+    "Omega_2 = F_2(domain_log_2)\n",
+    "\n",
+    "# Join the patches\n",
+    "\n",
+    "from sympde import Domain\n",
+    "connectivity = [((0,1,1),(1,1,-1))]\n",
+    "patches = [Omega_1,Omega_2]\n",
+    "Omega = Domain.join(patches, connectivity, 'domain')\n",
+    "\n",
+    "# Example of a complex multi-patch domain\n",
+    "# Omega = build_pretzel()\n",
+    "\n",
+    "# Simple visualization of the topological domain\n",
+    "plot_domain(Omega, draw=False, isolines=True)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 2: Defining the Abstract PDE model using SymPDE"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from sympde.calculus import grad, dot\n",
+    "from sympde.calculus import minus, plus\n",
+    "\n",
+    "from sympde.expr.expr          import LinearForm, BilinearForm\n",
+    "from sympde.expr.expr          import integral              \n",
+    "from sympde.expr.expr          import Norm                       \n",
+    "from sympde.expr               import find, EssentialBC\n",
+    "\n",
+    "from sympde.topology import ScalarFunctionSpace\n",
+    "from sympde.topology import elements_of\n",
+    "from sympde.topology import NormalVector\n",
+    "\n",
+    "# Define the abstract model to solve Poisson's equation using the manufactured solution method\n",
+    "x,y       = Omega.coordinates\n",
+    "solution  = x**2 + y**2\n",
+    "f         = -4\n",
+    "\n",
+    "V   = ScalarFunctionSpace('V', Omega, kind=None)\n",
+    "\n",
+    "u, v = elements_of(V, names='u, v')\n",
+    "nn   = NormalVector('nn')\n",
+    "\n",
+    "bc   = EssentialBC(u, solution, Omega.boundary)\n",
+    "\n",
+    "error  = u - solution\n",
+    "\n",
+    "I = Omega.interfaces\n",
+    "\n",
+    "kappa  = 10**3\n",
+    "\n",
+    "expr_I =- 0.5*dot(grad(plus(u)),nn)*minus(v)  + 0.5*dot(grad(minus(v)),nn)*plus(u)  - kappa*plus(u)*minus(v)\\\n",
+    "        + 0.5*dot(grad(minus(u)),nn)*plus(v)  - 0.5*dot(grad(plus(v)),nn)*minus(u)  - kappa*plus(v)*minus(u)\\\n",
+    "        - 0.5*dot(grad(minus(v)),nn)*minus(u) - 0.5*dot(grad(minus(u)),nn)*minus(v) + kappa*minus(u)*minus(v)\\\n",
+    "        + 0.5*dot(grad(plus(v)),nn)*plus(u)   + 0.5*dot(grad(plus(u)),nn)*plus(v)   + kappa*plus(u)*plus(v)\n",
+    "\n",
+    "expr   = dot(grad(u),grad(v))\n",
+    "\n",
+    "a = BilinearForm((u,v),  integral(Omega, expr) + integral(I, expr_I))\n",
+    "l = LinearForm(v, integral(Omega, f*v))\n",
+    "\n",
+    "equation = find(u, forall=v, lhs=a(u,v), rhs=l(v), bc=bc)\n",
+    "\n",
+    "l2norm = Norm(error, Omega, kind='l2')\n",
+    "h1norm = Norm(error, Omega, kind='h1')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 3: Discretizing the domain, spaces and equations"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 41,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Discretize the geometry and equation\n",
+    "from psydac.api.discretization import discretize\n",
+    "from psydac.api.settings import PSYDAC_BACKENDS\n",
+    "\n",
+    "# The backends are as follows and can all be run with OpenMP by setting backend['omp'] = True.\n",
+    "# For all but the Python backend, compilation flags can be accessed and changed using backend['flags'].\n",
+    "# - Python backend: Generate and runs python files. Accessed via PSYDAC_BACKENDS['python']\n",
+    "# - Pyccel GCC backend: Accessed via PSYDAC_BACKENDS['pyccel-gcc']\n",
+    "# - Pyccel Intel backend: Accessed via PSYDAC_BACKENDS['pyccel-intel']\n",
+    "# - Pyccel PGI backend: Accessed via PSYDAC_BACKENDS['pyccel-pgi']\n",
+    "# - Pyccel Numba backend: Accessed via PSYDAC_BACKENDS['numba']\n",
+    "\n",
+    "backend = PSYDAC_BACKENDS['python']\n",
+    "\n",
+    "# Uncomment to use OpenMp\n",
+    "# import os\n",
+    "# os.environ['OMP_NUM_THREADS'] = \"4\"\n",
+    "# backend['omp'] = True\n",
+    "\n",
+    "ncells = [10, 10]\n",
+    "degree = [2, 2]\n",
+    "periodic = [False, False]\n",
+    "\n",
+    "# MPI version\n",
+    "# from mpi4py import MPI\n",
+    "# comm = MPI.COMM_WORLD\n",
+    "# Omega_h = discretize(Omega, ncells=ncells, comm=comm)\n",
+    "Omega_h = discretize(Omega, ncells=ncells, periodic=periodic)\n",
+    "\n",
+    "Vh         = discretize(V, Omega_h, degree=degree)\n",
+    "equation_h = discretize(equation, Omega_h, [Vh, Vh], backend=backend)\n",
+    "l2norm_h   = discretize(l2norm, Omega_h, Vh, backend=backend)\n",
+    "h1norm_h   = discretize(h1norm, Omega_h, Vh, backend=backend)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 4: Solving the equation and computing the error norms"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 42,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "> CG info       ::  {'niter': 109, 'success': True, 'res_norm': 5.704506834276634e-15}\n",
+      "> L2 error      :: 9.14e-02\n",
+      "> H1 error      :: 1.62e+00\n",
+      "> Solution time :: 1.42e+01s\n",
+      "> Evaluat. time :: 4.73e-01s \n"
+     ]
+    }
+   ],
+   "source": [
+    "# Set the solver parameters\n",
+    "# 'cg' -> Conjugate gradient method\n",
+    "equation_h.set_solver('cg', info=True, tol=1e-14)\n",
+    "\n",
+    "import time\n",
+    "\n",
+    "t0_s     = time.time()\n",
+    "uh, info = equation_h.solve()\n",
+    "t1_s     = time.time()\n",
+    "\n",
+    "\n",
+    "t0_d     = time.time()\n",
+    "l2_error = l2norm_h.assemble(u=uh)\n",
+    "h1_error = h1norm_h.assemble(u=uh)\n",
+    "t1_d     = time.time()\n",
+    "\n",
+    "print( '> CG info       :: ',info )\n",
+    "print( '> L2 error      :: {:.2e}'.format(l2_error))\n",
+    "print( '> H1 error      :: {:.2e}'.format(h1_error))\n",
+    "print( '> Solution time :: {:.2e}s'.format(t1_s - t0_s))\n",
+    "print( '> Evaluat. time :: {:.2e}s '.format(t1_d - t0_d))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 5: Saving the results"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 43,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Save the results using OutputManager\n",
+    "from psydac.api.postprocessing import OutputManager\n",
+    "import os\n",
+    "\n",
+    "os.makedirs('results_poisson', exist_ok=True)\n",
+    "\n",
+    "Om = OutputManager(\n",
+    "    f'results_poisson/space_info_{Omega.name}',\n",
+    "    f'results_poisson/field_info_{Omega.name}',\n",
+    "# MPI version\n",
+    "#   comm=comm,    \n",
+    ")\n",
+    "\n",
+    "Om.add_spaces(V=Vh)\n",
+    "Om.export_space_info()\n",
+    "\n",
+    "Om.set_static()\n",
+    "Om.export_fields(u=uh)\n",
+    "\n",
+    "Om.close()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Step 6: Exporting the results to VTK and visualizing them with Paraview"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 44,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Export the results to VTK using PostProcessManager\n",
+    "from psydac.api.postprocessing import PostProcessManager\n",
+    "from sympy import lambdify\n",
+    "\n",
+    "Pm = PostProcessManager(\n",
+    "    domain=Omega,\n",
+    "    space_file=f'results_poisson/space_info_{Omega.name}.yaml',\n",
+    "    fields_file=f'results_poisson/field_info_{Omega.name}.h5',\n",
+    "# MPI version\n",
+    "#   comm=comm,    \n",
+    ")\n",
+    "\n",
+    "Pm.export_to_vtk(\n",
+    "    f'results_poisson/visu_{Omega.name}',\n",
+    "    grid=None,\n",
+    "    npts_per_cell=3,\n",
+    "    fields='u',\n",
+    "    additional_physical_functions={'exact_solution': lambdify(Omega.coordinates, solution, modules='numpy')}\n",
+    ")\n",
+    "\n",
+    "Pm.close()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Example: Results on the pretzel domain\n",
+    "\n",
+    "<p float=\"left\">\n",
+    "  <img src=\"./paraview_images/poisson_pretzel_u.png\" width=\"32%\" />\n",
+    "  <img src=\"./paraview_images/poisson_pretzel_exact.png\" width=\"32%\" />\n",
+    "  <img src=\"./paraview_images/poisson_pretzel_error.png\" width=\"32%\" />\n",
+    "</p>"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "v_psydac",
+   "language": "python",
+   "name": "v_psydac"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.6"
+  },
+  "vscode": {
+   "interpreter": {
+    "hash": "e7370f93d1d0cde622a1f8e1c04877d8463912d04d973331ad4851f04de6915a"
+   }
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/psydac/output.md b/_sources/psydac/output.md
new file mode 100644
index 0000000..3ce4bb5
--- /dev/null
+++ b/_sources/psydac/output.md
@@ -0,0 +1,167 @@
+# Psydac's outputs
+## Structure
+Psydac has a class meant to take care of outputing simulation results. This class, named `OuputManager` is located in `psydac/api/postprocessing.py`.
+It writes `FemSpace` related information in the Yaml syntax. The file looks like this:
+```yaml
+ndim: 2
+fields: file.h5 # Name of the fields file
+patches:
+- name: patch_0
+  breakpoints:
+  - [0.0, 0.25, 0.5, 0.75, 1.0]
+  - [0.0, 0.25, 0.5, 0.75, 1.0]
+  scalar_spaces:
+  - name: Scalar_space
+    ldim: 2
+    kind: l2
+    dtype: <class 'float'>
+    rational: false
+    periodic: [false, false]
+    degree: [1, 1]
+    multiplicity: [1, 1]
+    basis: [B, B]
+    knots:
+    - [0.0, 0.0, 0.25, 0.5, 0.75, 1.0, 1.0]
+    - [0.0, 0.0, 0.25, 0.5, 0.75, 1.0, 1.0]
+  - &id001
+    name: Vector_space[0]
+    ldim: 2
+    kind: undefined
+    dtype: <class 'float'>
+    rational: false
+    periodic: [false, false]
+    degree: [1, 2]
+    multiplicity: [1, 1]
+    basis: [B, B]
+    knots:
+    - [0.0, 0.0, 0.25, 0.5, 0.75, 1.0, 1.0]
+    - [0.0, 0.0, 0.0, 0.25, 0.5, 0.75, 1.0, 1.0, 1.0]
+  - &id002
+    name: Vector_space[1]
+    ldim: 2
+    kind: undefined
+    dtype: <class 'float'>
+    rational: false
+    periodic: [false, false]
+    degree: [2, 1]
+    multiplicity: [1, 1]
+    basis: [B, B]
+    knots:
+    - [0.0, 0.0, 0.0, 0.25, 0.5, 0.75, 1.0, 1.0, 1.0]
+    - [0.0, 0.0, 0.25, 0.5, 0.75, 1.0, 1.0]
+  vector_spaces:
+  - name: Vector_space
+    kind: hcurl
+    components:
+    - *id001
+    - *id002
+- name: patch_1
+  breakpoints:
+  - [1.0, 1.25, 1.5, 1.75, 2.0]
+  - [0.0, 0.25, 0.5, 0.75, 1.0]
+  scalar_spaces:
+  - name: Scalar_space
+    ldim: 2
+    kind: l2
+    dtype: <class 'float'>
+    rational: false
+    periodic: [false, false]
+    degree: [1, 1]
+    multiplicity: [1, 1]
+    basis: [B, B]
+    knots:
+    - [1.0, 1.0, 1.25, 1.5, 1.75, 2.0, 2.0]
+    - [0.0, 0.0, 0.25, 0.5, 0.75, 1.0, 1.0]
+
+```
+The field coefficients are saved to the `HDF5` format in the following manner :
+```bash
+file.h5
+    attribute: spaces # name of the aforementioned Yaml file
+    static/
+        scalar_space_1/
+            field_s1_1
+            field_s1_2
+            ....
+            field_s1_n
+        vector_space_1_[0]/
+            attribute: parent_space # 'vector_space_1'
+            field_v1_1_[0]
+                attribute: parent_field # 'field_v1_1'
+        vector_space_1_[1]/
+            attribute: parent_space # 'vector_space_1'
+            field_v1_1_[1]
+                attribute: parent_field # 'field_v1_1'
+        ...
+    snapshot_1/
+        attribute: t
+        attribute: ts
+        space_1/
+        ...
+        space_n/
+    ...
+    snapshot_n/
+```
+In addition to that, Psydac also features the `PostProcessManager` class to read those files, recreate all the `FemSpace` and `FemField` objects and export them to `VTK`.
+
+## Usage of class `OutputManager`
+
+An instance of the `OutputManager` class is created at the beginning of the simulation, by specifying the following:
+
+1.  The name of the YAML file (e.g. `spaces.yaml`) where the information about all FEM spaces will be written, and
+2.  The name of the HDF5 file (e.g. `fields.h5`) where the coefficients of all FEM fields will be written.
+
+References to the available FEM spaces are given to the OutputManager object through the `add_spaces(**kwargs)` method, and the corresponding YAML file is created upon calling the method `export_space_info()`. In order to inform the OutputManager object that the next fields to be exported are time-independent, the user should call the `set_static()` method. In the case of time-dependent fields, the user should prepare a time snapshot (which is defined for a specific integer time step `ts` and time value `t`) by calling the method `add_snapshot(t, ts)`. In both cases the fields are exported to the HDF5 file through a call to the method `export_fields(**kwargs)`. Here is a usage example:
+
+```python
+# SymPDE Layer
+# Discretization
+# V0h and V1h are discretized SymPDE Space
+# u0 and u1 are FemFields belonging to either of those spaces
+output_m = OutputManager('spaces.yml', 'fields.h5')
+
+output_m.add_spaces(V0=V0h, V1=V1h)
+output_m.export_space_info() # Writes the space information to Yaml
+
+output_m.set_static() # Tells the object to save in /static/
+output_m.export_fields(u0_static=u0, u1_static=u1) # Actually does the saving
+
+output_m.add_snapshot(t=0., ts=0)
+# The line above tells the object to:
+# 1. create the group snapshot_x with attribute t and ts
+# 2. save in this snapshot
+output_m.export_fields(u0=u0, u1=u1)
+```
+
+## Usage of class `PostProcessManager`
+
+Typically the `PostProcessManager` class is used in a separate post-processing script, which is run after the simulation has finished. In essence it evaluates the FEM fields over a uniform grid (applying the appropriate push-forward operations) and exports the values to a VTK file (or a sequence of files in the case of a time series). An instance of the `PostProcessManager` class is created by specifying the following:
+
+1.  The name of the geometry file (in HDF5 format) which defines the geometry or the topological domain from which the geometry is derived.
+2.  The name of the YAML file that contains the information about the FEM spaces
+3.  The name of the HDF5 file that contains the coefficients of all the FEM fields
+
+In order to export the fields to a VTK file, the user needs to call the method `export_to_vtk(base_name, grid, npts_per_cell, snapshots, fields)`, where:
+1.  `base_name` is the base name for the VTK output files.
+2.  `grid` is either a user specified evaluation grid or `None`.
+3.  `npts_per_cell` specifies the refinement in the case of a uniform grid.
+4.  `snapshots` specifies which time snapshots should be extracted from the HDF5 file (`none` in the case of static fields)
+5.  `fields` is a tuple  of `h5_field_name`.
+
+Here is a usage example:
+
+```python
+# geometry.h5 is where the domain comes from. See PostProcessManager's docstring for more information
+post = PostProcessManager(geometry_file='geometry.h5', space_file='spaces.yml', fields_file='fields.h5')
+
+# See PostProcessManager.export_to_vtk's and TensorFemSpace.eval_fields' docstrings for more information
+post.export_to_vtk('filename_vtk', grid=grid, npts_per_cell=npts_per_cell, snapshots='all', fields = ('u0', 'u1'))
+```
+
+## Further Examples
+Further examples are present in the following files:
+
+*   `examples/poisson_3d_target_torus.py`
+*   `examples/sample_multipatch_parallel.py`
+*   `examples/notebooks/Poisson_non_periodic.ipynb`
+*   `psydac/api/tests/test_postprocessing.py`
\ No newline at end of file
diff --git a/_sphinx_design_static/design-style.1e8bd061cd6da7fc9cf755528e8ffc24.min.css b/_sphinx_design_static/design-style.1e8bd061cd6da7fc9cf755528e8ffc24.min.css
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diff --git a/_sphinx_design_static/design-tabs.js b/_sphinx_design_static/design-tabs.js
new file mode 100644
index 0000000..36b38cf
--- /dev/null
+++ b/_sphinx_design_static/design-tabs.js
@@ -0,0 +1,27 @@
+var sd_labels_by_text = {};
+
+function ready() {
+  const li = document.getElementsByClassName("sd-tab-label");
+  for (const label of li) {
+    syncId = label.getAttribute("data-sync-id");
+    if (syncId) {
+      label.onclick = onLabelClick;
+      if (!sd_labels_by_text[syncId]) {
+        sd_labels_by_text[syncId] = [];
+      }
+      sd_labels_by_text[syncId].push(label);
+    }
+  }
+}
+
+function onLabelClick() {
+  // Activate other inputs with the same sync id.
+  syncId = this.getAttribute("data-sync-id");
+  for (label of sd_labels_by_text[syncId]) {
+    if (label === this) continue;
+    label.previousElementSibling.checked = true;
+  }
+  window.localStorage.setItem("sphinx-design-last-tab", syncId);
+}
+
+document.addEventListener("DOMContentLoaded", ready, false);
diff --git a/_static/basic.css b/_static/basic.css
new file mode 100644
index 0000000..e760386
--- /dev/null
+++ b/_static/basic.css
@@ -0,0 +1,925 @@
+/*
+ * basic.css
+ * ~~~~~~~~~
+ *
+ * Sphinx stylesheet -- basic theme.
+ *
+ * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+
+/* -- main layout ----------------------------------------------------------- */
+
+div.clearer {
+    clear: both;
+}
+
+div.section::after {
+    display: block;
+    content: '';
+    clear: left;
+}
+
+/* -- relbar ---------------------------------------------------------------- */
+
+div.related {
+    width: 100%;
+    font-size: 90%;
+}
+
+div.related h3 {
+    display: none;
+}
+
+div.related ul {
+    margin: 0;
+    padding: 0 0 0 10px;
+    list-style: none;
+}
+
+div.related li {
+    display: inline;
+}
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+div.related li.right {
+    float: right;
+    margin-right: 5px;
+}
+
+/* -- sidebar --------------------------------------------------------------- */
+
+div.sphinxsidebarwrapper {
+    padding: 10px 5px 0 10px;
+}
+
+div.sphinxsidebar {
+    float: left;
+    width: 270px;
+    margin-left: -100%;
+    font-size: 90%;
+    word-wrap: break-word;
+    overflow-wrap : break-word;
+}
+
+div.sphinxsidebar ul {
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+}
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+div.sphinxsidebar ul ul,
+div.sphinxsidebar ul.want-points {
+    margin-left: 20px;
+    list-style: square;
+}
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+    margin-bottom: 0;
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+div.sphinxsidebar input {
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+    font-size: 1em;
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+div.sphinxsidebar #searchbox form.search {
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+div.sphinxsidebar #searchbox input[type="text"] {
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+    padding: 0.25em;
+    box-sizing: border-box;
+}
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+div.sphinxsidebar #searchbox input[type="submit"] {
+    float: left;
+    width: 20%;
+    border-left: none;
+    padding: 0.25em;
+    box-sizing: border-box;
+}
+
+
+img {
+    border: 0;
+    max-width: 100%;
+}
+
+/* -- search page ----------------------------------------------------------- */
+
+ul.search {
+    margin: 10px 0 0 20px;
+    padding: 0;
+}
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+ul.search li {
+    padding: 5px 0 5px 20px;
+    background-image: url(file.png);
+    background-repeat: no-repeat;
+    background-position: 0 7px;
+}
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+ul.search li a {
+    font-weight: bold;
+}
+
+ul.search li p.context {
+    color: #888;
+    margin: 2px 0 0 30px;
+    text-align: left;
+}
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+ul.keywordmatches li.goodmatch a {
+    font-weight: bold;
+}
+
+/* -- index page ------------------------------------------------------------ */
+
+table.contentstable {
+    width: 90%;
+    margin-left: auto;
+    margin-right: auto;
+}
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+}
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+}
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+span.linkdescr {
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+    padding-top: 5px;
+    font-size: 90%;
+}
+
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+
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+    width: 100%;
+}
+
+table.indextable td {
+    text-align: left;
+    vertical-align: top;
+}
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+}
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+    background-color: #f2f2f2;
+}
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+img.toggler {
+    margin-right: 3px;
+    margin-top: 3px;
+    cursor: pointer;
+}
+
+div.modindex-jumpbox {
+    border-top: 1px solid #ddd;
+    border-bottom: 1px solid #ddd;
+    margin: 1em 0 1em 0;
+    padding: 0.4em;
+}
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+div.genindex-jumpbox {
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+    border-bottom: 1px solid #ddd;
+    margin: 1em 0 1em 0;
+    padding: 0.4em;
+}
+
+/* -- domain module index --------------------------------------------------- */
+
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+    border-collapse: collapse;
+}
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+div.body {
+    min-width: 360px;
+    max-width: 800px;
+}
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+}
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+}
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+}
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+    font-weight: bold;
+}
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+    float: left;
+    margin-right: 1em;
+}
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+    float: right;
+    margin-left: 1em;
+}
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+  margin-left: auto;
+  margin-right: auto;
+}
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+}
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+}
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+}
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+.align-default {
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+}
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+}
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+/* -- sidebars -------------------------------------------------------------- */
+
+div.sidebar,
+aside.sidebar {
+    margin: 0 0 0.5em 1em;
+    border: 1px solid #ddb;
+    padding: 7px;
+    background-color: #ffe;
+    width: 40%;
+    float: right;
+    clear: right;
+    overflow-x: auto;
+}
+
+p.sidebar-title {
+    font-weight: bold;
+}
+
+nav.contents,
+aside.topic,
+div.admonition, div.topic, blockquote {
+    clear: left;
+}
+
+/* -- topics ---------------------------------------------------------------- */
+
+nav.contents,
+aside.topic,
+div.topic {
+    border: 1px solid #ccc;
+    padding: 7px;
+    margin: 10px 0 10px 0;
+}
+
+p.topic-title {
+    font-size: 1.1em;
+    font-weight: bold;
+    margin-top: 10px;
+}
+
+/* -- admonitions ----------------------------------------------------------- */
+
+div.admonition {
+    margin-top: 10px;
+    margin-bottom: 10px;
+    padding: 7px;
+}
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+div.admonition dt {
+    font-weight: bold;
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+    font-weight: bold;
+}
+
+div.body p.centered {
+    text-align: center;
+    margin-top: 25px;
+}
+
+/* -- content of sidebars/topics/admonitions -------------------------------- */
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+div.sidebar > :last-child,
+aside.sidebar > :last-child,
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+    content: '';
+    clear: both;
+}
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+/* -- tables ---------------------------------------------------------------- */
+
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+    margin-bottom: 10px;
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+}
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+}
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+}
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+}
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+}
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+    margin-left: 1px;
+}
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+}
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+}
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+}
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+    padding: 0.5em;
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+    font-style: italic;
+}
+
+div.figure p.caption span.caption-text,
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+}
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+
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+}
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+}
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+    -ms-hyphens: manual;
+    -webkit-hyphens: manual;
+    hyphens: manual;
+}
+
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+    margin: 1em 0;
+}
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+}
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+}
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+}
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+}
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+}
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+}
+
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+	color: #067D17;
+}
+
+
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+    display: flex;
+}
+
+table.highlighttable td {
+    margin: 0;
+    padding: 0;
+}
+
+table.highlighttable td.linenos {
+    padding-right: 0.5em;
+}
+
+table.highlighttable td.code {
+    flex: 1;
+    overflow: hidden;
+}
+
+.highlight .hll {
+    display: block;
+}
+
+div.highlight pre,
+table.highlighttable pre {
+    margin: 0;
+}
+
+div.code-block-caption + div {
+    margin-top: 0;
+}
+
+div.code-block-caption {
+    margin-top: 1em;
+    padding: 2px 5px;
+    font-size: small;
+}
+
+div.code-block-caption code {
+    background-color: transparent;
+}
+
+table.highlighttable td.linenos,
+span.linenos,
+div.highlight span.gp {  /* gp: Generic.Prompt */
+  user-select: none;
+  -webkit-user-select: text; /* Safari fallback only */
+  -webkit-user-select: none; /* Chrome/Safari */
+  -moz-user-select: none; /* Firefox */
+  -ms-user-select: none; /* IE10+ */
+}
+
+div.code-block-caption span.caption-number {
+    padding: 0.1em 0.3em;
+    font-style: italic;
+}
+
+div.code-block-caption span.caption-text {
+}
+
+div.literal-block-wrapper {
+    margin: 1em 0;
+}
+
+code.xref, a code {
+    background-color: transparent;
+    font-weight: bold;
+}
+
+h1 code, h2 code, h3 code, h4 code, h5 code, h6 code {
+    background-color: transparent;
+}
+
+.viewcode-link {
+    float: right;
+}
+
+.viewcode-back {
+    float: right;
+    font-family: sans-serif;
+}
+
+div.viewcode-block:target {
+    margin: -1px -10px;
+    padding: 0 10px;
+}
+
+/* -- math display ---------------------------------------------------------- */
+
+img.math {
+    vertical-align: middle;
+}
+
+div.body div.math p {
+    text-align: center;
+}
+
+span.eqno {
+    float: right;
+}
+
+span.eqno a.headerlink {
+    position: absolute;
+    z-index: 1;
+}
+
+div.math:hover a.headerlink {
+    visibility: visible;
+}
+
+/* -- printout stylesheet --------------------------------------------------- */
+
+@media print {
+    div.document,
+    div.documentwrapper,
+    div.bodywrapper {
+        margin: 0 !important;
+        width: 100%;
+    }
+
+    div.sphinxsidebar,
+    div.related,
+    div.footer,
+    #top-link {
+        display: none;
+    }
+}
\ No newline at end of file
diff --git a/_static/check-solid.svg b/_static/check-solid.svg
new file mode 100644
index 0000000..92fad4b
--- /dev/null
+++ b/_static/check-solid.svg
@@ -0,0 +1,4 @@
+<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-check" width="44" height="44" viewBox="0 0 24 24" stroke-width="2" stroke="#22863a" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <path d="M5 12l5 5l10 -10" />
+</svg>
diff --git a/_static/clipboard.min.js b/_static/clipboard.min.js
new file mode 100644
index 0000000..54b3c46
--- /dev/null
+++ b/_static/clipboard.min.js
@@ -0,0 +1,7 @@
+/*!
+ * clipboard.js v2.0.8
+ * https://clipboardjs.com/
+ *
+ * Licensed MIT © Zeno Rocha
+ */
+!function(t,e){"object"==typeof exports&&"object"==typeof module?module.exports=e():"function"==typeof define&&define.amd?define([],e):"object"==typeof exports?exports.ClipboardJS=e():t.ClipboardJS=e()}(this,function(){return n={686:function(t,e,n){"use strict";n.d(e,{default:function(){return o}});var e=n(279),i=n.n(e),e=n(370),u=n.n(e),e=n(817),c=n.n(e);function a(t){try{return document.execCommand(t)}catch(t){return}}var f=function(t){t=c()(t);return a("cut"),t};var l=function(t){var e,n,o,r=1<arguments.length&&void 0!==arguments[1]?arguments[1]:{container:document.body},i="";return"string"==typeof t?(e=t,n="rtl"===document.documentElement.getAttribute("dir"),(o=document.createElement("textarea")).style.fontSize="12pt",o.style.border="0",o.style.padding="0",o.style.margin="0",o.style.position="absolute",o.style[n?"right":"left"]="-9999px",n=window.pageYOffset||document.documentElement.scrollTop,o.style.top="".concat(n,"px"),o.setAttribute("readonly",""),o.value=e,o=o,r.container.appendChild(o),i=c()(o),a("copy"),o.remove()):(i=c()(t),a("copy")),i};function r(t){return(r="function"==typeof Symbol&&"symbol"==typeof Symbol.iterator?function(t){return typeof t}:function(t){return t&&"function"==typeof Symbol&&t.constructor===Symbol&&t!==Symbol.prototype?"symbol":typeof t})(t)}var s=function(){var t=0<arguments.length&&void 0!==arguments[0]?arguments[0]:{},e=t.action,n=void 0===e?"copy":e,o=t.container,e=t.target,t=t.text;if("copy"!==n&&"cut"!==n)throw new Error('Invalid "action" value, use either "copy" or "cut"');if(void 0!==e){if(!e||"object"!==r(e)||1!==e.nodeType)throw new Error('Invalid "target" value, use a valid Element');if("copy"===n&&e.hasAttribute("disabled"))throw new Error('Invalid "target" attribute. 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You can\'t cut text from elements with "readonly" or "disabled" attributes')}return t?l(t,{container:o}):e?"cut"===n?f(e):l(e,{container:o}):void 0};function d(t){return(d="function"==typeof Symbol&&"symbol"==typeof Symbol.iterator?function(t){return typeof t}:function(t){return t&&"function"==typeof Symbol&&t.constructor===Symbol&&t!==Symbol.prototype?"symbol":typeof t})(t)}function p(t,e){for(var n=0;n<e.length;n++){var o=e[n];o.enumerable=o.enumerable||!1,o.configurable=!0,"value"in o&&(o.writable=!0),Object.defineProperty(t,o.key,o)}}function y(t,e){return(y=Object.setPrototypeOf||function(t,e){return t.__proto__=e,t})(t,e)}function h(n){var o=function(){if("undefined"==typeof Reflect||!Reflect.construct)return!1;if(Reflect.construct.sham)return!1;if("function"==typeof Proxy)return!0;try{return Date.prototype.toString.call(Reflect.construct(Date,[],function(){})),!0}catch(t){return!1}}();return function(){var t,e=m(n);return t=o?(t=m(this).constructor,Reflect.construct(e,arguments,t)):e.apply(this,arguments),e=this,!(t=t)||"object"!==d(t)&&"function"!=typeof t?function(t){if(void 0!==t)return t;throw new ReferenceError("this hasn't been initialised - super() hasn't been called")}(e):t}}function m(t){return(m=Object.setPrototypeOf?Object.getPrototypeOf:function(t){return t.__proto__||Object.getPrototypeOf(t)})(t)}function v(t,e){t="data-clipboard-".concat(t);if(e.hasAttribute(t))return e.getAttribute(t)}var o=function(){!function(t,e){if("function"!=typeof e&&null!==e)throw new TypeError("Super expression must either be null or a function");t.prototype=Object.create(e&&e.prototype,{constructor:{value:t,writable:!0,configurable:!0}}),e&&y(t,e)}(r,i());var t,e,n,o=h(r);function r(t,e){var n;return function(t){if(!(t instanceof r))throw new TypeError("Cannot call a class as a function")}(this),(n=o.call(this)).resolveOptions(e),n.listenClick(t),n}return t=r,n=[{key:"copy",value:function(t){var e=1<arguments.length&&void 0!==arguments[1]?arguments[1]:{container:document.body};return l(t,e)}},{key:"cut",value:function(t){return f(t)}},{key:"isSupported",value:function(){var t=0<arguments.length&&void 0!==arguments[0]?arguments[0]:["copy","cut"],t="string"==typeof t?[t]:t,e=!!document.queryCommandSupported;return t.forEach(function(t){e=e&&!!document.queryCommandSupported(t)}),e}}],(e=[{key:"resolveOptions",value:function(){var t=0<arguments.length&&void 0!==arguments[0]?arguments[0]:{};this.action="function"==typeof t.action?t.action:this.defaultAction,this.target="function"==typeof t.target?t.target:this.defaultTarget,this.text="function"==typeof t.text?t.text:this.defaultText,this.container="object"===d(t.container)?t.container:document.body}},{key:"listenClick",value:function(t){var e=this;this.listener=u()(t,"click",function(t){return e.onClick(t)})}},{key:"onClick",value:function(t){var e=t.delegateTarget||t.currentTarget,t=s({action:this.action(e),container:this.container,target:this.target(e),text:this.text(e)});this.emit(t?"success":"error",{action:this.action,text:t,trigger:e,clearSelection:function(){e&&e.focus(),document.activeElement.blur(),window.getSelection().removeAllRanges()}})}},{key:"defaultAction",value:function(t){return v("action",t)}},{key:"defaultTarget",value:function(t){t=v("target",t);if(t)return document.querySelector(t)}},{key:"defaultText",value:function(t){return v("text",t)}},{key:"destroy",value:function(){this.listener.destroy()}}])&&p(t.prototype,e),n&&p(t,n),r}()},828:function(t){var e;"undefined"==typeof Element||Element.prototype.matches||((e=Element.prototype).matches=e.matchesSelector||e.mozMatchesSelector||e.msMatchesSelector||e.oMatchesSelector||e.webkitMatchesSelector),t.exports=function(t,e){for(;t&&9!==t.nodeType;){if("function"==typeof t.matches&&t.matches(e))return t;t=t.parentNode}}},438:function(t,e,n){var u=n(828);function i(t,e,n,o,r){var i=function(e,n,t,o){return function(t){t.delegateTarget=u(t.target,n),t.delegateTarget&&o.call(e,t)}}.apply(this,arguments);return t.addEventListener(n,i,r),{destroy:function(){t.removeEventListener(n,i,r)}}}t.exports=function(t,e,n,o,r){return"function"==typeof t.addEventListener?i.apply(null,arguments):"function"==typeof n?i.bind(null,document).apply(null,arguments):("string"==typeof t&&(t=document.querySelectorAll(t)),Array.prototype.map.call(t,function(t){return i(t,e,n,o,r)}))}},879:function(t,n){n.node=function(t){return void 0!==t&&t instanceof HTMLElement&&1===t.nodeType},n.nodeList=function(t){var e=Object.prototype.toString.call(t);return void 0!==t&&("[object NodeList]"===e||"[object HTMLCollection]"===e)&&"length"in t&&(0===t.length||n.node(t[0]))},n.string=function(t){return"string"==typeof t||t instanceof String},n.fn=function(t){return"[object Function]"===Object.prototype.toString.call(t)}},370:function(t,e,n){var f=n(879),l=n(438);t.exports=function(t,e,n){if(!t&&!e&&!n)throw new Error("Missing required arguments");if(!f.string(e))throw new TypeError("Second argument must be a String");if(!f.fn(n))throw new TypeError("Third argument must be a Function");if(f.node(t))return c=e,a=n,(u=t).addEventListener(c,a),{destroy:function(){u.removeEventListener(c,a)}};if(f.nodeList(t))return o=t,r=e,i=n,Array.prototype.forEach.call(o,function(t){t.addEventListener(r,i)}),{destroy:function(){Array.prototype.forEach.call(o,function(t){t.removeEventListener(r,i)})}};if(f.string(t))return t=t,e=e,n=n,l(document.body,t,e,n);throw new TypeError("First argument must be a String, HTMLElement, HTMLCollection, or NodeList");var o,r,i,u,c,a}},817:function(t){t.exports=function(t){var e,n="SELECT"===t.nodeName?(t.focus(),t.value):"INPUT"===t.nodeName||"TEXTAREA"===t.nodeName?((e=t.hasAttribute("readonly"))||t.setAttribute("readonly",""),t.select(),t.setSelectionRange(0,t.value.length),e||t.removeAttribute("readonly"),t.value):(t.hasAttribute("contenteditable")&&t.focus(),n=window.getSelection(),(e=document.createRange()).selectNodeContents(t),n.removeAllRanges(),n.addRange(e),n.toString());return n}},279:function(t){function e(){}e.prototype={on:function(t,e,n){var o=this.e||(this.e={});return(o[t]||(o[t]=[])).push({fn:e,ctx:n}),this},once:function(t,e,n){var o=this;function r(){o.off(t,r),e.apply(n,arguments)}return r._=e,this.on(t,r,n)},emit:function(t){for(var e=[].slice.call(arguments,1),n=((this.e||(this.e={}))[t]||[]).slice(),o=0,r=n.length;o<r;o++)n[o].fn.apply(n[o].ctx,e);return this},off:function(t,e){var n=this.e||(this.e={}),o=n[t],r=[];if(o&&e)for(var i=0,u=o.length;i<u;i++)o[i].fn!==e&&o[i].fn._!==e&&r.push(o[i]);return r.length?n[t]=r:delete n[t],this}},t.exports=e,t.exports.TinyEmitter=e}},r={},o.n=function(t){var e=t&&t.__esModule?function(){return t.default}:function(){return t};return o.d(e,{a:e}),e},o.d=function(t,e){for(var n in e)o.o(e,n)&&!o.o(t,n)&&Object.defineProperty(t,n,{enumerable:!0,get:e[n]})},o.o=function(t,e){return Object.prototype.hasOwnProperty.call(t,e)},o(686).default;function o(t){if(r[t])return r[t].exports;var e=r[t]={exports:{}};return n[t](e,e.exports,o),e.exports}var n,r});
\ No newline at end of file
diff --git a/_static/copy-button.svg b/_static/copy-button.svg
new file mode 100644
index 0000000..9c074da
--- /dev/null
+++ b/_static/copy-button.svg
@@ -0,0 +1,5 @@
+<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-copy" width="44" height="44" viewBox="0 0 24 24" stroke-width="1.5" stroke="#000000" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <rect x="8" y="8" width="12" height="12" rx="2" />
+  <path d="M16 8v-2a2 2 0 0 0 -2 -2h-8a2 2 0 0 0 -2 2v8a2 2 0 0 0 2 2h2" />
+</svg>
diff --git a/_static/copybutton.css b/_static/copybutton.css
new file mode 100644
index 0000000..f1916ec
--- /dev/null
+++ b/_static/copybutton.css
@@ -0,0 +1,94 @@
+/* Copy buttons */
+button.copybtn {
+    position: absolute;
+    display: flex;
+    top: .3em;
+    right: .3em;
+    width: 1.7em;
+    height: 1.7em;
+	opacity: 0;
+    transition: opacity 0.3s, border .3s, background-color .3s;
+    user-select: none;
+    padding: 0;
+    border: none;
+    outline: none;
+    border-radius: 0.4em;
+    /* The colors that GitHub uses */
+    border: #1b1f2426 1px solid;
+    background-color: #f6f8fa;
+    color: #57606a;
+}
+
+button.copybtn.success {
+    border-color: #22863a;
+    color: #22863a;
+}
+
+button.copybtn svg {
+    stroke: currentColor;
+    width: 1.5em;
+    height: 1.5em;
+    padding: 0.1em;
+}
+
+div.highlight  {
+    position: relative;
+}
+
+/* Show the copybutton */
+.highlight:hover button.copybtn, button.copybtn.success {
+	opacity: 1;
+}
+
+.highlight button.copybtn:hover {
+    background-color: rgb(235, 235, 235);
+}
+
+.highlight button.copybtn:active {
+    background-color: rgb(187, 187, 187);
+}
+
+/**
+ * A minimal CSS-only tooltip copied from:
+ *   https://codepen.io/mildrenben/pen/rVBrpK
+ *
+ * To use, write HTML like the following:
+ *
+ * <p class="o-tooltip--left" data-tooltip="Hey">Short</p>
+ */
+ .o-tooltip--left {
+  position: relative;
+ }
+
+ .o-tooltip--left:after {
+    opacity: 0;
+    visibility: hidden;
+    position: absolute;
+    content: attr(data-tooltip);
+    padding: .2em;
+    font-size: .8em;
+    left: -.2em;
+    background: grey;
+    color: white;
+    white-space: nowrap;
+    z-index: 2;
+    border-radius: 2px;
+    transform: translateX(-102%) translateY(0);
+    transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1);
+}
+
+.o-tooltip--left:hover:after {
+    display: block;
+    opacity: 1;
+    visibility: visible;
+    transform: translateX(-100%) translateY(0);
+    transition: opacity 0.2s cubic-bezier(0.64, 0.09, 0.08, 1), transform 0.2s cubic-bezier(0.64, 0.09, 0.08, 1);
+    transition-delay: .5s;
+}
+
+/* By default the copy button shouldn't show up when printing a page */
+@media print {
+    button.copybtn {
+        display: none;
+    }
+}
diff --git a/_static/copybutton.js b/_static/copybutton.js
new file mode 100644
index 0000000..2ea7ff3
--- /dev/null
+++ b/_static/copybutton.js
@@ -0,0 +1,248 @@
+// Localization support
+const messages = {
+  'en': {
+    'copy': 'Copy',
+    'copy_to_clipboard': 'Copy to clipboard',
+    'copy_success': 'Copied!',
+    'copy_failure': 'Failed to copy',
+  },
+  'es' : {
+    'copy': 'Copiar',
+    'copy_to_clipboard': 'Copiar al portapapeles',
+    'copy_success': '¡Copiado!',
+    'copy_failure': 'Error al copiar',
+  },
+  'de' : {
+    'copy': 'Kopieren',
+    'copy_to_clipboard': 'In die Zwischenablage kopieren',
+    'copy_success': 'Kopiert!',
+    'copy_failure': 'Fehler beim Kopieren',
+  },
+  'fr' : {
+    'copy': 'Copier',
+    'copy_to_clipboard': 'Copier dans le presse-papier',
+    'copy_success': 'Copié !',
+    'copy_failure': 'Échec de la copie',
+  },
+  'ru': {
+    'copy': 'Скопировать',
+    'copy_to_clipboard': 'Скопировать в буфер',
+    'copy_success': 'Скопировано!',
+    'copy_failure': 'Не удалось скопировать',
+  },
+  'zh-CN': {
+    'copy': '复制',
+    'copy_to_clipboard': '复制到剪贴板',
+    'copy_success': '复制成功!',
+    'copy_failure': '复制失败',
+  },
+  'it' : {
+    'copy': 'Copiare',
+    'copy_to_clipboard': 'Copiato negli appunti',
+    'copy_success': 'Copiato!',
+    'copy_failure': 'Errore durante la copia',
+  }
+}
+
+let locale = 'en'
+if( document.documentElement.lang !== undefined
+    && messages[document.documentElement.lang] !== undefined ) {
+  locale = document.documentElement.lang
+}
+
+let doc_url_root = DOCUMENTATION_OPTIONS.URL_ROOT;
+if (doc_url_root == '#') {
+    doc_url_root = '';
+}
+
+/**
+ * SVG files for our copy buttons
+ */
+let iconCheck = `<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-check" width="44" height="44" viewBox="0 0 24 24" stroke-width="2" stroke="#22863a" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <title>${messages[locale]['copy_success']}</title>
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <path d="M5 12l5 5l10 -10" />
+</svg>`
+
+// If the user specified their own SVG use that, otherwise use the default
+let iconCopy = ``;
+if (!iconCopy) {
+  iconCopy = `<svg xmlns="http://www.w3.org/2000/svg" class="icon icon-tabler icon-tabler-copy" width="44" height="44" viewBox="0 0 24 24" stroke-width="1.5" stroke="#000000" fill="none" stroke-linecap="round" stroke-linejoin="round">
+  <title>${messages[locale]['copy_to_clipboard']}</title>
+  <path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+  <rect x="8" y="8" width="12" height="12" rx="2" />
+  <path d="M16 8v-2a2 2 0 0 0 -2 -2h-8a2 2 0 0 0 -2 2v8a2 2 0 0 0 2 2h2" />
+</svg>`
+}
+
+/**
+ * Set up copy/paste for code blocks
+ */
+
+const runWhenDOMLoaded = cb => {
+  if (document.readyState != 'loading') {
+    cb()
+  } else if (document.addEventListener) {
+    document.addEventListener('DOMContentLoaded', cb)
+  } else {
+    document.attachEvent('onreadystatechange', function() {
+      if (document.readyState == 'complete') cb()
+    })
+  }
+}
+
+const codeCellId = index => `codecell${index}`
+
+// Clears selected text since ClipboardJS will select the text when copying
+const clearSelection = () => {
+  if (window.getSelection) {
+    window.getSelection().removeAllRanges()
+  } else if (document.selection) {
+    document.selection.empty()
+  }
+}
+
+// Changes tooltip text for a moment, then changes it back
+// We want the timeout of our `success` class to be a bit shorter than the
+// tooltip and icon change, so that we can hide the icon before changing back.
+var timeoutIcon = 2000;
+var timeoutSuccessClass = 1500;
+
+const temporarilyChangeTooltip = (el, oldText, newText) => {
+  el.setAttribute('data-tooltip', newText)
+  el.classList.add('success')
+  // Remove success a little bit sooner than we change the tooltip
+  // So that we can use CSS to hide the copybutton first
+  setTimeout(() => el.classList.remove('success'), timeoutSuccessClass)
+  setTimeout(() => el.setAttribute('data-tooltip', oldText), timeoutIcon)
+}
+
+// Changes the copy button icon for two seconds, then changes it back
+const temporarilyChangeIcon = (el) => {
+  el.innerHTML = iconCheck;
+  setTimeout(() => {el.innerHTML = iconCopy}, timeoutIcon)
+}
+
+const addCopyButtonToCodeCells = () => {
+  // If ClipboardJS hasn't loaded, wait a bit and try again. This
+  // happens because we load ClipboardJS asynchronously.
+  if (window.ClipboardJS === undefined) {
+    setTimeout(addCopyButtonToCodeCells, 250)
+    return
+  }
+
+  // Add copybuttons to all of our code cells
+  const COPYBUTTON_SELECTOR = 'div.highlight pre';
+  const codeCells = document.querySelectorAll(COPYBUTTON_SELECTOR)
+  codeCells.forEach((codeCell, index) => {
+    const id = codeCellId(index)
+    codeCell.setAttribute('id', id)
+
+    const clipboardButton = id =>
+    `<button class="copybtn o-tooltip--left" data-tooltip="${messages[locale]['copy']}" data-clipboard-target="#${id}">
+      ${iconCopy}
+    </button>`
+    codeCell.insertAdjacentHTML('afterend', clipboardButton(id))
+  })
+
+function escapeRegExp(string) {
+    return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string
+}
+
+/**
+ * Removes excluded text from a Node.
+ *
+ * @param {Node} target Node to filter.
+ * @param {string} exclude CSS selector of nodes to exclude.
+ * @returns {DOMString} Text from `target` with text removed.
+ */
+function filterText(target, exclude) {
+    const clone = target.cloneNode(true);  // clone as to not modify the live DOM
+    if (exclude) {
+        // remove excluded nodes
+        clone.querySelectorAll(exclude).forEach(node => node.remove());
+    }
+    return clone.innerText;
+}
+
+// Callback when a copy button is clicked. Will be passed the node that was clicked
+// should then grab the text and replace pieces of text that shouldn't be used in output
+function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") {
+    var regexp;
+    var match;
+
+    // Do we check for line continuation characters and "HERE-documents"?
+    var useLineCont = !!lineContinuationChar
+    var useHereDoc = !!hereDocDelim
+
+    // create regexp to capture prompt and remaining line
+    if (isRegexp) {
+        regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)')
+    } else {
+        regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)')
+    }
+
+    const outputLines = [];
+    var promptFound = false;
+    var gotLineCont = false;
+    var gotHereDoc = false;
+    const lineGotPrompt = [];
+    for (const line of textContent.split('\n')) {
+        match = line.match(regexp)
+        if (match || gotLineCont || gotHereDoc) {
+            promptFound = regexp.test(line)
+            lineGotPrompt.push(promptFound)
+            if (removePrompts && promptFound) {
+                outputLines.push(match[2])
+            } else {
+                outputLines.push(line)
+            }
+            gotLineCont = line.endsWith(lineContinuationChar) & useLineCont
+            if (line.includes(hereDocDelim) & useHereDoc)
+                gotHereDoc = !gotHereDoc
+        } else if (!onlyCopyPromptLines) {
+            outputLines.push(line)
+        } else if (copyEmptyLines && line.trim() === '') {
+            outputLines.push(line)
+        }
+    }
+
+    // If no lines with the prompt were found then just use original lines
+    if (lineGotPrompt.some(v => v === true)) {
+        textContent = outputLines.join('\n');
+    }
+
+    // Remove a trailing newline to avoid auto-running when pasting
+    if (textContent.endsWith("\n")) {
+        textContent = textContent.slice(0, -1)
+    }
+    return textContent
+}
+
+
+var copyTargetText = (trigger) => {
+  var target = document.querySelector(trigger.attributes['data-clipboard-target'].value);
+
+  // get filtered text
+  let exclude = '.linenos';
+
+  let text = filterText(target, exclude);
+  return formatCopyText(text, '', false, true, true, true, '', '')
+}
+
+  // Initialize with a callback so we can modify the text before copy
+  const clipboard = new ClipboardJS('.copybtn', {text: copyTargetText})
+
+  // Update UI with error/success messages
+  clipboard.on('success', event => {
+    clearSelection()
+    temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_success'])
+    temporarilyChangeIcon(event.trigger)
+  })
+
+  clipboard.on('error', event => {
+    temporarilyChangeTooltip(event.trigger, messages[locale]['copy'], messages[locale]['copy_failure'])
+  })
+}
+
+runWhenDOMLoaded(addCopyButtonToCodeCells)
\ No newline at end of file
diff --git a/_static/copybutton_funcs.js b/_static/copybutton_funcs.js
new file mode 100644
index 0000000..dbe1aaa
--- /dev/null
+++ b/_static/copybutton_funcs.js
@@ -0,0 +1,73 @@
+function escapeRegExp(string) {
+    return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string
+}
+
+/**
+ * Removes excluded text from a Node.
+ *
+ * @param {Node} target Node to filter.
+ * @param {string} exclude CSS selector of nodes to exclude.
+ * @returns {DOMString} Text from `target` with text removed.
+ */
+export function filterText(target, exclude) {
+    const clone = target.cloneNode(true);  // clone as to not modify the live DOM
+    if (exclude) {
+        // remove excluded nodes
+        clone.querySelectorAll(exclude).forEach(node => node.remove());
+    }
+    return clone.innerText;
+}
+
+// Callback when a copy button is clicked. Will be passed the node that was clicked
+// should then grab the text and replace pieces of text that shouldn't be used in output
+export function formatCopyText(textContent, copybuttonPromptText, isRegexp = false, onlyCopyPromptLines = true, removePrompts = true, copyEmptyLines = true, lineContinuationChar = "", hereDocDelim = "") {
+    var regexp;
+    var match;
+
+    // Do we check for line continuation characters and "HERE-documents"?
+    var useLineCont = !!lineContinuationChar
+    var useHereDoc = !!hereDocDelim
+
+    // create regexp to capture prompt and remaining line
+    if (isRegexp) {
+        regexp = new RegExp('^(' + copybuttonPromptText + ')(.*)')
+    } else {
+        regexp = new RegExp('^(' + escapeRegExp(copybuttonPromptText) + ')(.*)')
+    }
+
+    const outputLines = [];
+    var promptFound = false;
+    var gotLineCont = false;
+    var gotHereDoc = false;
+    const lineGotPrompt = [];
+    for (const line of textContent.split('\n')) {
+        match = line.match(regexp)
+        if (match || gotLineCont || gotHereDoc) {
+            promptFound = regexp.test(line)
+            lineGotPrompt.push(promptFound)
+            if (removePrompts && promptFound) {
+                outputLines.push(match[2])
+            } else {
+                outputLines.push(line)
+            }
+            gotLineCont = line.endsWith(lineContinuationChar) & useLineCont
+            if (line.includes(hereDocDelim) & useHereDoc)
+                gotHereDoc = !gotHereDoc
+        } else if (!onlyCopyPromptLines) {
+            outputLines.push(line)
+        } else if (copyEmptyLines && line.trim() === '') {
+            outputLines.push(line)
+        }
+    }
+
+    // If no lines with the prompt were found then just use original lines
+    if (lineGotPrompt.some(v => v === true)) {
+        textContent = outputLines.join('\n');
+    }
+
+    // Remove a trailing newline to avoid auto-running when pasting
+    if (textContent.endsWith("\n")) {
+        textContent = textContent.slice(0, -1)
+    }
+    return textContent
+}
diff --git a/_static/design-style.1e8bd061cd6da7fc9cf755528e8ffc24.min.css b/_static/design-style.1e8bd061cd6da7fc9cf755528e8ffc24.min.css
new file mode 100644
index 0000000..eb19f69
--- /dev/null
+++ b/_static/design-style.1e8bd061cd6da7fc9cf755528e8ffc24.min.css
@@ -0,0 +1 @@
+.sd-bg-primary{background-color:var(--sd-color-primary) !important}.sd-bg-text-primary{color:var(--sd-color-primary-text) !important}button.sd-bg-primary:focus,button.sd-bg-primary:hover{background-color:var(--sd-color-primary-highlight) !important}a.sd-bg-primary:focus,a.sd-bg-primary:hover{background-color:var(--sd-color-primary-highlight) !important}.sd-bg-secondary{background-color:var(--sd-color-secondary) !important}.sd-bg-text-secondary{color:var(--sd-color-secondary-text) !important}button.sd-bg-secondary:focus,button.sd-bg-secondary:hover{background-color:var(--sd-color-secondary-highlight) !important}a.sd-bg-secondary:focus,a.sd-bg-secondary:hover{background-color:var(--sd-color-secondary-highlight) !important}.sd-bg-success{background-color:var(--sd-color-success) !important}.sd-bg-text-success{color:var(--sd-color-success-text) !important}button.sd-bg-success:focus,button.sd-bg-success:hover{background-color:var(--sd-color-success-highlight) !important}a.sd-bg-success:focus,a.sd-bg-success:hover{background-color:var(--sd-color-success-highlight) !important}.sd-bg-info{background-color:var(--sd-color-info) !important}.sd-bg-text-info{color:var(--sd-color-info-text) !important}button.sd-bg-info:focus,button.sd-bg-info:hover{background-color:var(--sd-color-info-highlight) !important}a.sd-bg-info:focus,a.sd-bg-info:hover{background-color:var(--sd-color-info-highlight) !important}.sd-bg-warning{background-color:var(--sd-color-warning) !important}.sd-bg-text-warning{color:var(--sd-color-warning-text) !important}button.sd-bg-warning:focus,button.sd-bg-warning:hover{background-color:var(--sd-color-warning-highlight) !important}a.sd-bg-warning:focus,a.sd-bg-warning:hover{background-color:var(--sd-color-warning-highlight) !important}.sd-bg-danger{background-color:var(--sd-color-danger) !important}.sd-bg-text-danger{color:var(--sd-color-danger-text) !important}button.sd-bg-danger:focus,button.sd-bg-danger:hover{background-color:var(--sd-color-danger-highlight) !important}a.sd-bg-danger:focus,a.sd-bg-danger:hover{background-color:var(--sd-color-danger-highlight) !important}.sd-bg-light{background-color:var(--sd-color-light) !important}.sd-bg-text-light{color:var(--sd-color-light-text) !important}button.sd-bg-light:focus,button.sd-bg-light:hover{background-color:var(--sd-color-light-highlight) !important}a.sd-bg-light:focus,a.sd-bg-light:hover{background-color:var(--sd-color-light-highlight) !important}.sd-bg-muted{background-color:var(--sd-color-muted) !important}.sd-bg-text-muted{color:var(--sd-color-muted-text) !important}button.sd-bg-muted:focus,button.sd-bg-muted:hover{background-color:var(--sd-color-muted-highlight) !important}a.sd-bg-muted:focus,a.sd-bg-muted:hover{background-color:var(--sd-color-muted-highlight) !important}.sd-bg-dark{background-color:var(--sd-color-dark) 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!important}.sd-text-primary,.sd-text-primary>p{color:var(--sd-color-primary) !important}a.sd-text-primary:focus,a.sd-text-primary:hover{color:var(--sd-color-primary-highlight) !important}.sd-text-secondary,.sd-text-secondary>p{color:var(--sd-color-secondary) !important}a.sd-text-secondary:focus,a.sd-text-secondary:hover{color:var(--sd-color-secondary-highlight) !important}.sd-text-success,.sd-text-success>p{color:var(--sd-color-success) !important}a.sd-text-success:focus,a.sd-text-success:hover{color:var(--sd-color-success-highlight) !important}.sd-text-info,.sd-text-info>p{color:var(--sd-color-info) !important}a.sd-text-info:focus,a.sd-text-info:hover{color:var(--sd-color-info-highlight) !important}.sd-text-warning,.sd-text-warning>p{color:var(--sd-color-warning) !important}a.sd-text-warning:focus,a.sd-text-warning:hover{color:var(--sd-color-warning-highlight) !important}.sd-text-danger,.sd-text-danger>p{color:var(--sd-color-danger) 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!important;border-style:solid !important;border-width:1px !important}a.sd-outline-primary:focus,a.sd-outline-primary:hover{border-color:var(--sd-color-primary-highlight) !important}.sd-outline-secondary{border-color:var(--sd-color-secondary) !important;border-style:solid !important;border-width:1px !important}a.sd-outline-secondary:focus,a.sd-outline-secondary:hover{border-color:var(--sd-color-secondary-highlight) !important}.sd-outline-success{border-color:var(--sd-color-success) !important;border-style:solid !important;border-width:1px !important}a.sd-outline-success:focus,a.sd-outline-success:hover{border-color:var(--sd-color-success-highlight) !important}.sd-outline-info{border-color:var(--sd-color-info) !important;border-style:solid !important;border-width:1px !important}a.sd-outline-info:focus,a.sd-outline-info:hover{border-color:var(--sd-color-info-highlight) !important}.sd-outline-warning{border-color:var(--sd-color-warning) !important;border-style:solid 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diff --git a/_static/design-tabs.js b/_static/design-tabs.js
new file mode 100644
index 0000000..36b38cf
--- /dev/null
+++ b/_static/design-tabs.js
@@ -0,0 +1,27 @@
+var sd_labels_by_text = {};
+
+function ready() {
+  const li = document.getElementsByClassName("sd-tab-label");
+  for (const label of li) {
+    syncId = label.getAttribute("data-sync-id");
+    if (syncId) {
+      label.onclick = onLabelClick;
+      if (!sd_labels_by_text[syncId]) {
+        sd_labels_by_text[syncId] = [];
+      }
+      sd_labels_by_text[syncId].push(label);
+    }
+  }
+}
+
+function onLabelClick() {
+  // Activate other inputs with the same sync id.
+  syncId = this.getAttribute("data-sync-id");
+  for (label of sd_labels_by_text[syncId]) {
+    if (label === this) continue;
+    label.previousElementSibling.checked = true;
+  }
+  window.localStorage.setItem("sphinx-design-last-tab", syncId);
+}
+
+document.addEventListener("DOMContentLoaded", ready, false);
diff --git a/_static/doctools.js b/_static/doctools.js
new file mode 100644
index 0000000..d06a71d
--- /dev/null
+++ b/_static/doctools.js
@@ -0,0 +1,156 @@
+/*
+ * doctools.js
+ * ~~~~~~~~~~~
+ *
+ * Base JavaScript utilities for all Sphinx HTML documentation.
+ *
+ * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+"use strict";
+
+const BLACKLISTED_KEY_CONTROL_ELEMENTS = new Set([
+  "TEXTAREA",
+  "INPUT",
+  "SELECT",
+  "BUTTON",
+]);
+
+const _ready = (callback) => {
+  if (document.readyState !== "loading") {
+    callback();
+  } else {
+    document.addEventListener("DOMContentLoaded", callback);
+  }
+};
+
+/**
+ * Small JavaScript module for the documentation.
+ */
+const Documentation = {
+  init: () => {
+    Documentation.initDomainIndexTable();
+    Documentation.initOnKeyListeners();
+  },
+
+  /**
+   * i18n support
+   */
+  TRANSLATIONS: {},
+  PLURAL_EXPR: (n) => (n === 1 ? 0 : 1),
+  LOCALE: "unknown",
+
+  // gettext and ngettext don't access this so that the functions
+  // can safely bound to a different name (_ = Documentation.gettext)
+  gettext: (string) => {
+    const translated = Documentation.TRANSLATIONS[string];
+    switch (typeof translated) {
+      case "undefined":
+        return string; // no translation
+      case "string":
+        return translated; // translation exists
+      default:
+        return translated[0]; // (singular, plural) translation tuple exists
+    }
+  },
+
+  ngettext: (singular, plural, n) => {
+    const translated = Documentation.TRANSLATIONS[singular];
+    if (typeof translated !== "undefined")
+      return translated[Documentation.PLURAL_EXPR(n)];
+    return n === 1 ? singular : plural;
+  },
+
+  addTranslations: (catalog) => {
+    Object.assign(Documentation.TRANSLATIONS, catalog.messages);
+    Documentation.PLURAL_EXPR = new Function(
+      "n",
+      `return (${catalog.plural_expr})`
+    );
+    Documentation.LOCALE = catalog.locale;
+  },
+
+  /**
+   * helper function to focus on search bar
+   */
+  focusSearchBar: () => {
+    document.querySelectorAll("input[name=q]")[0]?.focus();
+  },
+
+  /**
+   * Initialise the domain index toggle buttons
+   */
+  initDomainIndexTable: () => {
+    const toggler = (el) => {
+      const idNumber = el.id.substr(7);
+      const toggledRows = document.querySelectorAll(`tr.cg-${idNumber}`);
+      if (el.src.substr(-9) === "minus.png") {
+        el.src = `${el.src.substr(0, el.src.length - 9)}plus.png`;
+        toggledRows.forEach((el) => (el.style.display = "none"));
+      } else {
+        el.src = `${el.src.substr(0, el.src.length - 8)}minus.png`;
+        toggledRows.forEach((el) => (el.style.display = ""));
+      }
+    };
+
+    const togglerElements = document.querySelectorAll("img.toggler");
+    togglerElements.forEach((el) =>
+      el.addEventListener("click", (event) => toggler(event.currentTarget))
+    );
+    togglerElements.forEach((el) => (el.style.display = ""));
+    if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) togglerElements.forEach(toggler);
+  },
+
+  initOnKeyListeners: () => {
+    // only install a listener if it is really needed
+    if (
+      !DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS &&
+      !DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS
+    )
+      return;
+
+    document.addEventListener("keydown", (event) => {
+      // bail for input elements
+      if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return;
+      // bail with special keys
+      if (event.altKey || event.ctrlKey || event.metaKey) return;
+
+      if (!event.shiftKey) {
+        switch (event.key) {
+          case "ArrowLeft":
+            if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break;
+
+            const prevLink = document.querySelector('link[rel="prev"]');
+            if (prevLink && prevLink.href) {
+              window.location.href = prevLink.href;
+              event.preventDefault();
+            }
+            break;
+          case "ArrowRight":
+            if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break;
+
+            const nextLink = document.querySelector('link[rel="next"]');
+            if (nextLink && nextLink.href) {
+              window.location.href = nextLink.href;
+              event.preventDefault();
+            }
+            break;
+        }
+      }
+
+      // some keyboard layouts may need Shift to get /
+      switch (event.key) {
+        case "/":
+          if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break;
+          Documentation.focusSearchBar();
+          event.preventDefault();
+      }
+    });
+  },
+};
+
+// quick alias for translations
+const _ = Documentation.gettext;
+
+_ready(Documentation.init);
diff --git a/_static/documentation_options.js b/_static/documentation_options.js
new file mode 100644
index 0000000..dab586c
--- /dev/null
+++ b/_static/documentation_options.js
@@ -0,0 +1,13 @@
+const DOCUMENTATION_OPTIONS = {
+    VERSION: '',
+    LANGUAGE: 'en',
+    COLLAPSE_INDEX: false,
+    BUILDER: 'html',
+    FILE_SUFFIX: '.html',
+    LINK_SUFFIX: '.html',
+    HAS_SOURCE: true,
+    SOURCELINK_SUFFIX: '',
+    NAVIGATION_WITH_KEYS: false,
+    SHOW_SEARCH_SUMMARY: true,
+    ENABLE_SEARCH_SHORTCUTS: true,
+};
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diff --git a/_static/images/logo_binder.svg b/_static/images/logo_binder.svg
new file mode 100644
index 0000000..45fecf7
--- /dev/null
+++ b/_static/images/logo_binder.svg
@@ -0,0 +1,19 @@
+<?xml version="1.0" encoding="utf-8"?>
+<!-- Generator: Adobe Illustrator 23.0.1, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
+<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
+	 viewBox="0 0 44.4 44.4" style="enable-background:new 0 0 44.4 44.4;" xml:space="preserve">
+<style type="text/css">
+	.st0{fill:none;stroke:#F5A252;stroke-width:5;stroke-miterlimit:10;}
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+<title>logo</title>
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+	<path class="st2" d="M25.9,17.8c2.6,4.6,1.1,10.4-3.5,13s-10.4,1.1-13-3.5s-1.1-10.4,3.5-13l0,0C17.5,11.7,23.3,13.2,25.9,17.8z"/>
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diff --git a/_static/images/logo_deepnote.svg b/_static/images/logo_deepnote.svg
new file mode 100644
index 0000000..fa77ebf
--- /dev/null
+++ b/_static/images/logo_deepnote.svg
@@ -0,0 +1 @@
+<svg viewBox="0 0 128 128" fill="none" xmlns="http://www.w3.org/2000/svg"><path fill-rule="evenodd" clip-rule="evenodd" d="M0 128h52.512l29.539-11.077-11.077-43.487-34.051 3.693L0 128Z" fill="#0076D4"/><path fill-rule="evenodd" clip-rule="evenodd" d="M52.513 128s16.6-8.759 19.673-24.277c3.072-15.517-12.091-26.594-35.263-26.594 0-.41 20.343-28.718 20.343-28.718l49.4 1.435L95.71 107.7l-20.452 15.978L52.513 128Z" fill="#002868"/><path fill-rule="evenodd" clip-rule="evenodd" d="M0 60.718 41.025.001s1.006.01 3.282 0c16.082-.068 81.23 3.12 81.23 60.368 0 65.352-73.025 67.631-73.025 67.631s30.495-5.839 30.495-34.816c0-28.978-27.541-32.466-45.264-32.466H0Z" fill="#00A9FF"/></svg>
diff --git a/_static/images/logo_jupyterhub.svg b/_static/images/logo_jupyterhub.svg
new file mode 100644
index 0000000..60cfe9f
--- /dev/null
+++ b/_static/images/logo_jupyterhub.svg
@@ -0,0 +1 @@
+<svg id="Layer_1" data-name="Layer 1" xmlns="http://www.w3.org/2000/svg" width="38.73" height="50" viewBox="0 0 38.73 50"><defs><style>.cls-1{fill:#767677;}.cls-2{fill:#f37726;}.cls-3{fill:#9e9e9e;}.cls-4{fill:#616262;}.cls-5{font-size:17.07px;fill:#fff;font-family:Roboto-Regular, Roboto;}</style></defs><title>logo_jupyterhub</title><g id="Canvas"><path id="path7_fill" data-name="path7 fill" class="cls-1" d="M39.51,3.53a3,3,0,0,1-1.7,2.9A3,3,0,0,1,34.48,6a3,3,0,0,1-.82-3.26,3,3,0,0,1,1.05-1.41A3,3,0,0,1,37.52.86a2.88,2.88,0,0,1,1,.6,3,3,0,0,1,.7.93,3.18,3.18,0,0,1,.28,1.14Z" transform="translate(-1.87 -0.69)"/><path id="path8_fill" data-name="path8 fill" class="cls-2" d="M21.91,38.39c-8,0-15.06-2.87-18.7-7.12a19.93,19.93,0,0,0,37.39,0C37,35.52,30,38.39,21.91,38.39Z" transform="translate(-1.87 -0.69)"/><path id="path9_fill" data-name="path9 fill" class="cls-2" d="M21.91,10.78c8,0,15.05,2.87,18.69,7.12a19.93,19.93,0,0,0-37.39,0C6.85,13.64,13.86,10.78,21.91,10.78Z" transform="translate(-1.87 -0.69)"/><path id="path10_fill" data-name="path10 fill" class="cls-3" d="M10.88,46.66a3.86,3.86,0,0,1-.52,2.15,3.81,3.81,0,0,1-1.62,1.51,3.93,3.93,0,0,1-2.19.34,3.79,3.79,0,0,1-2-.94,3.73,3.73,0,0,1-1.14-1.9,3.79,3.79,0,0,1,.1-2.21,3.86,3.86,0,0,1,1.33-1.78,3.92,3.92,0,0,1,3.54-.53,3.85,3.85,0,0,1,2.14,1.93,3.74,3.74,0,0,1,.37,1.43Z" transform="translate(-1.87 -0.69)"/><path id="path11_fill" data-name="path11 fill" class="cls-4" d="M4.12,9.81A2.18,2.18,0,0,1,2.9,9.48a2.23,2.23,0,0,1-.84-1A2.26,2.26,0,0,1,1.9,7.26a2.13,2.13,0,0,1,.56-1.13,2.18,2.18,0,0,1,2.36-.56,2.13,2.13,0,0,1,1,.76,2.18,2.18,0,0,1,.42,1.2A2.22,2.22,0,0,1,4.12,9.81Z" transform="translate(-1.87 -0.69)"/></g><text class="cls-5" transform="translate(5.24 30.01)">Hub</text></svg>
diff --git a/_static/language_data.js b/_static/language_data.js
new file mode 100644
index 0000000..250f566
--- /dev/null
+++ b/_static/language_data.js
@@ -0,0 +1,199 @@
+/*
+ * language_data.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * This script contains the language-specific data used by searchtools.js,
+ * namely the list of stopwords, stemmer, scorer and splitter.
+ *
+ * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+
+var stopwords = ["a", "and", "are", "as", "at", "be", "but", "by", "for", "if", "in", "into", "is", "it", "near", "no", "not", "of", "on", "or", "such", "that", "the", "their", "then", "there", "these", "they", "this", "to", "was", "will", "with"];
+
+
+/* Non-minified version is copied as a separate JS file, is available */
+
+/**
+ * Porter Stemmer
+ */
+var Stemmer = function() {
+
+  var step2list = {
+    ational: 'ate',
+    tional: 'tion',
+    enci: 'ence',
+    anci: 'ance',
+    izer: 'ize',
+    bli: 'ble',
+    alli: 'al',
+    entli: 'ent',
+    eli: 'e',
+    ousli: 'ous',
+    ization: 'ize',
+    ation: 'ate',
+    ator: 'ate',
+    alism: 'al',
+    iveness: 'ive',
+    fulness: 'ful',
+    ousness: 'ous',
+    aliti: 'al',
+    iviti: 'ive',
+    biliti: 'ble',
+    logi: 'log'
+  };
+
+  var step3list = {
+    icate: 'ic',
+    ative: '',
+    alize: 'al',
+    iciti: 'ic',
+    ical: 'ic',
+    ful: '',
+    ness: ''
+  };
+
+  var c = "[^aeiou]";          // consonant
+  var v = "[aeiouy]";          // vowel
+  var C = c + "[^aeiouy]*";    // consonant sequence
+  var V = v + "[aeiou]*";      // vowel sequence
+
+  var mgr0 = "^(" + C + ")?" + V + C;                      // [C]VC... is m>0
+  var meq1 = "^(" + C + ")?" + V + C + "(" + V + ")?$";    // [C]VC[V] is m=1
+  var mgr1 = "^(" + C + ")?" + V + C + V + C;              // [C]VCVC... is m>1
+  var s_v   = "^(" + C + ")?" + v;                         // vowel in stem
+
+  this.stemWord = function (w) {
+    var stem;
+    var suffix;
+    var firstch;
+    var origword = w;
+
+    if (w.length < 3)
+      return w;
+
+    var re;
+    var re2;
+    var re3;
+    var re4;
+
+    firstch = w.substr(0,1);
+    if (firstch == "y")
+      w = firstch.toUpperCase() + w.substr(1);
+
+    // Step 1a
+    re = /^(.+?)(ss|i)es$/;
+    re2 = /^(.+?)([^s])s$/;
+
+    if (re.test(w))
+      w = w.replace(re,"$1$2");
+    else if (re2.test(w))
+      w = w.replace(re2,"$1$2");
+
+    // Step 1b
+    re = /^(.+?)eed$/;
+    re2 = /^(.+?)(ed|ing)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      re = new RegExp(mgr0);
+      if (re.test(fp[1])) {
+        re = /.$/;
+        w = w.replace(re,"");
+      }
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1];
+      re2 = new RegExp(s_v);
+      if (re2.test(stem)) {
+        w = stem;
+        re2 = /(at|bl|iz)$/;
+        re3 = new RegExp("([^aeiouylsz])\\1$");
+        re4 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+        if (re2.test(w))
+          w = w + "e";
+        else if (re3.test(w)) {
+          re = /.$/;
+          w = w.replace(re,"");
+        }
+        else if (re4.test(w))
+          w = w + "e";
+      }
+    }
+
+    // Step 1c
+    re = /^(.+?)y$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(s_v);
+      if (re.test(stem))
+        w = stem + "i";
+    }
+
+    // Step 2
+    re = /^(.+?)(ational|tional|enci|anci|izer|bli|alli|entli|eli|ousli|ization|ation|ator|alism|iveness|fulness|ousness|aliti|iviti|biliti|logi)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step2list[suffix];
+    }
+
+    // Step 3
+    re = /^(.+?)(icate|ative|alize|iciti|ical|ful|ness)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      suffix = fp[2];
+      re = new RegExp(mgr0);
+      if (re.test(stem))
+        w = stem + step3list[suffix];
+    }
+
+    // Step 4
+    re = /^(.+?)(al|ance|ence|er|ic|able|ible|ant|ement|ment|ent|ou|ism|ate|iti|ous|ive|ize)$/;
+    re2 = /^(.+?)(s|t)(ion)$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      if (re.test(stem))
+        w = stem;
+    }
+    else if (re2.test(w)) {
+      var fp = re2.exec(w);
+      stem = fp[1] + fp[2];
+      re2 = new RegExp(mgr1);
+      if (re2.test(stem))
+        w = stem;
+    }
+
+    // Step 5
+    re = /^(.+?)e$/;
+    if (re.test(w)) {
+      var fp = re.exec(w);
+      stem = fp[1];
+      re = new RegExp(mgr1);
+      re2 = new RegExp(meq1);
+      re3 = new RegExp("^" + C + v + "[^aeiouwxy]$");
+      if (re.test(stem) || (re2.test(stem) && !(re3.test(stem))))
+        w = stem;
+    }
+    re = /ll$/;
+    re2 = new RegExp(mgr1);
+    if (re.test(w) && re2.test(w)) {
+      re = /.$/;
+      w = w.replace(re,"");
+    }
+
+    // and turn initial Y back to y
+    if (firstch == "y")
+      w = firstch.toLowerCase() + w.substr(1);
+    return w;
+  }
+}
+
diff --git a/_static/locales/ar/LC_MESSAGES/booktheme.mo b/_static/locales/ar/LC_MESSAGES/booktheme.mo
new file mode 100644
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HcmV?d00001

diff --git a/_static/locales/ar/LC_MESSAGES/booktheme.po b/_static/locales/ar/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..edae2ec
--- /dev/null
+++ b/_static/locales/ar/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ar\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "موضوع بواسطة"
+
+msgid "Open an issue"
+msgstr "افتح قضية"
+
+msgid "Contents"
+msgstr "محتويات"
+
+msgid "Download notebook file"
+msgstr "تنزيل ملف دفتر الملاحظات"
+
+msgid "Sphinx Book Theme"
+msgstr "موضوع كتاب أبو الهول"
+
+msgid "Fullscreen mode"
+msgstr "وضع ملء الشاشة"
+
+msgid "Edit this page"
+msgstr "قم بتحرير هذه الصفحة"
+
+msgid "By"
+msgstr "بواسطة"
+
+msgid "Copyright"
+msgstr "حقوق النشر"
+
+msgid "Source repository"
+msgstr "مستودع المصدر"
+
+msgid "previous page"
+msgstr "الصفحة السابقة"
+
+msgid "next page"
+msgstr "الصفحة التالية"
+
+msgid "Toggle navigation"
+msgstr "تبديل التنقل"
+
+msgid "repository"
+msgstr "مخزن"
+
+msgid "suggest edit"
+msgstr "أقترح تحرير"
+
+msgid "open issue"
+msgstr "قضية مفتوحة"
+
+msgid "Launch"
+msgstr "إطلاق"
+
+msgid "Print to PDF"
+msgstr "طباعة إلى PDF"
+
+msgid "By the"
+msgstr "بواسطة"
+
+msgid "Last updated on"
+msgstr "آخر تحديث في"
+
+msgid "Download source file"
+msgstr "تنزيل ملف المصدر"
+
+msgid "Download this page"
+msgstr "قم بتنزيل هذه الصفحة"
diff --git a/_static/locales/bg/LC_MESSAGES/booktheme.mo b/_static/locales/bg/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..da95120037f8d7c52ad1f166a7682ea55ca47a01
GIT binary patch
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literal 0
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diff --git a/_static/locales/bg/LC_MESSAGES/booktheme.po b/_static/locales/bg/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..1f363b9
--- /dev/null
+++ b/_static/locales/bg/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: bg\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Тема от"
+
+msgid "Open an issue"
+msgstr "Отворете проблем"
+
+msgid "Contents"
+msgstr "Съдържание"
+
+msgid "Download notebook file"
+msgstr "Изтеглете файла на бележника"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема на книгата Sphinx"
+
+msgid "Fullscreen mode"
+msgstr "Режим на цял екран"
+
+msgid "Edit this page"
+msgstr "Редактирайте тази страница"
+
+msgid "By"
+msgstr "От"
+
+msgid "Copyright"
+msgstr "Авторско право"
+
+msgid "Source repository"
+msgstr "Хранилище на източника"
+
+msgid "previous page"
+msgstr "предишна страница"
+
+msgid "next page"
+msgstr "Следваща страница"
+
+msgid "Toggle navigation"
+msgstr "Превключване на навигацията"
+
+msgid "repository"
+msgstr "хранилище"
+
+msgid "suggest edit"
+msgstr "предложи редактиране"
+
+msgid "open issue"
+msgstr "отворен брой"
+
+msgid "Launch"
+msgstr "Стартиране"
+
+msgid "Print to PDF"
+msgstr "Печат в PDF"
+
+msgid "By the"
+msgstr "По"
+
+msgid "Last updated on"
+msgstr "Последна актуализация на"
+
+msgid "Download source file"
+msgstr "Изтеглете изходния файл"
+
+msgid "Download this page"
+msgstr "Изтеглете тази страница"
diff --git a/_static/locales/bn/LC_MESSAGES/booktheme.mo b/_static/locales/bn/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..6b96639b726a2fa959a6419b6f8b7e0dfcce33ae
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literal 0
HcmV?d00001

diff --git a/_static/locales/bn/LC_MESSAGES/booktheme.po b/_static/locales/bn/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..fa54372
--- /dev/null
+++ b/_static/locales/bn/LC_MESSAGES/booktheme.po
@@ -0,0 +1,63 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: bn\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "থিম দ্বারা"
+
+msgid "Open an issue"
+msgstr "একটি সমস্যা খুলুন"
+
+msgid "Download notebook file"
+msgstr "নোটবুক ফাইল ডাউনলোড করুন"
+
+msgid "Sphinx Book Theme"
+msgstr "স্পিনিক্স বুক থিম"
+
+msgid "Edit this page"
+msgstr "এই পৃষ্ঠাটি সম্পাদনা করুন"
+
+msgid "By"
+msgstr "দ্বারা"
+
+msgid "Copyright"
+msgstr "কপিরাইট"
+
+msgid "Source repository"
+msgstr "উত্স সংগ্রহস্থল"
+
+msgid "previous page"
+msgstr "আগের পৃষ্ঠা"
+
+msgid "next page"
+msgstr "পরবর্তী পৃষ্ঠা"
+
+msgid "Toggle navigation"
+msgstr "নেভিগেশন টগল করুন"
+
+msgid "open issue"
+msgstr "খোলা সমস্যা"
+
+msgid "Launch"
+msgstr "শুরু করা"
+
+msgid "Print to PDF"
+msgstr "পিডিএফ প্রিন্ট করুন"
+
+msgid "By the"
+msgstr "দ্বারা"
+
+msgid "Last updated on"
+msgstr "সর্বশেষ আপডেট"
+
+msgid "Download source file"
+msgstr "উত্স ফাইল ডাউনলোড করুন"
+
+msgid "Download this page"
+msgstr "এই পৃষ্ঠাটি ডাউনলোড করুন"
diff --git a/_static/locales/ca/LC_MESSAGES/booktheme.mo b/_static/locales/ca/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..a4dd30e9bd762578b119c4401d082ea4fa430613
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literal 0
HcmV?d00001

diff --git a/_static/locales/ca/LC_MESSAGES/booktheme.po b/_static/locales/ca/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..22f1569
--- /dev/null
+++ b/_static/locales/ca/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ca\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema del"
+
+msgid "Open an issue"
+msgstr "Obriu un número"
+
+msgid "Download notebook file"
+msgstr "Descarregar fitxer de quadern"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del llibre Esfinx"
+
+msgid "Edit this page"
+msgstr "Editeu aquesta pàgina"
+
+msgid "By"
+msgstr "Per"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Source repository"
+msgstr "Dipòsit de fonts"
+
+msgid "previous page"
+msgstr "Pàgina anterior"
+
+msgid "next page"
+msgstr "pàgina següent"
+
+msgid "Toggle navigation"
+msgstr "Commuta la navegació"
+
+msgid "suggest edit"
+msgstr "suggerir edició"
+
+msgid "open issue"
+msgstr "número obert"
+
+msgid "Launch"
+msgstr "Llançament"
+
+msgid "Print to PDF"
+msgstr "Imprimeix a PDF"
+
+msgid "By the"
+msgstr "Per la"
+
+msgid "Last updated on"
+msgstr "Darrera actualització el"
+
+msgid "Download source file"
+msgstr "Baixeu el fitxer font"
+
+msgid "Download this page"
+msgstr "Descarregueu aquesta pàgina"
diff --git a/_static/locales/cs/LC_MESSAGES/booktheme.mo b/_static/locales/cs/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..c39e01a6ae9ae5242f04566ffa2cda96b328ca18
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/cs/LC_MESSAGES/booktheme.po b/_static/locales/cs/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..afecd9e
--- /dev/null
+++ b/_static/locales/cs/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: cs\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Téma od"
+
+msgid "Open an issue"
+msgstr "Otevřete problém"
+
+msgid "Contents"
+msgstr "Obsah"
+
+msgid "Download notebook file"
+msgstr "Stáhnout soubor poznámkového bloku"
+
+msgid "Sphinx Book Theme"
+msgstr "Téma knihy Sfinga"
+
+msgid "Fullscreen mode"
+msgstr "Režim celé obrazovky"
+
+msgid "Edit this page"
+msgstr "Upravit tuto stránku"
+
+msgid "By"
+msgstr "Podle"
+
+msgid "Copyright"
+msgstr "autorská práva"
+
+msgid "Source repository"
+msgstr "Zdrojové úložiště"
+
+msgid "previous page"
+msgstr "předchozí stránka"
+
+msgid "next page"
+msgstr "další strana"
+
+msgid "Toggle navigation"
+msgstr "Přepnout navigaci"
+
+msgid "repository"
+msgstr "úložiště"
+
+msgid "suggest edit"
+msgstr "navrhnout úpravy"
+
+msgid "open issue"
+msgstr "otevřené číslo"
+
+msgid "Launch"
+msgstr "Zahájení"
+
+msgid "Print to PDF"
+msgstr "Tisk do PDF"
+
+msgid "By the"
+msgstr "Podle"
+
+msgid "Last updated on"
+msgstr "Naposledy aktualizováno"
+
+msgid "Download source file"
+msgstr "Stáhněte si zdrojový soubor"
+
+msgid "Download this page"
+msgstr "Stáhněte si tuto stránku"
diff --git a/_static/locales/da/LC_MESSAGES/booktheme.mo b/_static/locales/da/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..f43157d70c63ff21b4385dc36cb73f5b14eb6b01
GIT binary patch
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literal 0
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diff --git a/_static/locales/da/LC_MESSAGES/booktheme.po b/_static/locales/da/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..649c78a
--- /dev/null
+++ b/_static/locales/da/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: da\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema af"
+
+msgid "Open an issue"
+msgstr "Åbn et problem"
+
+msgid "Contents"
+msgstr "Indhold"
+
+msgid "Download notebook file"
+msgstr "Download notesbog-fil"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx bogtema"
+
+msgid "Fullscreen mode"
+msgstr "Fuldskærmstilstand"
+
+msgid "Edit this page"
+msgstr "Rediger denne side"
+
+msgid "By"
+msgstr "Ved"
+
+msgid "Copyright"
+msgstr "ophavsret"
+
+msgid "Source repository"
+msgstr "Kildelager"
+
+msgid "previous page"
+msgstr "forrige side"
+
+msgid "next page"
+msgstr "Næste side"
+
+msgid "Toggle navigation"
+msgstr "Skift navigation"
+
+msgid "repository"
+msgstr "lager"
+
+msgid "suggest edit"
+msgstr "foreslå redigering"
+
+msgid "open issue"
+msgstr "åbent nummer"
+
+msgid "Launch"
+msgstr "Start"
+
+msgid "Print to PDF"
+msgstr "Udskriv til PDF"
+
+msgid "By the"
+msgstr "Ved"
+
+msgid "Last updated on"
+msgstr "Sidst opdateret den"
+
+msgid "Download source file"
+msgstr "Download kildefil"
+
+msgid "Download this page"
+msgstr "Download denne side"
diff --git a/_static/locales/de/LC_MESSAGES/booktheme.mo b/_static/locales/de/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..648b565c7829535bc5f26f86909f458d0cd0fe1b
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/de/LC_MESSAGES/booktheme.po b/_static/locales/de/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..f51d2ec
--- /dev/null
+++ b/_static/locales/de/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: de\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Thema von der"
+
+msgid "Open an issue"
+msgstr "Öffnen Sie ein Problem"
+
+msgid "Contents"
+msgstr "Inhalt"
+
+msgid "Download notebook file"
+msgstr "Notebook-Datei herunterladen"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-Buch-Thema"
+
+msgid "Fullscreen mode"
+msgstr "Vollbildmodus"
+
+msgid "Edit this page"
+msgstr "Bearbeite diese Seite"
+
+msgid "By"
+msgstr "Durch"
+
+msgid "Copyright"
+msgstr "Urheberrechte ©"
+
+msgid "Source repository"
+msgstr "Quell-Repository"
+
+msgid "previous page"
+msgstr "vorherige Seite"
+
+msgid "next page"
+msgstr "Nächste Seite"
+
+msgid "Toggle navigation"
+msgstr "Navigation umschalten"
+
+msgid "repository"
+msgstr "Repository"
+
+msgid "suggest edit"
+msgstr "vorschlagen zu bearbeiten"
+
+msgid "open issue"
+msgstr "offenes Thema"
+
+msgid "Launch"
+msgstr "Starten"
+
+msgid "Print to PDF"
+msgstr "In PDF drucken"
+
+msgid "By the"
+msgstr "Bis zum"
+
+msgid "Last updated on"
+msgstr "Zuletzt aktualisiert am"
+
+msgid "Download source file"
+msgstr "Quelldatei herunterladen"
+
+msgid "Download this page"
+msgstr "Laden Sie diese Seite herunter"
diff --git a/_static/locales/el/LC_MESSAGES/booktheme.mo b/_static/locales/el/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..fca6e9355f314677f2890f5679fd79dd67a7a5a2
GIT binary patch
literal 1722
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literal 0
HcmV?d00001

diff --git a/_static/locales/el/LC_MESSAGES/booktheme.po b/_static/locales/el/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..8bec790
--- /dev/null
+++ b/_static/locales/el/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: el\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Θέμα από το"
+
+msgid "Open an issue"
+msgstr "Ανοίξτε ένα ζήτημα"
+
+msgid "Contents"
+msgstr "Περιεχόμενα"
+
+msgid "Download notebook file"
+msgstr "Λήψη αρχείου σημειωματάριου"
+
+msgid "Sphinx Book Theme"
+msgstr "Θέμα βιβλίου Sphinx"
+
+msgid "Fullscreen mode"
+msgstr "ΛΕΙΤΟΥΡΓΙΑ ΠΛΗΡΟΥΣ ΟΘΟΝΗΣ"
+
+msgid "Edit this page"
+msgstr "Επεξεργαστείτε αυτήν τη σελίδα"
+
+msgid "By"
+msgstr "Με"
+
+msgid "Copyright"
+msgstr "Πνευματική ιδιοκτησία"
+
+msgid "Source repository"
+msgstr "Αποθήκη πηγής"
+
+msgid "previous page"
+msgstr "προηγούμενη σελίδα"
+
+msgid "next page"
+msgstr "επόμενη σελίδα"
+
+msgid "Toggle navigation"
+msgstr "Εναλλαγή πλοήγησης"
+
+msgid "repository"
+msgstr "αποθήκη"
+
+msgid "suggest edit"
+msgstr "προτείνω επεξεργασία"
+
+msgid "open issue"
+msgstr "ανοιχτό ζήτημα"
+
+msgid "Launch"
+msgstr "Εκτόξευση"
+
+msgid "Print to PDF"
+msgstr "Εκτύπωση σε PDF"
+
+msgid "By the"
+msgstr "Από το"
+
+msgid "Last updated on"
+msgstr "Τελευταία ενημέρωση στις"
+
+msgid "Download source file"
+msgstr "Λήψη αρχείου προέλευσης"
+
+msgid "Download this page"
+msgstr "Λήψη αυτής της σελίδας"
diff --git a/_static/locales/eo/LC_MESSAGES/booktheme.mo b/_static/locales/eo/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..d1072bbec62959524a87953d3f41b7c272ecbe37
GIT binary patch
literal 1337
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literal 0
HcmV?d00001

diff --git a/_static/locales/eo/LC_MESSAGES/booktheme.po b/_static/locales/eo/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..d72a048
--- /dev/null
+++ b/_static/locales/eo/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: eo\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Temo de la"
+
+msgid "Open an issue"
+msgstr "Malfermu numeron"
+
+msgid "Contents"
+msgstr "Enhavo"
+
+msgid "Download notebook file"
+msgstr "Elŝutu kajeran dosieron"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksa Libro-Temo"
+
+msgid "Fullscreen mode"
+msgstr "Plenekrana reĝimo"
+
+msgid "Edit this page"
+msgstr "Redaktu ĉi tiun paĝon"
+
+msgid "By"
+msgstr "De"
+
+msgid "Copyright"
+msgstr "Kopirajto"
+
+msgid "Source repository"
+msgstr "Fonto-deponejo"
+
+msgid "previous page"
+msgstr "antaŭa paĝo"
+
+msgid "next page"
+msgstr "sekva paĝo"
+
+msgid "Toggle navigation"
+msgstr "Ŝalti navigadon"
+
+msgid "repository"
+msgstr "deponejo"
+
+msgid "suggest edit"
+msgstr "sugesti redaktadon"
+
+msgid "open issue"
+msgstr "malferma numero"
+
+msgid "Launch"
+msgstr "Lanĉo"
+
+msgid "Print to PDF"
+msgstr "Presi al PDF"
+
+msgid "By the"
+msgstr "Per la"
+
+msgid "Last updated on"
+msgstr "Laste ĝisdatigita la"
+
+msgid "Download source file"
+msgstr "Elŝutu fontodosieron"
+
+msgid "Download this page"
+msgstr "Elŝutu ĉi tiun paĝon"
diff --git a/_static/locales/es/LC_MESSAGES/booktheme.mo b/_static/locales/es/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..ba2ee4dc22148ed53f2aeba32c5965654d4a5a2f
GIT binary patch
literal 1396
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literal 0
HcmV?d00001

diff --git a/_static/locales/es/LC_MESSAGES/booktheme.po b/_static/locales/es/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..611834b
--- /dev/null
+++ b/_static/locales/es/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: es\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema por el"
+
+msgid "Open an issue"
+msgstr "Abrir un problema"
+
+msgid "Contents"
+msgstr "Contenido"
+
+msgid "Download notebook file"
+msgstr "Descargar archivo de cuaderno"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del libro de la esfinge"
+
+msgid "Fullscreen mode"
+msgstr "Modo de pantalla completa"
+
+msgid "Edit this page"
+msgstr "Edita esta página"
+
+msgid "By"
+msgstr "Por"
+
+msgid "Copyright"
+msgstr "Derechos de autor"
+
+msgid "Source repository"
+msgstr "Repositorio de origen"
+
+msgid "previous page"
+msgstr "pagina anterior"
+
+msgid "next page"
+msgstr "siguiente página"
+
+msgid "Toggle navigation"
+msgstr "Navegación de palanca"
+
+msgid "repository"
+msgstr "repositorio"
+
+msgid "suggest edit"
+msgstr "sugerir editar"
+
+msgid "open issue"
+msgstr "Tema abierto"
+
+msgid "Launch"
+msgstr "Lanzamiento"
+
+msgid "Print to PDF"
+msgstr "Imprimir en PDF"
+
+msgid "By the"
+msgstr "Por el"
+
+msgid "Last updated on"
+msgstr "Ultima actualización en"
+
+msgid "Download source file"
+msgstr "Descargar archivo fuente"
+
+msgid "Download this page"
+msgstr "Descarga esta pagina"
diff --git a/_static/locales/et/LC_MESSAGES/booktheme.mo b/_static/locales/et/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..983b82391f499b67a9046c15d0dd8744650ad925
GIT binary patch
literal 1341
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literal 0
HcmV?d00001

diff --git a/_static/locales/et/LC_MESSAGES/booktheme.po b/_static/locales/et/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..345088f
--- /dev/null
+++ b/_static/locales/et/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: et\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Teema"
+
+msgid "Open an issue"
+msgstr "Avage probleem"
+
+msgid "Contents"
+msgstr "Sisu"
+
+msgid "Download notebook file"
+msgstr "Laadige sülearvuti fail alla"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksiraamatu teema"
+
+msgid "Fullscreen mode"
+msgstr "Täisekraanirežiim"
+
+msgid "Edit this page"
+msgstr "Muutke seda lehte"
+
+msgid "By"
+msgstr "Kõrval"
+
+msgid "Copyright"
+msgstr "Autoriõigus"
+
+msgid "Source repository"
+msgstr "Allikahoidla"
+
+msgid "previous page"
+msgstr "eelmine leht"
+
+msgid "next page"
+msgstr "järgmine leht"
+
+msgid "Toggle navigation"
+msgstr "Lülita navigeerimine sisse"
+
+msgid "repository"
+msgstr "hoidla"
+
+msgid "suggest edit"
+msgstr "soovita muuta"
+
+msgid "open issue"
+msgstr "avatud küsimus"
+
+msgid "Launch"
+msgstr "Käivitage"
+
+msgid "Print to PDF"
+msgstr "Prindi PDF-i"
+
+msgid "By the"
+msgstr "Autor"
+
+msgid "Last updated on"
+msgstr "Viimati uuendatud"
+
+msgid "Download source file"
+msgstr "Laadige alla lähtefail"
+
+msgid "Download this page"
+msgstr "Laadige see leht alla"
diff --git a/_static/locales/fi/LC_MESSAGES/booktheme.mo b/_static/locales/fi/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..d8ac054597c924e3010f629caeac1c748b7211cd
GIT binary patch
literal 1368
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literal 0
HcmV?d00001

diff --git a/_static/locales/fi/LC_MESSAGES/booktheme.po b/_static/locales/fi/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..d97a08d
--- /dev/null
+++ b/_static/locales/fi/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: fi\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Teeman tekijä"
+
+msgid "Open an issue"
+msgstr "Avaa ongelma"
+
+msgid "Contents"
+msgstr "Sisällys"
+
+msgid "Download notebook file"
+msgstr "Lataa muistikirjatiedosto"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-kirjan teema"
+
+msgid "Fullscreen mode"
+msgstr "Koko näytön tila"
+
+msgid "Edit this page"
+msgstr "Muokkaa tätä sivua"
+
+msgid "By"
+msgstr "Tekijä"
+
+msgid "Copyright"
+msgstr "Tekijänoikeus"
+
+msgid "Source repository"
+msgstr "Lähteen arkisto"
+
+msgid "previous page"
+msgstr "Edellinen sivu"
+
+msgid "next page"
+msgstr "seuraava sivu"
+
+msgid "Toggle navigation"
+msgstr "Vaihda navigointia"
+
+msgid "repository"
+msgstr "arkisto"
+
+msgid "suggest edit"
+msgstr "ehdottaa muokkausta"
+
+msgid "open issue"
+msgstr "avoin ongelma"
+
+msgid "Launch"
+msgstr "Tuoda markkinoille"
+
+msgid "Print to PDF"
+msgstr "Tulosta PDF-tiedostoon"
+
+msgid "By the"
+msgstr "Mukaan"
+
+msgid "Last updated on"
+msgstr "Viimeksi päivitetty"
+
+msgid "Download source file"
+msgstr "Lataa lähdetiedosto"
+
+msgid "Download this page"
+msgstr "Lataa tämä sivu"
diff --git a/_static/locales/fr/LC_MESSAGES/booktheme.mo b/_static/locales/fr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..f663d39f0faa76c5b9bd504c51252eef74cca5de
GIT binary patch
literal 1412
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T<A7>A2+cw{Vw>h7Yg+MtLz81?

literal 0
HcmV?d00001

diff --git a/_static/locales/fr/LC_MESSAGES/booktheme.po b/_static/locales/fr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..88f3517
--- /dev/null
+++ b/_static/locales/fr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: fr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Thème par le"
+
+msgid "Open an issue"
+msgstr "Ouvrez un problème"
+
+msgid "Contents"
+msgstr "Contenu"
+
+msgid "Download notebook file"
+msgstr "Télécharger le fichier notebook"
+
+msgid "Sphinx Book Theme"
+msgstr "Thème du livre Sphinx"
+
+msgid "Fullscreen mode"
+msgstr "Mode plein écran"
+
+msgid "Edit this page"
+msgstr "Modifier cette page"
+
+msgid "By"
+msgstr "Par"
+
+msgid "Copyright"
+msgstr "droits d'auteur"
+
+msgid "Source repository"
+msgstr "Dépôt source"
+
+msgid "previous page"
+msgstr "page précédente"
+
+msgid "next page"
+msgstr "page suivante"
+
+msgid "Toggle navigation"
+msgstr "Basculer la navigation"
+
+msgid "repository"
+msgstr "dépôt"
+
+msgid "suggest edit"
+msgstr "suggestion de modification"
+
+msgid "open issue"
+msgstr "signaler un problème"
+
+msgid "Launch"
+msgstr "lancement"
+
+msgid "Print to PDF"
+msgstr "Imprimer au format PDF"
+
+msgid "By the"
+msgstr "Par le"
+
+msgid "Last updated on"
+msgstr "Dernière mise à jour le"
+
+msgid "Download source file"
+msgstr "Télécharger le fichier source"
+
+msgid "Download this page"
+msgstr "Téléchargez cette page"
diff --git a/_static/locales/hr/LC_MESSAGES/booktheme.mo b/_static/locales/hr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..eca4a1a2842830f06bd5f6235bf01d07bdd313d2
GIT binary patch
literal 1402
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z_QwY`4=BRGdR=K_Hz+H$4;+Vk<5B<S={~P*xX-@7X)~7_TMX#tzKOj%+7=0Q9Z>0e
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k(#nYgw(f}olqHrfw@Z7L48TfWxy6y8<oWoKnK!Qg2ay|IDF6Tf

literal 0
HcmV?d00001

diff --git a/_static/locales/hr/LC_MESSAGES/booktheme.po b/_static/locales/hr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..fb9440a
--- /dev/null
+++ b/_static/locales/hr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: hr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema autora"
+
+msgid "Open an issue"
+msgstr "Otvorite izdanje"
+
+msgid "Contents"
+msgstr "Sadržaj"
+
+msgid "Download notebook file"
+msgstr "Preuzmi datoteku bilježnice"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema knjige Sphinx"
+
+msgid "Fullscreen mode"
+msgstr "Način preko cijelog zaslona"
+
+msgid "Edit this page"
+msgstr "Uredite ovu stranicu"
+
+msgid "By"
+msgstr "Po"
+
+msgid "Copyright"
+msgstr "Autorska prava"
+
+msgid "Source repository"
+msgstr "Izvorno spremište"
+
+msgid "previous page"
+msgstr "Prethodna stranica"
+
+msgid "next page"
+msgstr "sljedeća stranica"
+
+msgid "Toggle navigation"
+msgstr "Uključi / isključi navigaciju"
+
+msgid "repository"
+msgstr "spremište"
+
+msgid "suggest edit"
+msgstr "predloži uređivanje"
+
+msgid "open issue"
+msgstr "otvoreno izdanje"
+
+msgid "Launch"
+msgstr "Pokrenite"
+
+msgid "Print to PDF"
+msgstr "Ispis u PDF"
+
+msgid "By the"
+msgstr "Od strane"
+
+msgid "Last updated on"
+msgstr "Posljednje ažuriranje:"
+
+msgid "Download source file"
+msgstr "Preuzmi izvornu datoteku"
+
+msgid "Download this page"
+msgstr "Preuzmite ovu stranicu"
diff --git a/_static/locales/id/LC_MESSAGES/booktheme.mo b/_static/locales/id/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..d07a06a9d27aaf6d67234c227c22d61d5a23203b
GIT binary patch
literal 1323
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zENnvJmhwTyGEspNX$|aX2AQ~1;({_#dml-xI&Bcd%GHG>aj(wjsI*=O>qg>UnQ5~j
z=eV|BMg@s#m;L%pJ)5RENuwU=R0WMBhBiW7o#i}rlG4#5ZDUjRSa=`Plu#w%B`C$N
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z_q>s3y7KCU*_ODI6J+IW*=v!hp&aNkC~@B;F{`@wufgMIJ{Em)%}J(mRd82~uKPNE
zu!45VT#b~I5H)6^M@|hh?PQ`$a((5ZmR_kKRwzf4qZ=wC=HgJvLU}~a#5@JmHKeM~
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T@Rc+s5Lgrh10CRyRonjp5+F85

literal 0
HcmV?d00001

diff --git a/_static/locales/id/LC_MESSAGES/booktheme.po b/_static/locales/id/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..9ffb56f
--- /dev/null
+++ b/_static/locales/id/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: id\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema oleh"
+
+msgid "Open an issue"
+msgstr "Buka masalah"
+
+msgid "Contents"
+msgstr "Isi"
+
+msgid "Download notebook file"
+msgstr "Unduh file notebook"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Buku Sphinx"
+
+msgid "Fullscreen mode"
+msgstr "Mode layar penuh"
+
+msgid "Edit this page"
+msgstr "Edit halaman ini"
+
+msgid "By"
+msgstr "Oleh"
+
+msgid "Copyright"
+msgstr "hak cipta"
+
+msgid "Source repository"
+msgstr "Repositori sumber"
+
+msgid "previous page"
+msgstr "halaman sebelumnya"
+
+msgid "next page"
+msgstr "halaman selanjutnya"
+
+msgid "Toggle navigation"
+msgstr "Alihkan navigasi"
+
+msgid "repository"
+msgstr "gudang"
+
+msgid "suggest edit"
+msgstr "menyarankan edit"
+
+msgid "open issue"
+msgstr "masalah terbuka"
+
+msgid "Launch"
+msgstr "Meluncurkan"
+
+msgid "Print to PDF"
+msgstr "Cetak ke PDF"
+
+msgid "By the"
+msgstr "Oleh"
+
+msgid "Last updated on"
+msgstr "Terakhir diperbarui saat"
+
+msgid "Download source file"
+msgstr "Unduh file sumber"
+
+msgid "Download this page"
+msgstr "Unduh halaman ini"
diff --git a/_static/locales/it/LC_MESSAGES/booktheme.mo b/_static/locales/it/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..53ba476edd2df2a802917e9df402257ceca1a130
GIT binary patch
literal 1403
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literal 0
HcmV?d00001

diff --git a/_static/locales/it/LC_MESSAGES/booktheme.po b/_static/locales/it/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..04308dd
--- /dev/null
+++ b/_static/locales/it/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: it\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema di"
+
+msgid "Open an issue"
+msgstr "Apri un problema"
+
+msgid "Contents"
+msgstr "Contenuti"
+
+msgid "Download notebook file"
+msgstr "Scarica il file del taccuino"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema del libro della Sfinge"
+
+msgid "Fullscreen mode"
+msgstr "Modalità schermo intero"
+
+msgid "Edit this page"
+msgstr "Modifica questa pagina"
+
+msgid "By"
+msgstr "Di"
+
+msgid "Copyright"
+msgstr "Diritto d'autore"
+
+msgid "Source repository"
+msgstr "Repository di origine"
+
+msgid "previous page"
+msgstr "pagina precedente"
+
+msgid "next page"
+msgstr "pagina successiva"
+
+msgid "Toggle navigation"
+msgstr "Attiva / disattiva la navigazione"
+
+msgid "repository"
+msgstr "repository"
+
+msgid "suggest edit"
+msgstr "suggerisci modifica"
+
+msgid "open issue"
+msgstr "questione aperta"
+
+msgid "Launch"
+msgstr "Lanciare"
+
+msgid "Print to PDF"
+msgstr "Stampa in PDF"
+
+msgid "By the"
+msgstr "Dal"
+
+msgid "Last updated on"
+msgstr "Ultimo aggiornamento il"
+
+msgid "Download source file"
+msgstr "Scarica il file sorgente"
+
+msgid "Download this page"
+msgstr "Scarica questa pagina"
diff --git a/_static/locales/iw/LC_MESSAGES/booktheme.mo b/_static/locales/iw/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..a45c6575e469914582f8223ac54793afc2f26575
GIT binary patch
literal 1445
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literal 0
HcmV?d00001

diff --git a/_static/locales/iw/LC_MESSAGES/booktheme.po b/_static/locales/iw/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..4ea190d
--- /dev/null
+++ b/_static/locales/iw/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: iw\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "נושא מאת"
+
+msgid "Open an issue"
+msgstr "פתח גיליון"
+
+msgid "Contents"
+msgstr "תוכן"
+
+msgid "Download notebook file"
+msgstr "הורד קובץ מחברת"
+
+msgid "Sphinx Book Theme"
+msgstr "נושא ספר ספינקס"
+
+msgid "Fullscreen mode"
+msgstr "מצב מסך מלא"
+
+msgid "Edit this page"
+msgstr "ערוך דף זה"
+
+msgid "By"
+msgstr "על ידי"
+
+msgid "Copyright"
+msgstr "זכויות יוצרים"
+
+msgid "Source repository"
+msgstr "מאגר המקורות"
+
+msgid "previous page"
+msgstr "עמוד קודם"
+
+msgid "next page"
+msgstr "עמוד הבא"
+
+msgid "Toggle navigation"
+msgstr "החלף ניווט"
+
+msgid "repository"
+msgstr "מאגר"
+
+msgid "suggest edit"
+msgstr "מציע לערוך"
+
+msgid "open issue"
+msgstr "בעיה פתוחה"
+
+msgid "Launch"
+msgstr "לְהַשִׁיק"
+
+msgid "Print to PDF"
+msgstr "הדפס לקובץ PDF"
+
+msgid "By the"
+msgstr "דרך"
+
+msgid "Last updated on"
+msgstr "עודכן לאחרונה ב"
+
+msgid "Download source file"
+msgstr "הורד את קובץ המקור"
+
+msgid "Download this page"
+msgstr "הורד דף זה"
diff --git a/_static/locales/ja/LC_MESSAGES/booktheme.mo b/_static/locales/ja/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..1cefd29ce3cc08792667a82dc7ff47e9843107be
GIT binary patch
literal 1471
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literal 0
HcmV?d00001

diff --git a/_static/locales/ja/LC_MESSAGES/booktheme.po b/_static/locales/ja/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..77d5a09
--- /dev/null
+++ b/_static/locales/ja/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ja\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "のテーマ"
+
+msgid "Open an issue"
+msgstr "問題を報告"
+
+msgid "Contents"
+msgstr "目次"
+
+msgid "Download notebook file"
+msgstr "ノートブックファイルをダウンロード"
+
+msgid "Sphinx Book Theme"
+msgstr "スフィンクスの本のテーマ"
+
+msgid "Fullscreen mode"
+msgstr "全画面モード"
+
+msgid "Edit this page"
+msgstr "このページを編集"
+
+msgid "By"
+msgstr "著者"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Source repository"
+msgstr "ソースリポジトリ"
+
+msgid "previous page"
+msgstr "前のページ"
+
+msgid "next page"
+msgstr "次のページ"
+
+msgid "Toggle navigation"
+msgstr "ナビゲーションを切り替え"
+
+msgid "repository"
+msgstr "リポジトリ"
+
+msgid "suggest edit"
+msgstr "編集を提案する"
+
+msgid "open issue"
+msgstr "未解決の問題"
+
+msgid "Launch"
+msgstr "起動"
+
+msgid "Print to PDF"
+msgstr "PDFに印刷"
+
+msgid "By the"
+msgstr "によって"
+
+msgid "Last updated on"
+msgstr "最終更新日"
+
+msgid "Download source file"
+msgstr "ソースファイルをダウンロード"
+
+msgid "Download this page"
+msgstr "このページをダウンロード"
diff --git a/_static/locales/ko/LC_MESSAGES/booktheme.mo b/_static/locales/ko/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..06c7ec938b4975c721caff60b956e343eef11570
GIT binary patch
literal 1375
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literal 0
HcmV?d00001

diff --git a/_static/locales/ko/LC_MESSAGES/booktheme.po b/_static/locales/ko/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..6ee3d78
--- /dev/null
+++ b/_static/locales/ko/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ko\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "테마별"
+
+msgid "Open an issue"
+msgstr "이슈 열기"
+
+msgid "Contents"
+msgstr "내용"
+
+msgid "Download notebook file"
+msgstr "노트북 파일 다운로드"
+
+msgid "Sphinx Book Theme"
+msgstr "스핑크스 도서 테마"
+
+msgid "Fullscreen mode"
+msgstr "전체 화면으로보기"
+
+msgid "Edit this page"
+msgstr "이 페이지 편집"
+
+msgid "By"
+msgstr "으로"
+
+msgid "Copyright"
+msgstr "저작권"
+
+msgid "Source repository"
+msgstr "소스 저장소"
+
+msgid "previous page"
+msgstr "이전 페이지"
+
+msgid "next page"
+msgstr "다음 페이지"
+
+msgid "Toggle navigation"
+msgstr "탐색 전환"
+
+msgid "repository"
+msgstr "저장소"
+
+msgid "suggest edit"
+msgstr "편집 제안"
+
+msgid "open issue"
+msgstr "열린 문제"
+
+msgid "Launch"
+msgstr "시작하다"
+
+msgid "Print to PDF"
+msgstr "PDF로 인쇄"
+
+msgid "By the"
+msgstr "에 의해"
+
+msgid "Last updated on"
+msgstr "마지막 업데이트"
+
+msgid "Download source file"
+msgstr "소스 파일 다운로드"
+
+msgid "Download this page"
+msgstr "이 페이지 다운로드"
diff --git a/_static/locales/lt/LC_MESSAGES/booktheme.mo b/_static/locales/lt/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..4468ba04bc134a84fea5e3c973461cf02c9c2da3
GIT binary patch
literal 1413
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literal 0
HcmV?d00001

diff --git a/_static/locales/lt/LC_MESSAGES/booktheme.po b/_static/locales/lt/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..01be267
--- /dev/null
+++ b/_static/locales/lt/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: lt\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema"
+
+msgid "Open an issue"
+msgstr "Atidarykite problemą"
+
+msgid "Contents"
+msgstr "Turinys"
+
+msgid "Download notebook file"
+msgstr "Atsisiųsti nešiojamojo kompiuterio failą"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinkso knygos tema"
+
+msgid "Fullscreen mode"
+msgstr "Pilno ekrano režimas"
+
+msgid "Edit this page"
+msgstr "Redaguoti šį puslapį"
+
+msgid "By"
+msgstr "Iki"
+
+msgid "Copyright"
+msgstr "Autorių teisės"
+
+msgid "Source repository"
+msgstr "Šaltinio saugykla"
+
+msgid "previous page"
+msgstr "Ankstesnis puslapis"
+
+msgid "next page"
+msgstr "Kitas puslapis"
+
+msgid "Toggle navigation"
+msgstr "Perjungti naršymą"
+
+msgid "repository"
+msgstr "saugykla"
+
+msgid "suggest edit"
+msgstr "pasiūlyti redaguoti"
+
+msgid "open issue"
+msgstr "atviras klausimas"
+
+msgid "Launch"
+msgstr "Paleiskite"
+
+msgid "Print to PDF"
+msgstr "Spausdinti į PDF"
+
+msgid "By the"
+msgstr "Prie"
+
+msgid "Last updated on"
+msgstr "Paskutinį kartą atnaujinta"
+
+msgid "Download source file"
+msgstr "Atsisiųsti šaltinio failą"
+
+msgid "Download this page"
+msgstr "Atsisiųskite šį puslapį"
diff --git a/_static/locales/lv/LC_MESSAGES/booktheme.mo b/_static/locales/lv/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..74aa4d8985d96a07c4c9be96f006f3b58d884342
GIT binary patch
literal 1404
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A2mk;8

literal 0
HcmV?d00001

diff --git a/_static/locales/lv/LC_MESSAGES/booktheme.po b/_static/locales/lv/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..993a1e4
--- /dev/null
+++ b/_static/locales/lv/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: lv\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Autora tēma"
+
+msgid "Open an issue"
+msgstr "Atveriet problēmu"
+
+msgid "Contents"
+msgstr "Saturs"
+
+msgid "Download notebook file"
+msgstr "Lejupielādēt piezīmju grāmatiņu"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfinksa grāmatas tēma"
+
+msgid "Fullscreen mode"
+msgstr "Pilnekrāna režīms"
+
+msgid "Edit this page"
+msgstr "Rediģēt šo lapu"
+
+msgid "By"
+msgstr "Autors"
+
+msgid "Copyright"
+msgstr "Autortiesības"
+
+msgid "Source repository"
+msgstr "Avota krātuve"
+
+msgid "previous page"
+msgstr "iepriekšējā lapa"
+
+msgid "next page"
+msgstr "nākamā lapaspuse"
+
+msgid "Toggle navigation"
+msgstr "Pārslēgt navigāciju"
+
+msgid "repository"
+msgstr "krātuve"
+
+msgid "suggest edit"
+msgstr "ieteikt rediģēt"
+
+msgid "open issue"
+msgstr "atklāts jautājums"
+
+msgid "Launch"
+msgstr "Uzsākt"
+
+msgid "Print to PDF"
+msgstr "Drukāt PDF formātā"
+
+msgid "By the"
+msgstr "Ar"
+
+msgid "Last updated on"
+msgstr "Pēdējoreiz atjaunināts"
+
+msgid "Download source file"
+msgstr "Lejupielādēt avota failu"
+
+msgid "Download this page"
+msgstr "Lejupielādējiet šo lapu"
diff --git a/_static/locales/ml/LC_MESSAGES/booktheme.mo b/_static/locales/ml/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..2736e8fcf6f9e923c2403307f0b366086d37b335
GIT binary patch
literal 1883
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literal 0
HcmV?d00001

diff --git a/_static/locales/ml/LC_MESSAGES/booktheme.po b/_static/locales/ml/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..81daf7c
--- /dev/null
+++ b/_static/locales/ml/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ml\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "പ്രമേയം"
+
+msgid "Open an issue"
+msgstr "ഒരു പ്രശ്നം തുറക്കുക"
+
+msgid "Download notebook file"
+msgstr "നോട്ട്ബുക്ക് ഫയൽ ഡൺലോഡ് ചെയ്യുക"
+
+msgid "Sphinx Book Theme"
+msgstr "സ്ഫിങ്ക്സ് പുസ്തക തീം"
+
+msgid "Edit this page"
+msgstr "ഈ പേജ് എഡിറ്റുചെയ്യുക"
+
+msgid "By"
+msgstr "എഴുതിയത്"
+
+msgid "Copyright"
+msgstr "പകർപ്പവകാശം"
+
+msgid "Source repository"
+msgstr "ഉറവിട ശേഖരം"
+
+msgid "previous page"
+msgstr "മുൻപത്തെ താൾ"
+
+msgid "next page"
+msgstr "അടുത്ത പേജ്"
+
+msgid "Toggle navigation"
+msgstr "നാവിഗേഷൻ ടോഗിൾ ചെയ്യുക"
+
+msgid "suggest edit"
+msgstr "എഡിറ്റുചെയ്യാൻ നിർദ്ദേശിക്കുക"
+
+msgid "open issue"
+msgstr "തുറന്ന പ്രശ്നം"
+
+msgid "Launch"
+msgstr "സമാരംഭിക്കുക"
+
+msgid "Print to PDF"
+msgstr "PDF- ലേക്ക് പ്രിന്റുചെയ്യുക"
+
+msgid "By the"
+msgstr "എഴുതിയത്"
+
+msgid "Last updated on"
+msgstr "അവസാനം അപ്‌ഡേറ്റുചെയ്‌തത്"
+
+msgid "Download source file"
+msgstr "ഉറവിട ഫയൽ ഡൗൺലോഡുചെയ്യുക"
+
+msgid "Download this page"
+msgstr "ഈ പേജ് ഡൗൺലോഡുചെയ്യുക"
diff --git a/_static/locales/mr/LC_MESSAGES/booktheme.mo b/_static/locales/mr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..fe530100d7715cdc19a6f9db33a971665835f3c4
GIT binary patch
literal 1674
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A6951J

literal 0
HcmV?d00001

diff --git a/_static/locales/mr/LC_MESSAGES/booktheme.po b/_static/locales/mr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..fd857bf
--- /dev/null
+++ b/_static/locales/mr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: mr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "द्वारा थीम"
+
+msgid "Open an issue"
+msgstr "एक मुद्दा उघडा"
+
+msgid "Download notebook file"
+msgstr "नोटबुक फाईल डाउनलोड करा"
+
+msgid "Sphinx Book Theme"
+msgstr "स्फिंक्स बुक थीम"
+
+msgid "Edit this page"
+msgstr "हे पृष्ठ संपादित करा"
+
+msgid "By"
+msgstr "द्वारा"
+
+msgid "Copyright"
+msgstr "कॉपीराइट"
+
+msgid "Source repository"
+msgstr "स्त्रोत भांडार"
+
+msgid "previous page"
+msgstr "मागील पान"
+
+msgid "next page"
+msgstr "पुढील पृष्ठ"
+
+msgid "Toggle navigation"
+msgstr "नेव्हिगेशन टॉगल करा"
+
+msgid "suggest edit"
+msgstr "संपादन सुचवा"
+
+msgid "open issue"
+msgstr "खुला मुद्दा"
+
+msgid "Launch"
+msgstr "लाँच करा"
+
+msgid "Print to PDF"
+msgstr "पीडीएफवर मुद्रित करा"
+
+msgid "By the"
+msgstr "द्वारा"
+
+msgid "Last updated on"
+msgstr "अखेरचे अद्यतनित"
+
+msgid "Download source file"
+msgstr "स्त्रोत फाइल डाउनलोड करा"
+
+msgid "Download this page"
+msgstr "हे पृष्ठ डाउनलोड करा"
diff --git a/_static/locales/ms/LC_MESSAGES/booktheme.mo b/_static/locales/ms/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..f02603fa2522a40060bd3f1b5d65052c77530de6
GIT binary patch
literal 1213
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zHb`+^fGzM9NcL|*vVRZ02ETwK;L`&&w?VSo0ms4TAm#b4;b-tP&Og8l;4hH$$Kizh
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zzNQMfckN8DvHa)KXUg$Pq>{}g%EROCaZxzWLzM_G6K*tH6hX%sTdt($LUZL@kZi>&
z4I{%>X6M;z?cd7EIOUBUu+=J4dW+9cF1<{ak_~s>uT$@tG%Y08Vne0ED+JMU%h%Z$
zic<%vtlUs0G-<k!rV^=1q>K1wUB*6INTU1FI=E=_I%q^eqhYw#(!$c*-uvW@iB{KW
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z(zRBKU_}wl8(X?I*VXXoI=7~I{oA>w`Q;9uyguDx%Y{TQY0E^;V^w*<mV!VpgAJOm
z303fZ$OFyt{}&g^zO;WEWTFryjYX?^kbhtFq!oFlEKgJ}OW}A9HMBVqn5vG2&4p$&
zAr~BvaFnsG^hhh0!@1tw9jM|>;2yki7=)(TY<OX)9L&@d%}AYMC3$t|i|6R7w880t
eV9EtF-J(E&`s>V{?8_pQ^c*8Iog%4v$iD&q$RjNP

literal 0
HcmV?d00001

diff --git a/_static/locales/ms/LC_MESSAGES/booktheme.po b/_static/locales/ms/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b616d70
--- /dev/null
+++ b/_static/locales/ms/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ms\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema oleh"
+
+msgid "Open an issue"
+msgstr "Buka masalah"
+
+msgid "Download notebook file"
+msgstr "Muat turun fail buku nota"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Buku Sphinx"
+
+msgid "Edit this page"
+msgstr "Edit halaman ini"
+
+msgid "By"
+msgstr "Oleh"
+
+msgid "Copyright"
+msgstr "hak cipta"
+
+msgid "Source repository"
+msgstr "Repositori sumber"
+
+msgid "previous page"
+msgstr "halaman sebelumnya"
+
+msgid "next page"
+msgstr "muka surat seterusnya"
+
+msgid "Toggle navigation"
+msgstr "Togol navigasi"
+
+msgid "suggest edit"
+msgstr "cadangkan edit"
+
+msgid "open issue"
+msgstr "isu terbuka"
+
+msgid "Launch"
+msgstr "Lancarkan"
+
+msgid "Print to PDF"
+msgstr "Cetak ke PDF"
+
+msgid "By the"
+msgstr "Oleh"
+
+msgid "Last updated on"
+msgstr "Terakhir dikemas kini pada"
+
+msgid "Download source file"
+msgstr "Muat turun fail sumber"
+
+msgid "Download this page"
+msgstr "Muat turun halaman ini"
diff --git a/_static/locales/nl/LC_MESSAGES/booktheme.mo b/_static/locales/nl/LC_MESSAGES/booktheme.mo
new file mode 100644
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GIT binary patch
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HcmV?d00001

diff --git a/_static/locales/nl/LC_MESSAGES/booktheme.po b/_static/locales/nl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..f16f4bc
--- /dev/null
+++ b/_static/locales/nl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: nl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Thema door de"
+
+msgid "Open an issue"
+msgstr "Open een probleem"
+
+msgid "Contents"
+msgstr "Inhoud"
+
+msgid "Download notebook file"
+msgstr "Download notebookbestand"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx-boekthema"
+
+msgid "Fullscreen mode"
+msgstr "Volledig scherm"
+
+msgid "Edit this page"
+msgstr "bewerk deze pagina"
+
+msgid "By"
+msgstr "Door"
+
+msgid "Copyright"
+msgstr "auteursrechten"
+
+msgid "Source repository"
+msgstr "Bronopslagplaats"
+
+msgid "previous page"
+msgstr "vorige pagina"
+
+msgid "next page"
+msgstr "volgende bladzijde"
+
+msgid "Toggle navigation"
+msgstr "Schakel navigatie"
+
+msgid "repository"
+msgstr "repository"
+
+msgid "suggest edit"
+msgstr "suggereren bewerken"
+
+msgid "open issue"
+msgstr "open probleem"
+
+msgid "Launch"
+msgstr "Lancering"
+
+msgid "Print to PDF"
+msgstr "Afdrukken naar pdf"
+
+msgid "By the"
+msgstr "Door de"
+
+msgid "Last updated on"
+msgstr "Laatst geupdate op"
+
+msgid "Download source file"
+msgstr "Download het bronbestand"
+
+msgid "Download this page"
+msgstr "Download deze pagina"
diff --git a/_static/locales/no/LC_MESSAGES/booktheme.mo b/_static/locales/no/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..6cd15c88de675226c00ca4d0430171075e5559ff
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/no/LC_MESSAGES/booktheme.po b/_static/locales/no/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b1d304e
--- /dev/null
+++ b/_static/locales/no/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: no\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema av"
+
+msgid "Open an issue"
+msgstr "Åpne et problem"
+
+msgid "Contents"
+msgstr "Innhold"
+
+msgid "Download notebook file"
+msgstr "Last ned notatbokfilen"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx boktema"
+
+msgid "Fullscreen mode"
+msgstr "Fullskjerm-modus"
+
+msgid "Edit this page"
+msgstr "Rediger denne siden"
+
+msgid "By"
+msgstr "Av"
+
+msgid "Copyright"
+msgstr "opphavsrett"
+
+msgid "Source repository"
+msgstr "Kildedepot"
+
+msgid "previous page"
+msgstr "forrige side"
+
+msgid "next page"
+msgstr "neste side"
+
+msgid "Toggle navigation"
+msgstr "Bytt navigasjon"
+
+msgid "repository"
+msgstr "oppbevaringssted"
+
+msgid "suggest edit"
+msgstr "foreslå redigering"
+
+msgid "open issue"
+msgstr "åpent nummer"
+
+msgid "Launch"
+msgstr "Start"
+
+msgid "Print to PDF"
+msgstr "Skriv ut til PDF"
+
+msgid "By the"
+msgstr "Ved"
+
+msgid "Last updated on"
+msgstr "Sist oppdatert den"
+
+msgid "Download source file"
+msgstr "Last ned kildefilen"
+
+msgid "Download this page"
+msgstr "Last ned denne siden"
diff --git a/_static/locales/pl/LC_MESSAGES/booktheme.mo b/_static/locales/pl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..9ebb584f77091f5666db4194ecd4f707396cbbca
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/pl/LC_MESSAGES/booktheme.po b/_static/locales/pl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..80d2c89
--- /dev/null
+++ b/_static/locales/pl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: pl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Motyw autorstwa"
+
+msgid "Open an issue"
+msgstr "Otwórz problem"
+
+msgid "Contents"
+msgstr "Zawartość"
+
+msgid "Download notebook file"
+msgstr "Pobierz plik notatnika"
+
+msgid "Sphinx Book Theme"
+msgstr "Motyw książki Sphinx"
+
+msgid "Fullscreen mode"
+msgstr "Pełny ekran"
+
+msgid "Edit this page"
+msgstr "Edytuj tę strone"
+
+msgid "By"
+msgstr "Przez"
+
+msgid "Copyright"
+msgstr "prawa autorskie"
+
+msgid "Source repository"
+msgstr "Repozytorium źródłowe"
+
+msgid "previous page"
+msgstr "Poprzednia strona"
+
+msgid "next page"
+msgstr "Następna strona"
+
+msgid "Toggle navigation"
+msgstr "Przełącz nawigację"
+
+msgid "repository"
+msgstr "magazyn"
+
+msgid "suggest edit"
+msgstr "zaproponuj edycję"
+
+msgid "open issue"
+msgstr "otwarty problem"
+
+msgid "Launch"
+msgstr "Uruchomić"
+
+msgid "Print to PDF"
+msgstr "Drukuj do PDF"
+
+msgid "By the"
+msgstr "Przez"
+
+msgid "Last updated on"
+msgstr "Ostatnia aktualizacja"
+
+msgid "Download source file"
+msgstr "Pobierz plik źródłowy"
+
+msgid "Download this page"
+msgstr "Pobierz tę stronę"
diff --git a/_static/locales/pt/LC_MESSAGES/booktheme.mo b/_static/locales/pt/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..d0ddb8728e1d5ef72dddc1537f66aa32251a83c8
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/pt/LC_MESSAGES/booktheme.po b/_static/locales/pt/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..45ac847
--- /dev/null
+++ b/_static/locales/pt/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: pt\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema por"
+
+msgid "Open an issue"
+msgstr "Abra um problema"
+
+msgid "Contents"
+msgstr "Conteúdo"
+
+msgid "Download notebook file"
+msgstr "Baixar arquivo de notebook"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema do livro Sphinx"
+
+msgid "Fullscreen mode"
+msgstr "Modo tela cheia"
+
+msgid "Edit this page"
+msgstr "Edite essa página"
+
+msgid "By"
+msgstr "De"
+
+msgid "Copyright"
+msgstr "direito autoral"
+
+msgid "Source repository"
+msgstr "Repositório fonte"
+
+msgid "previous page"
+msgstr "página anterior"
+
+msgid "next page"
+msgstr "próxima página"
+
+msgid "Toggle navigation"
+msgstr "Alternar de navegação"
+
+msgid "repository"
+msgstr "repositório"
+
+msgid "suggest edit"
+msgstr "sugerir edição"
+
+msgid "open issue"
+msgstr "questão aberta"
+
+msgid "Launch"
+msgstr "Lançamento"
+
+msgid "Print to PDF"
+msgstr "Imprimir em PDF"
+
+msgid "By the"
+msgstr "Pelo"
+
+msgid "Last updated on"
+msgstr "Última atualização em"
+
+msgid "Download source file"
+msgstr "Baixar arquivo fonte"
+
+msgid "Download this page"
+msgstr "Baixe esta página"
diff --git a/_static/locales/ro/LC_MESSAGES/booktheme.mo b/_static/locales/ro/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..3c36ab1df7c589ad82614599b8c91f072a25d4a6
GIT binary patch
literal 1390
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K1txIYaExD}s9q8P

literal 0
HcmV?d00001

diff --git a/_static/locales/ro/LC_MESSAGES/booktheme.po b/_static/locales/ro/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..532b3b8
--- /dev/null
+++ b/_static/locales/ro/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ro\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema de"
+
+msgid "Open an issue"
+msgstr "Deschideți o problemă"
+
+msgid "Contents"
+msgstr "Cuprins"
+
+msgid "Download notebook file"
+msgstr "Descărcați fișierul notebook"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema Sphinx Book"
+
+msgid "Fullscreen mode"
+msgstr "Modul ecran întreg"
+
+msgid "Edit this page"
+msgstr "Editați această pagină"
+
+msgid "By"
+msgstr "De"
+
+msgid "Copyright"
+msgstr "Drepturi de autor"
+
+msgid "Source repository"
+msgstr "Depozit sursă"
+
+msgid "previous page"
+msgstr "pagina anterioară"
+
+msgid "next page"
+msgstr "pagina următoare"
+
+msgid "Toggle navigation"
+msgstr "Comutare navigare"
+
+msgid "repository"
+msgstr "repertoriu"
+
+msgid "suggest edit"
+msgstr "sugerează editare"
+
+msgid "open issue"
+msgstr "problema deschisă"
+
+msgid "Launch"
+msgstr "Lansa"
+
+msgid "Print to PDF"
+msgstr "Imprimați în PDF"
+
+msgid "By the"
+msgstr "Langa"
+
+msgid "Last updated on"
+msgstr "Ultima actualizare la"
+
+msgid "Download source file"
+msgstr "Descărcați fișierul sursă"
+
+msgid "Download this page"
+msgstr "Descarcă această pagină"
diff --git a/_static/locales/ru/LC_MESSAGES/booktheme.mo b/_static/locales/ru/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..6b8ca41f36ebf869818399a9584cdb15619bea78
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/ru/LC_MESSAGES/booktheme.po b/_static/locales/ru/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b718b48
--- /dev/null
+++ b/_static/locales/ru/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ru\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Тема от"
+
+msgid "Open an issue"
+msgstr "Открыть вопрос"
+
+msgid "Contents"
+msgstr "Содержание"
+
+msgid "Download notebook file"
+msgstr "Скачать файл записной книжки"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема книги Сфинкс"
+
+msgid "Fullscreen mode"
+msgstr "Полноэкранный режим"
+
+msgid "Edit this page"
+msgstr "Редактировать эту страницу"
+
+msgid "By"
+msgstr "По"
+
+msgid "Copyright"
+msgstr "авторское право"
+
+msgid "Source repository"
+msgstr "Исходный репозиторий"
+
+msgid "previous page"
+msgstr "Предыдущая страница"
+
+msgid "next page"
+msgstr "Следующая страница"
+
+msgid "Toggle navigation"
+msgstr "Переключить навигацию"
+
+msgid "repository"
+msgstr "хранилище"
+
+msgid "suggest edit"
+msgstr "предложить редактировать"
+
+msgid "open issue"
+msgstr "открытый вопрос"
+
+msgid "Launch"
+msgstr "Запуск"
+
+msgid "Print to PDF"
+msgstr "Распечатать в PDF"
+
+msgid "By the"
+msgstr "Посредством"
+
+msgid "Last updated on"
+msgstr "Последнее обновление"
+
+msgid "Download source file"
+msgstr "Скачать исходный файл"
+
+msgid "Download this page"
+msgstr "Загрузите эту страницу"
diff --git a/_static/locales/sk/LC_MESSAGES/booktheme.mo b/_static/locales/sk/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..59bd0ddfa341477cf5ccfdc88f4c3e6127046f7a
GIT binary patch
literal 1393
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Di}h+h

literal 0
HcmV?d00001

diff --git a/_static/locales/sk/LC_MESSAGES/booktheme.po b/_static/locales/sk/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..f6c423b
--- /dev/null
+++ b/_static/locales/sk/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sk\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Téma od"
+
+msgid "Open an issue"
+msgstr "Otvorte problém"
+
+msgid "Contents"
+msgstr "Obsah"
+
+msgid "Download notebook file"
+msgstr "Stiahnite si zošit"
+
+msgid "Sphinx Book Theme"
+msgstr "Téma knihy Sfinga"
+
+msgid "Fullscreen mode"
+msgstr "Režim celej obrazovky"
+
+msgid "Edit this page"
+msgstr "Upraviť túto stránku"
+
+msgid "By"
+msgstr "Autor:"
+
+msgid "Copyright"
+msgstr "Autorské práva"
+
+msgid "Source repository"
+msgstr "Zdrojové úložisko"
+
+msgid "previous page"
+msgstr "predchádzajúca strana"
+
+msgid "next page"
+msgstr "ďalšia strana"
+
+msgid "Toggle navigation"
+msgstr "Prepnúť navigáciu"
+
+msgid "repository"
+msgstr "Úložisko"
+
+msgid "suggest edit"
+msgstr "navrhnúť úpravu"
+
+msgid "open issue"
+msgstr "otvorené vydanie"
+
+msgid "Launch"
+msgstr "Spustiť"
+
+msgid "Print to PDF"
+msgstr "Tlač do PDF"
+
+msgid "By the"
+msgstr "Podľa"
+
+msgid "Last updated on"
+msgstr "Posledná aktualizácia dňa"
+
+msgid "Download source file"
+msgstr "Stiahnite si zdrojový súbor"
+
+msgid "Download this page"
+msgstr "Stiahnite si túto stránku"
diff --git a/_static/locales/sl/LC_MESSAGES/booktheme.mo b/_static/locales/sl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..87bf26de683cb18c73bd23c2b4c57ef6a02545ec
GIT binary patch
literal 1374
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L??Y>WljW5^x#&@4

literal 0
HcmV?d00001

diff --git a/_static/locales/sl/LC_MESSAGES/booktheme.po b/_static/locales/sl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..9822dc5
--- /dev/null
+++ b/_static/locales/sl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema avtorja"
+
+msgid "Open an issue"
+msgstr "Odprite številko"
+
+msgid "Contents"
+msgstr "Vsebina"
+
+msgid "Download notebook file"
+msgstr "Prenesite datoteko zvezka"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema knjige Sphinx"
+
+msgid "Fullscreen mode"
+msgstr "Celozaslonski način"
+
+msgid "Edit this page"
+msgstr "Uredite to stran"
+
+msgid "By"
+msgstr "Avtor"
+
+msgid "Copyright"
+msgstr "avtorske pravice"
+
+msgid "Source repository"
+msgstr "Izvorno skladišče"
+
+msgid "previous page"
+msgstr "Prejšnja stran"
+
+msgid "next page"
+msgstr "Naslednja stran"
+
+msgid "Toggle navigation"
+msgstr "Preklopi navigacijo"
+
+msgid "repository"
+msgstr "odlagališče"
+
+msgid "suggest edit"
+msgstr "predlagajte urejanje"
+
+msgid "open issue"
+msgstr "odprto vprašanje"
+
+msgid "Launch"
+msgstr "Kosilo"
+
+msgid "Print to PDF"
+msgstr "Natisni v PDF"
+
+msgid "By the"
+msgstr "Avtor"
+
+msgid "Last updated on"
+msgstr "Nazadnje posodobljeno dne"
+
+msgid "Download source file"
+msgstr "Prenesite izvorno datoteko"
+
+msgid "Download this page"
+msgstr "Prenesite to stran"
diff --git a/_static/locales/sr/LC_MESSAGES/booktheme.mo b/_static/locales/sr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..ec740f4852f3973fa72ed0c7f7cf59273b8dba41
GIT binary patch
literal 1679
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literal 0
HcmV?d00001

diff --git a/_static/locales/sr/LC_MESSAGES/booktheme.po b/_static/locales/sr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..e809230
--- /dev/null
+++ b/_static/locales/sr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Тхеме би"
+
+msgid "Open an issue"
+msgstr "Отворите издање"
+
+msgid "Contents"
+msgstr "Садржај"
+
+msgid "Download notebook file"
+msgstr "Преузмите датотеку бележнице"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема књиге Спхинк"
+
+msgid "Fullscreen mode"
+msgstr "Режим целог екрана"
+
+msgid "Edit this page"
+msgstr "Уредите ову страницу"
+
+msgid "By"
+msgstr "Од стране"
+
+msgid "Copyright"
+msgstr "Ауторско право"
+
+msgid "Source repository"
+msgstr "Изворно спремиште"
+
+msgid "previous page"
+msgstr "Претходна страница"
+
+msgid "next page"
+msgstr "Следећа страна"
+
+msgid "Toggle navigation"
+msgstr "Укључи / искључи навигацију"
+
+msgid "repository"
+msgstr "спремиште"
+
+msgid "suggest edit"
+msgstr "предложи уређивање"
+
+msgid "open issue"
+msgstr "отворено издање"
+
+msgid "Launch"
+msgstr "Лансирање"
+
+msgid "Print to PDF"
+msgstr "Испис у ПДФ"
+
+msgid "By the"
+msgstr "Од"
+
+msgid "Last updated on"
+msgstr "Последње ажурирање"
+
+msgid "Download source file"
+msgstr "Преузми изворну датотеку"
+
+msgid "Download this page"
+msgstr "Преузмите ову страницу"
diff --git a/_static/locales/sv/LC_MESSAGES/booktheme.mo b/_static/locales/sv/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..b07dc76ff21128244172d2e415cc899555d8b49d
GIT binary patch
literal 1365
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literal 0
HcmV?d00001

diff --git a/_static/locales/sv/LC_MESSAGES/booktheme.po b/_static/locales/sv/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..2421b00
--- /dev/null
+++ b/_static/locales/sv/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: sv\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema av"
+
+msgid "Open an issue"
+msgstr "Öppna en problemrapport"
+
+msgid "Contents"
+msgstr "Innehåll"
+
+msgid "Download notebook file"
+msgstr "Ladda ner notebook-fil"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx Boktema"
+
+msgid "Fullscreen mode"
+msgstr "Fullskärmsläge"
+
+msgid "Edit this page"
+msgstr "Redigera den här sidan"
+
+msgid "By"
+msgstr "Av"
+
+msgid "Copyright"
+msgstr "Upphovsrätt"
+
+msgid "Source repository"
+msgstr "Källkodsrepositorium"
+
+msgid "previous page"
+msgstr "föregående sida"
+
+msgid "next page"
+msgstr "nästa sida"
+
+msgid "Toggle navigation"
+msgstr "Växla navigering"
+
+msgid "repository"
+msgstr "repositorium"
+
+msgid "suggest edit"
+msgstr "föreslå ändring"
+
+msgid "open issue"
+msgstr "öppna problemrapport"
+
+msgid "Launch"
+msgstr "Öppna"
+
+msgid "Print to PDF"
+msgstr "Skriv ut till PDF"
+
+msgid "By the"
+msgstr "Av den"
+
+msgid "Last updated on"
+msgstr "Senast uppdaterad den"
+
+msgid "Download source file"
+msgstr "Ladda ner källfil"
+
+msgid "Download this page"
+msgstr "Ladda ner den här sidan"
diff --git a/_static/locales/ta/LC_MESSAGES/booktheme.mo b/_static/locales/ta/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..29f52e1f6f1ec96aceb0f883d8ccc82c23eb35c2
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/ta/LC_MESSAGES/booktheme.po b/_static/locales/ta/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..500042f
--- /dev/null
+++ b/_static/locales/ta/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ta\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "வழங்கிய தீம்"
+
+msgid "Open an issue"
+msgstr "சிக்கலைத் திறக்கவும்"
+
+msgid "Download notebook file"
+msgstr "நோட்புக் கோப்பைப் பதிவிறக்கவும்"
+
+msgid "Sphinx Book Theme"
+msgstr "ஸ்பிங்க்ஸ் புத்தக தீம்"
+
+msgid "Edit this page"
+msgstr "இந்தப் பக்கத்தைத் திருத்தவும்"
+
+msgid "By"
+msgstr "வழங்கியவர்"
+
+msgid "Copyright"
+msgstr "பதிப்புரிமை"
+
+msgid "Source repository"
+msgstr "மூல களஞ்சியம்"
+
+msgid "previous page"
+msgstr "முந்தைய பக்கம்"
+
+msgid "next page"
+msgstr "அடுத்த பக்கம்"
+
+msgid "Toggle navigation"
+msgstr "வழிசெலுத்தலை நிலைமாற்று"
+
+msgid "suggest edit"
+msgstr "திருத்த பரிந்துரைக்கவும்"
+
+msgid "open issue"
+msgstr "திறந்த பிரச்சினை"
+
+msgid "Launch"
+msgstr "தொடங்க"
+
+msgid "Print to PDF"
+msgstr "PDF இல் அச்சிடுக"
+
+msgid "By the"
+msgstr "மூலம்"
+
+msgid "Last updated on"
+msgstr "கடைசியாக புதுப்பிக்கப்பட்டது"
+
+msgid "Download source file"
+msgstr "மூல கோப்பைப் பதிவிறக்குக"
+
+msgid "Download this page"
+msgstr "இந்தப் பக்கத்தைப் பதிவிறக்கவும்"
diff --git a/_static/locales/te/LC_MESSAGES/booktheme.mo b/_static/locales/te/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..0a5f4b46adfda1551ae6791c3c3782525b61be04
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/te/LC_MESSAGES/booktheme.po b/_static/locales/te/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..b1afebb
--- /dev/null
+++ b/_static/locales/te/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: te\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "ద్వారా థీమ్"
+
+msgid "Open an issue"
+msgstr "సమస్యను తెరవండి"
+
+msgid "Download notebook file"
+msgstr "నోట్బుక్ ఫైల్ను డౌన్లోడ్ చేయండి"
+
+msgid "Sphinx Book Theme"
+msgstr "సింహిక పుస్తక థీమ్"
+
+msgid "Edit this page"
+msgstr "ఈ పేజీని సవరించండి"
+
+msgid "By"
+msgstr "ద్వారా"
+
+msgid "Copyright"
+msgstr "కాపీరైట్"
+
+msgid "Source repository"
+msgstr "మూల రిపోజిటరీ"
+
+msgid "previous page"
+msgstr "ముందు పేజి"
+
+msgid "next page"
+msgstr "తరువాతి పేజీ"
+
+msgid "Toggle navigation"
+msgstr "నావిగేషన్‌ను టోగుల్ చేయండి"
+
+msgid "suggest edit"
+msgstr "సవరించమని సూచించండి"
+
+msgid "open issue"
+msgstr "ఓపెన్ ఇష్యూ"
+
+msgid "Launch"
+msgstr "ప్రారంభించండి"
+
+msgid "Print to PDF"
+msgstr "PDF కి ముద్రించండి"
+
+msgid "By the"
+msgstr "ద్వారా"
+
+msgid "Last updated on"
+msgstr "చివరిగా నవీకరించబడింది"
+
+msgid "Download source file"
+msgstr "మూల ఫైల్‌ను డౌన్‌లోడ్ చేయండి"
+
+msgid "Download this page"
+msgstr "ఈ పేజీని డౌన్‌లోడ్ చేయండి"
diff --git a/_static/locales/tg/LC_MESSAGES/booktheme.mo b/_static/locales/tg/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..b21c6c6340194fdf35f7e5660deb42479c8dbff9
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/tg/LC_MESSAGES/booktheme.po b/_static/locales/tg/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..29b8237
--- /dev/null
+++ b/_static/locales/tg/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tg\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Мавзӯъи аз"
+
+msgid "Open an issue"
+msgstr "Масъаларо кушоед"
+
+msgid "Contents"
+msgstr "Мундариҷа"
+
+msgid "Download notebook file"
+msgstr "Файли дафтарро зеркашӣ кунед"
+
+msgid "Sphinx Book Theme"
+msgstr "Сфинкс Мавзӯи китоб"
+
+msgid "Fullscreen mode"
+msgstr "Ҳолати экрани пурра"
+
+msgid "Edit this page"
+msgstr "Ин саҳифаро таҳрир кунед"
+
+msgid "By"
+msgstr "Бо"
+
+msgid "Copyright"
+msgstr "Ҳуқуқи муаллиф"
+
+msgid "Source repository"
+msgstr "Анбори манбаъ"
+
+msgid "previous page"
+msgstr "саҳифаи қаблӣ"
+
+msgid "next page"
+msgstr "саҳифаи оянда"
+
+msgid "Toggle navigation"
+msgstr "Гузаришро иваз кунед"
+
+msgid "repository"
+msgstr "анбор"
+
+msgid "suggest edit"
+msgstr "пешниҳод вироиш"
+
+msgid "open issue"
+msgstr "барориши кушод"
+
+msgid "Launch"
+msgstr "Оғоз"
+
+msgid "Print to PDF"
+msgstr "Чоп ба PDF"
+
+msgid "By the"
+msgstr "Бо"
+
+msgid "Last updated on"
+msgstr "Last навсозӣ дар"
+
+msgid "Download source file"
+msgstr "Файли манбаъро зеркашӣ кунед"
+
+msgid "Download this page"
+msgstr "Ин саҳифаро зеркашӣ кунед"
diff --git a/_static/locales/th/LC_MESSAGES/booktheme.mo b/_static/locales/th/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..abede98aa11b163e580a26b545980ace31e61ccf
GIT binary patch
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zjT&US*r9DL+Oc6=XyNi4J`IP)4~3RpiSp}N5WkMWZ=`O(hjnrBo?i;ppCMASYj{fI
qV*6MBqwc03(drtRAFXAoSBd=B?a4S*sk|OB^ZqtY{YJQZ;_45RC*i9A

literal 0
HcmV?d00001

diff --git a/_static/locales/th/LC_MESSAGES/booktheme.po b/_static/locales/th/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..ac65ee0
--- /dev/null
+++ b/_static/locales/th/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: th\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "ธีมโดย"
+
+msgid "Open an issue"
+msgstr "เปิดปัญหา"
+
+msgid "Contents"
+msgstr "สารบัญ"
+
+msgid "Download notebook file"
+msgstr "ดาวน์โหลดไฟล์สมุดบันทึก"
+
+msgid "Sphinx Book Theme"
+msgstr "ธีมหนังสือสฟิงซ์"
+
+msgid "Fullscreen mode"
+msgstr "โหมดเต็มหน้าจอ"
+
+msgid "Edit this page"
+msgstr "แก้ไขหน้านี้"
+
+msgid "By"
+msgstr "โดย"
+
+msgid "Copyright"
+msgstr "ลิขสิทธิ์"
+
+msgid "Source repository"
+msgstr "ที่เก็บซอร์ส"
+
+msgid "previous page"
+msgstr "หน้าที่แล้ว"
+
+msgid "next page"
+msgstr "หน้าต่อไป"
+
+msgid "Toggle navigation"
+msgstr "ไม่ต้องสลับช่องทาง"
+
+msgid "repository"
+msgstr "ที่เก็บ"
+
+msgid "suggest edit"
+msgstr "แนะนำแก้ไข"
+
+msgid "open issue"
+msgstr "เปิดปัญหา"
+
+msgid "Launch"
+msgstr "เปิด"
+
+msgid "Print to PDF"
+msgstr "พิมพ์เป็น PDF"
+
+msgid "By the"
+msgstr "โดย"
+
+msgid "Last updated on"
+msgstr "ปรับปรุงล่าสุดเมื่อ"
+
+msgid "Download source file"
+msgstr "ดาวน์โหลดไฟล์ต้นฉบับ"
+
+msgid "Download this page"
+msgstr "ดาวน์โหลดหน้านี้"
diff --git a/_static/locales/tl/LC_MESSAGES/booktheme.mo b/_static/locales/tl/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..8df1b73310e0c606251d1abff7e6033d1f6b273f
GIT binary patch
literal 1273
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z_Gk|th%ew?a0@&DeghAKKfp8KZ;;mh1;@cLOg;`?0Z)LpKw38o9s}=zN5K_v43vGY
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literal 0
HcmV?d00001

diff --git a/_static/locales/tl/LC_MESSAGES/booktheme.po b/_static/locales/tl/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..662d66c
--- /dev/null
+++ b/_static/locales/tl/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tl\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tema ng"
+
+msgid "Open an issue"
+msgstr "Magbukas ng isyu"
+
+msgid "Download notebook file"
+msgstr "Mag-download ng file ng notebook"
+
+msgid "Sphinx Book Theme"
+msgstr "Tema ng Sphinx Book"
+
+msgid "Edit this page"
+msgstr "I-edit ang pahinang ito"
+
+msgid "By"
+msgstr "Ni"
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Source repository"
+msgstr "Pinagmulan ng imbakan"
+
+msgid "previous page"
+msgstr "Nakaraang pahina"
+
+msgid "next page"
+msgstr "Susunod na pahina"
+
+msgid "Toggle navigation"
+msgstr "I-toggle ang pag-navigate"
+
+msgid "suggest edit"
+msgstr "iminumungkahi i-edit"
+
+msgid "open issue"
+msgstr "bukas na isyu"
+
+msgid "Launch"
+msgstr "Ilunsad"
+
+msgid "Print to PDF"
+msgstr "I-print sa PDF"
+
+msgid "By the"
+msgstr "Sa pamamagitan ng"
+
+msgid "Last updated on"
+msgstr "Huling na-update noong"
+
+msgid "Download source file"
+msgstr "Mag-download ng file ng pinagmulan"
+
+msgid "Download this page"
+msgstr "I-download ang pahinang ito"
diff --git a/_static/locales/tr/LC_MESSAGES/booktheme.mo b/_static/locales/tr/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..029ae18afb5360a8c238a052e8d3971761759d3d
GIT binary patch
literal 1373
zcmZ{izi%8x6vqdWKynFz!~~?sQXUOIiX{o62)3@sVLOWA2n!$d<J)(;liizH&CG_Z
zMVT%slPZNJnur1kHxWf7in~bEh>C&=i7J19yL{j6T0W2%?d)gY?96-L@67yl{`g~s
z_CESD`XA_*&>y~r7uw0!mBOP=;q?Z120RHafv3P_@ICND@LjMEo(I1M&w>oT4aSc5
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zo6h$Ap{&BCl*;`;;xXGI&@B^nMMIv5kwyPFiiuV?c-PnoHwMdQd*;PnG+qT~k=xL=
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literal 0
HcmV?d00001

diff --git a/_static/locales/tr/LC_MESSAGES/booktheme.po b/_static/locales/tr/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..d1ae723
--- /dev/null
+++ b/_static/locales/tr/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: tr\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Tarafından tema"
+
+msgid "Open an issue"
+msgstr "Bir sorunu açın"
+
+msgid "Contents"
+msgstr "İçindekiler"
+
+msgid "Download notebook file"
+msgstr "Defter dosyasını indirin"
+
+msgid "Sphinx Book Theme"
+msgstr "Sfenks Kitap Teması"
+
+msgid "Fullscreen mode"
+msgstr "Tam ekran modu"
+
+msgid "Edit this page"
+msgstr "Bu sayfayı düzenle"
+
+msgid "By"
+msgstr "Tarafından"
+
+msgid "Copyright"
+msgstr "Telif hakkı"
+
+msgid "Source repository"
+msgstr "Kaynak kod deposu"
+
+msgid "previous page"
+msgstr "önceki sayfa"
+
+msgid "next page"
+msgstr "sonraki Sayfa"
+
+msgid "Toggle navigation"
+msgstr "Gezinmeyi değiştir"
+
+msgid "repository"
+msgstr "depo"
+
+msgid "suggest edit"
+msgstr "düzenleme öner"
+
+msgid "open issue"
+msgstr "Açık konu"
+
+msgid "Launch"
+msgstr "Başlatmak"
+
+msgid "Print to PDF"
+msgstr "PDF olarak yazdır"
+
+msgid "By the"
+msgstr "Tarafından"
+
+msgid "Last updated on"
+msgstr "Son güncelleme tarihi"
+
+msgid "Download source file"
+msgstr "Kaynak dosyayı indirin"
+
+msgid "Download this page"
+msgstr "Bu sayfayı indirin"
diff --git a/_static/locales/uk/LC_MESSAGES/booktheme.mo b/_static/locales/uk/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..16ab78909cfbaba7fa199f47fdfc2934426ac699
GIT binary patch
literal 1681
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zPJ@uD1fdWPh|e20*TKPc;zZ)a@r}d@A+B8DUJjM|f4dvvq>McK+j;xuef*#QJbLI0
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zZRr;9`hYeh-KP!O!kr#M<2%RQ<4zrzI}#B(?yg&<pWxY{2meKQ;zAaG(Je_Xjtj=E
z9(7$q(09~EfF`ZE2A7mZcK`2+9^rzjaHX3z)7!~B2h%!=*c9|lnghL|yA=s%zR*HT
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z<NofZ={oLHR_+Jb;Vx4(JG2JbhUt5^koP}7+~?BbLU@u{xGrtDCHKjW*gc*BJl2qW
z5mR0g^fg|gC#u^<Yt4P)a{4r6s#Q8N%xG_Ktmt;04^wo@PsZ5OQCrZ@sDuAWd=c9A
LPmM*`H%jps*u#;t

literal 0
HcmV?d00001

diff --git a/_static/locales/uk/LC_MESSAGES/booktheme.po b/_static/locales/uk/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..be49ab8
--- /dev/null
+++ b/_static/locales/uk/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: uk\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Тема від"
+
+msgid "Open an issue"
+msgstr "Відкрийте випуск"
+
+msgid "Contents"
+msgstr "Зміст"
+
+msgid "Download notebook file"
+msgstr "Завантажте файл блокнота"
+
+msgid "Sphinx Book Theme"
+msgstr "Тема книги \"Сфінкс\""
+
+msgid "Fullscreen mode"
+msgstr "Повноекранний режим"
+
+msgid "Edit this page"
+msgstr "Редагувати цю сторінку"
+
+msgid "By"
+msgstr "Автор"
+
+msgid "Copyright"
+msgstr "Авторське право"
+
+msgid "Source repository"
+msgstr "Джерело сховища"
+
+msgid "previous page"
+msgstr "Попередня сторінка"
+
+msgid "next page"
+msgstr "Наступна сторінка"
+
+msgid "Toggle navigation"
+msgstr "Переключити навігацію"
+
+msgid "repository"
+msgstr "сховище"
+
+msgid "suggest edit"
+msgstr "запропонувати редагувати"
+
+msgid "open issue"
+msgstr "відкритий випуск"
+
+msgid "Launch"
+msgstr "Запуск"
+
+msgid "Print to PDF"
+msgstr "Друк у форматі PDF"
+
+msgid "By the"
+msgstr "По"
+
+msgid "Last updated on"
+msgstr "Останнє оновлення:"
+
+msgid "Download source file"
+msgstr "Завантажити вихідний файл"
+
+msgid "Download this page"
+msgstr "Завантажте цю сторінку"
diff --git a/_static/locales/ur/LC_MESSAGES/booktheme.mo b/_static/locales/ur/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..de8c84b935ed5c3db23e304015d45b638a3d2db3
GIT binary patch
literal 1462
zcma)(&2Jk;7{&)0pqP)6LL-m>O`(TSv<;z`$dL~;A5u^%qBgBK^s=!h>rLI6)qdpT
zRwLUoA{Qc+BL{@CNd#$~fM}{TM=rgzQV%n0sfS7&;BP2WpLe~9L#0ZLy!)G(eLtS}
z@uTBCHyGACc;3gei{}G8S6;vi>pu7*_yBwvd<eb@w!uE|FOYJdfXBgJ1iuNM0*`@b
zLCPBe-vGY^UkASidqI(LA-n%JJO2PupEWQCZh@447o_}q;B8Q6=kJd&_8QJVf>d_{
zw7^~PH24qrDR`oXv3~FiknZO|ioXGV2*%(?;Gf{z;NRf8;J+FBUg~@w04eVRNO2|b
zJ#Z2H7+e8i$-yFT-pWp-6LJ_r(sdMXnvv>}ceKb;S{N^_PkZu%7R~=ky6bpCK9k3g
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z*MdBkUV01)pAYlSw$Ly6$pRnzWH`t4n!cr5?Pa~wuCipkz0CC=x@BmuW3xx+mX7tR
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z$~?F1e_~CYdU3r0$=I6uq5ePVO_Bn=P}riFSZ}qf6oE57w5w3K+FoX+hEW?y<4|&p
zx|p)Aw=sGx%}Vw#el49X^(ARiaRizwH+vA!fH?=_BQ=JATO9Uo!rm6_g(Z9Fvrf~J
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c^9?3>eEmE8GP@z_uy-`khW>?G>aA{%rxd0lq5uE@

literal 0
HcmV?d00001

diff --git a/_static/locales/ur/LC_MESSAGES/booktheme.po b/_static/locales/ur/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..94bcab3
--- /dev/null
+++ b/_static/locales/ur/LC_MESSAGES/booktheme.po
@@ -0,0 +1,66 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: ur\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "کے ذریعہ تھیم"
+
+msgid "Open an issue"
+msgstr "ایک مسئلہ کھولیں"
+
+msgid "Download notebook file"
+msgstr "نوٹ بک فائل ڈاؤن لوڈ کریں"
+
+msgid "Sphinx Book Theme"
+msgstr "سپنکس بک تھیم"
+
+msgid "Edit this page"
+msgstr "اس صفحے میں ترمیم کریں"
+
+msgid "By"
+msgstr "بذریعہ"
+
+msgid "Copyright"
+msgstr "کاپی رائٹ"
+
+msgid "Source repository"
+msgstr "ماخذ ذخیرہ"
+
+msgid "previous page"
+msgstr "سابقہ ​​صفحہ"
+
+msgid "next page"
+msgstr "اگلا صفحہ"
+
+msgid "Toggle navigation"
+msgstr "نیویگیشن ٹوگل کریں"
+
+msgid "suggest edit"
+msgstr "ترمیم کی تجویز کریں"
+
+msgid "open issue"
+msgstr "کھلا مسئلہ"
+
+msgid "Launch"
+msgstr "لانچ کریں"
+
+msgid "Print to PDF"
+msgstr "پی ڈی ایف پرنٹ کریں"
+
+msgid "By the"
+msgstr "کی طرف"
+
+msgid "Last updated on"
+msgstr "آخری بار تازہ کاری ہوئی"
+
+msgid "Download source file"
+msgstr "سورس فائل ڈاؤن لوڈ کریں"
+
+msgid "Download this page"
+msgstr "اس صفحے کو ڈاؤن لوڈ کریں"
diff --git a/_static/locales/vi/LC_MESSAGES/booktheme.mo b/_static/locales/vi/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..2bb32555c3ea0dab08ed37b71cb863fbe75698c6
GIT binary patch
literal 1431
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literal 0
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diff --git a/_static/locales/vi/LC_MESSAGES/booktheme.po b/_static/locales/vi/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..116236d
--- /dev/null
+++ b/_static/locales/vi/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: vi\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "Chủ đề của"
+
+msgid "Open an issue"
+msgstr "Mở một vấn đề"
+
+msgid "Contents"
+msgstr "Nội dung"
+
+msgid "Download notebook file"
+msgstr "Tải xuống tệp sổ tay"
+
+msgid "Sphinx Book Theme"
+msgstr "Chủ đề sách nhân sư"
+
+msgid "Fullscreen mode"
+msgstr "Chế độ toàn màn hình"
+
+msgid "Edit this page"
+msgstr "chỉnh sửa trang này"
+
+msgid "By"
+msgstr "Bởi"
+
+msgid "Copyright"
+msgstr "Bản quyền"
+
+msgid "Source repository"
+msgstr "Kho nguồn"
+
+msgid "previous page"
+msgstr "trang trước"
+
+msgid "next page"
+msgstr "Trang tiếp theo"
+
+msgid "Toggle navigation"
+msgstr "Chuyển đổi điều hướng thành"
+
+msgid "repository"
+msgstr "kho"
+
+msgid "suggest edit"
+msgstr "đề nghị chỉnh sửa"
+
+msgid "open issue"
+msgstr "vấn đề mở"
+
+msgid "Launch"
+msgstr "Phóng"
+
+msgid "Print to PDF"
+msgstr "In sang PDF"
+
+msgid "By the"
+msgstr "Bằng"
+
+msgid "Last updated on"
+msgstr "Cập nhật lần cuối vào"
+
+msgid "Download source file"
+msgstr "Tải xuống tệp nguồn"
+
+msgid "Download this page"
+msgstr "Tải xuống trang này"
diff --git a/_static/locales/zh_CN/LC_MESSAGES/booktheme.mo b/_static/locales/zh_CN/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..0e3235d090c2a05c0e278d38520f0437847dc79f
GIT binary patch
literal 1310
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literal 0
HcmV?d00001

diff --git a/_static/locales/zh_CN/LC_MESSAGES/booktheme.po b/_static/locales/zh_CN/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..4f4ab57
--- /dev/null
+++ b/_static/locales/zh_CN/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: zh_CN\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "主题作者："
+
+msgid "Open an issue"
+msgstr "创建议题"
+
+msgid "Contents"
+msgstr "目录"
+
+msgid "Download notebook file"
+msgstr "下载笔记本文件"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx Book 主题"
+
+msgid "Fullscreen mode"
+msgstr "全屏模式"
+
+msgid "Edit this page"
+msgstr "编辑此页面"
+
+msgid "By"
+msgstr "作者："
+
+msgid "Copyright"
+msgstr "版权"
+
+msgid "Source repository"
+msgstr "源码库"
+
+msgid "previous page"
+msgstr "上一页"
+
+msgid "next page"
+msgstr "下一页"
+
+msgid "Toggle navigation"
+msgstr "显示或隐藏导航栏"
+
+msgid "repository"
+msgstr "仓库"
+
+msgid "suggest edit"
+msgstr "提出修改建议"
+
+msgid "open issue"
+msgstr "创建议题"
+
+msgid "Launch"
+msgstr "启动"
+
+msgid "Print to PDF"
+msgstr "列印成 PDF"
+
+msgid "By the"
+msgstr "作者："
+
+msgid "Last updated on"
+msgstr "上次更新时间："
+
+msgid "Download source file"
+msgstr "下载源文件"
+
+msgid "Download this page"
+msgstr "下载此页面"
diff --git a/_static/locales/zh_TW/LC_MESSAGES/booktheme.mo b/_static/locales/zh_TW/LC_MESSAGES/booktheme.mo
new file mode 100644
index 0000000000000000000000000000000000000000..9116fa95d0a752665e0f1d59857a32abfc0c6bce
GIT binary patch
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literal 0
HcmV?d00001

diff --git a/_static/locales/zh_TW/LC_MESSAGES/booktheme.po b/_static/locales/zh_TW/LC_MESSAGES/booktheme.po
new file mode 100644
index 0000000..42b43b8
--- /dev/null
+++ b/_static/locales/zh_TW/LC_MESSAGES/booktheme.po
@@ -0,0 +1,75 @@
+
+msgid ""
+msgstr ""
+"Project-Id-Version: Sphinx-Book-Theme\n"
+"MIME-Version: 1.0\n"
+"Content-Type: text/plain; charset=UTF-8\n"
+"Content-Transfer-Encoding: 8bit\n"
+"Language: zh_TW\n"
+"Plural-Forms: nplurals=2; plural=(n != 1);\n"
+
+msgid "Theme by the"
+msgstr "佈景主題作者："
+
+msgid "Open an issue"
+msgstr "開啟議題"
+
+msgid "Contents"
+msgstr "目錄"
+
+msgid "Download notebook file"
+msgstr "下載 Notebook 檔案"
+
+msgid "Sphinx Book Theme"
+msgstr "Sphinx Book 佈景主題"
+
+msgid "Fullscreen mode"
+msgstr "全螢幕模式"
+
+msgid "Edit this page"
+msgstr "編輯此頁面"
+
+msgid "By"
+msgstr "作者："
+
+msgid "Copyright"
+msgstr "Copyright"
+
+msgid "Source repository"
+msgstr "來源儲存庫"
+
+msgid "previous page"
+msgstr "上一頁"
+
+msgid "next page"
+msgstr "下一頁"
+
+msgid "Toggle navigation"
+msgstr "顯示或隱藏導覽列"
+
+msgid "repository"
+msgstr "儲存庫"
+
+msgid "suggest edit"
+msgstr "提出修改建議"
+
+msgid "open issue"
+msgstr "公開的問題"
+
+msgid "Launch"
+msgstr "啟動"
+
+msgid "Print to PDF"
+msgstr "列印成 PDF"
+
+msgid "By the"
+msgstr "作者："
+
+msgid "Last updated on"
+msgstr "最後更新時間："
+
+msgid "Download source file"
+msgstr "下載原始檔"
+
+msgid "Download this page"
+msgstr "下載此頁面"
diff --git a/_static/logo.png b/_static/logo.png
new file mode 100644
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diff --git a/_static/minus.png b/_static/minus.png
new file mode 100644
index 0000000000000000000000000000000000000000..d96755fdaf8bb2214971e0db9c1fd3077d7c419d
GIT binary patch
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literal 0
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diff --git a/_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css b/_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css
new file mode 100644
index 0000000..3356631
--- /dev/null
+++ b/_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css
@@ -0,0 +1,2342 @@
+/* Variables */
+:root {
+  --mystnb-source-bg-color: #f7f7f7;
+  --mystnb-stdout-bg-color: #fcfcfc;
+  --mystnb-stderr-bg-color: #fdd;
+  --mystnb-traceback-bg-color: #fcfcfc;
+  --mystnb-source-border-color: #ccc;
+  --mystnb-source-margin-color: green;
+  --mystnb-stdout-border-color: #f7f7f7;
+  --mystnb-stderr-border-color: #f7f7f7;
+  --mystnb-traceback-border-color: #ffd6d6;
+  --mystnb-hide-prompt-opacity: 70%;
+  --mystnb-source-border-radius: .4em;
+  --mystnb-source-border-width: 1px;
+}
+
+/* Whole cell */
+div.container.cell {
+  padding-left: 0;
+  margin-bottom: 1em;
+}
+
+/* Removing all background formatting so we can control at the div level */
+.cell_input div.highlight,
+.cell_output pre,
+.cell_input pre,
+.cell_output .output {
+  border: none;
+  box-shadow: none;
+}
+
+.cell_output .output pre,
+.cell_input pre {
+  margin: 0px;
+}
+
+/* Input cells */
+div.cell div.cell_input,
+div.cell details.above-input>summary {
+  padding-left: 0em;
+  padding-right: 0em;
+  border: var(--mystnb-source-border-width) var(--mystnb-source-border-color) solid;
+  background-color: var(--mystnb-source-bg-color);
+  border-left-color: var(--mystnb-source-margin-color);
+  border-left-width: medium;
+  border-radius: var(--mystnb-source-border-radius);
+}
+
+div.cell_input>div,
+div.cell_output div.output>div.highlight {
+  margin: 0em !important;
+  border: none !important;
+}
+
+/* All cell outputs */
+.cell_output {
+  padding-left: 1em;
+  padding-right: 0em;
+  margin-top: 1em;
+}
+
+/* Text outputs from cells */
+.cell_output .output.text_plain,
+.cell_output .output.traceback,
+.cell_output .output.stream,
+.cell_output .output.stderr {
+  margin-top: 1em;
+  margin-bottom: 0em;
+  box-shadow: none;
+}
+
+.cell_output .output.text_plain,
+.cell_output .output.stream {
+  background: var(--mystnb-stdout-bg-color);
+  border: 1px solid var(--mystnb-stdout-border-color);
+}
+
+.cell_output .output.stderr {
+  background: var(--mystnb-stderr-bg-color);
+  border: 1px solid var(--mystnb-stderr-border-color);
+}
+
+.cell_output .output.traceback {
+  background: var(--mystnb-traceback-bg-color);
+  border: 1px solid var(--mystnb-traceback-border-color);
+}
+
+/* Collapsible cell content */
+div.cell details.above-input div.cell_input {
+  border-top-left-radius: 0;
+  border-top-right-radius: 0;
+  border-top: var(--mystnb-source-border-width) var(--mystnb-source-border-color) dashed;
+}
+
+div.cell div.cell_input.above-output-prompt {
+  border-bottom-left-radius: 0;
+  border-bottom-right-radius: 0;
+}
+
+div.cell details.above-input>summary {
+  border-bottom-left-radius: 0;
+  border-bottom-right-radius: 0;
+  border-bottom: var(--mystnb-source-border-width) var(--mystnb-source-border-color) dashed;
+  padding-left: 1em;
+  margin-bottom: 0;
+}
+
+div.cell details.above-output>summary {
+  background-color: var(--mystnb-source-bg-color);
+  padding-left: 1em;
+  padding-right: 0em;
+  border: var(--mystnb-source-border-width) var(--mystnb-source-border-color) solid;
+  border-radius: var(--mystnb-source-border-radius);
+  border-left-color: var(--mystnb-source-margin-color);
+  border-left-width: medium;
+}
+
+div.cell details.below-input>summary {
+  background-color: var(--mystnb-source-bg-color);
+  padding-left: 1em;
+  padding-right: 0em;
+  border: var(--mystnb-source-border-width) var(--mystnb-source-border-color) solid;
+  border-top: none;
+  border-bottom-left-radius: var(--mystnb-source-border-radius);
+  border-bottom-right-radius: var(--mystnb-source-border-radius);
+  border-left-color: var(--mystnb-source-margin-color);
+  border-left-width: medium;
+}
+
+div.cell details.hide>summary>span {
+  opacity: var(--mystnb-hide-prompt-opacity);
+}
+
+div.cell details.hide[open]>summary>span.collapsed {
+  display: none;
+}
+
+div.cell details.hide:not([open])>summary>span.expanded {
+  display: none;
+}
+
+@keyframes collapsed-fade-in {
+  0% {
+    opacity: 0;
+  }
+
+  100% {
+    opacity: 1;
+  }
+}
+div.cell details.hide[open]>summary~* {
+  -moz-animation: collapsed-fade-in 0.3s ease-in-out;
+  -webkit-animation: collapsed-fade-in 0.3s ease-in-out;
+  animation: collapsed-fade-in 0.3s ease-in-out;
+}
+
+/* Math align to the left */
+.cell_output .MathJax_Display {
+  text-align: left !important;
+}
+
+/* Pandas tables. Pulled from the Jupyter / nbsphinx CSS */
+div.cell_output table {
+  border: none;
+  border-collapse: collapse;
+  border-spacing: 0;
+  color: black;
+  font-size: 1em;
+  table-layout: fixed;
+}
+
+div.cell_output thead {
+  border-bottom: 1px solid black;
+  vertical-align: bottom;
+}
+
+div.cell_output tr,
+div.cell_output th,
+div.cell_output td {
+  text-align: right;
+  vertical-align: middle;
+  padding: 0.5em 0.5em;
+  line-height: normal;
+  white-space: normal;
+  max-width: none;
+  border: none;
+}
+
+div.cell_output th {
+  font-weight: bold;
+}
+
+div.cell_output tbody tr:nth-child(odd) {
+  background: #f5f5f5;
+}
+
+div.cell_output tbody tr:hover {
+  background: rgba(66, 165, 245, 0.2);
+}
+
+/** source code line numbers **/
+span.linenos {
+  opacity: 0.5;
+}
+
+/* Inline text from `paste` operation */
+
+span.pasted-text {
+  font-weight: bold;
+}
+
+span.pasted-inline img {
+  max-height: 2em;
+}
+
+tbody span.pasted-inline img {
+  max-height: none;
+}
+
+/* Font colors for translated ANSI escape sequences
+Color values are copied from Jupyter Notebook
+https://github.com/jupyter/notebook/blob/52581f8eda9b319eb0390ac77fe5903c38f81e3e/notebook/static/notebook/less/ansicolors.less#L14-L21
+Background colors from
+https://nbsphinx.readthedocs.io/en/latest/code-cells.html#ANSI-Colors
+*/
+div.highlight .-Color-Bold {
+  font-weight: bold;
+}
+
+div.highlight .-Color[class*=-Black] {
+  color: #3E424D
+}
+
+div.highlight .-Color[class*=-Red] {
+  color: #E75C58
+}
+
+div.highlight .-Color[class*=-Green] {
+  color: #00A250
+}
+
+div.highlight .-Color[class*=-Yellow] {
+  color: #DDB62B
+}
+
+div.highlight .-Color[class*=-Blue] {
+  color: #208FFB
+}
+
+div.highlight .-Color[class*=-Magenta] {
+  color: #D160C4
+}
+
+div.highlight .-Color[class*=-Cyan] {
+  color: #60C6C8
+}
+
+div.highlight .-Color[class*=-White] {
+  color: #C5C1B4
+}
+
+div.highlight .-Color[class*=-BGBlack] {
+  background-color: #3E424D
+}
+
+div.highlight .-Color[class*=-BGRed] {
+  background-color: #E75C58
+}
+
+div.highlight .-Color[class*=-BGGreen] {
+  background-color: #00A250
+}
+
+div.highlight .-Color[class*=-BGYellow] {
+  background-color: #DDB62B
+}
+
+div.highlight .-Color[class*=-BGBlue] {
+  background-color: #208FFB
+}
+
+div.highlight .-Color[class*=-BGMagenta] {
+  background-color: #D160C4
+}
+
+div.highlight .-Color[class*=-BGCyan] {
+  background-color: #60C6C8
+}
+
+div.highlight .-Color[class*=-BGWhite] {
+  background-color: #C5C1B4
+}
+
+/* Font colors for 8-bit ANSI */
+
+div.highlight .-Color[class*=-C0] {
+  color: #000000
+}
+
+div.highlight .-Color[class*=-BGC0] {
+  background-color: #000000
+}
+
+div.highlight .-Color[class*=-C1] {
+  color: #800000
+}
+
+div.highlight .-Color[class*=-BGC1] {
+  background-color: #800000
+}
+
+div.highlight .-Color[class*=-C2] {
+  color: #008000
+}
+
+div.highlight .-Color[class*=-BGC2] {
+  background-color: #008000
+}
+
+div.highlight .-Color[class*=-C3] {
+  color: #808000
+}
+
+div.highlight .-Color[class*=-BGC3] {
+  background-color: #808000
+}
+
+div.highlight .-Color[class*=-C4] {
+  color: #000080
+}
+
+div.highlight .-Color[class*=-BGC4] {
+  background-color: #000080
+}
+
+div.highlight .-Color[class*=-C5] {
+  color: #800080
+}
+
+div.highlight .-Color[class*=-BGC5] {
+  background-color: #800080
+}
+
+div.highlight .-Color[class*=-C6] {
+  color: #008080
+}
+
+div.highlight .-Color[class*=-BGC6] {
+  background-color: #008080
+}
+
+div.highlight .-Color[class*=-C7] {
+  color: #C0C0C0
+}
+
+div.highlight .-Color[class*=-BGC7] {
+  background-color: #C0C0C0
+}
+
+div.highlight .-Color[class*=-C8] {
+  color: #808080
+}
+
+div.highlight .-Color[class*=-BGC8] {
+  background-color: #808080
+}
+
+div.highlight .-Color[class*=-C9] {
+  color: #FF0000
+}
+
+div.highlight .-Color[class*=-BGC9] {
+  background-color: #FF0000
+}
+
+div.highlight .-Color[class*=-C10] {
+  color: #00FF00
+}
+
+div.highlight .-Color[class*=-BGC10] {
+  background-color: #00FF00
+}
+
+div.highlight .-Color[class*=-C11] {
+  color: #FFFF00
+}
+
+div.highlight .-Color[class*=-BGC11] {
+  background-color: #FFFF00
+}
+
+div.highlight .-Color[class*=-C12] {
+  color: #0000FF
+}
+
+div.highlight .-Color[class*=-BGC12] {
+  background-color: #0000FF
+}
+
+div.highlight .-Color[class*=-C13] {
+  color: #FF00FF
+}
+
+div.highlight .-Color[class*=-BGC13] {
+  background-color: #FF00FF
+}
+
+div.highlight .-Color[class*=-C14] {
+  color: #00FFFF
+}
+
+div.highlight .-Color[class*=-BGC14] {
+  background-color: #00FFFF
+}
+
+div.highlight .-Color[class*=-C15] {
+  color: #FFFFFF
+}
+
+div.highlight .-Color[class*=-BGC15] {
+  background-color: #FFFFFF
+}
+
+div.highlight .-Color[class*=-C16] {
+  color: #000000
+}
+
+div.highlight .-Color[class*=-BGC16] {
+  background-color: #000000
+}
+
+div.highlight .-Color[class*=-C17] {
+  color: #00005F
+}
+
+div.highlight .-Color[class*=-BGC17] {
+  background-color: #00005F
+}
+
+div.highlight .-Color[class*=-C18] {
+  color: #000087
+}
+
+div.highlight .-Color[class*=-BGC18] {
+  background-color: #000087
+}
+
+div.highlight .-Color[class*=-C19] {
+  color: #0000AF
+}
+
+div.highlight .-Color[class*=-BGC19] {
+  background-color: #0000AF
+}
+
+div.highlight .-Color[class*=-C20] {
+  color: #0000D7
+}
+
+div.highlight .-Color[class*=-BGC20] {
+  background-color: #0000D7
+}
+
+div.highlight .-Color[class*=-C21] {
+  color: #0000FF
+}
+
+div.highlight .-Color[class*=-BGC21] {
+  background-color: #0000FF
+}
+
+div.highlight .-Color[class*=-C22] {
+  color: #005F00
+}
+
+div.highlight .-Color[class*=-BGC22] {
+  background-color: #005F00
+}
+
+div.highlight .-Color[class*=-C23] {
+  color: #005F5F
+}
+
+div.highlight .-Color[class*=-BGC23] {
+  background-color: #005F5F
+}
+
+div.highlight .-Color[class*=-C24] {
+  color: #005F87
+}
+
+div.highlight .-Color[class*=-BGC24] {
+  background-color: #005F87
+}
+
+div.highlight .-Color[class*=-C25] {
+  color: #005FAF
+}
+
+div.highlight .-Color[class*=-BGC25] {
+  background-color: #005FAF
+}
+
+div.highlight .-Color[class*=-C26] {
+  color: #005FD7
+}
+
+div.highlight .-Color[class*=-BGC26] {
+  background-color: #005FD7
+}
+
+div.highlight .-Color[class*=-C27] {
+  color: #005FFF
+}
+
+div.highlight .-Color[class*=-BGC27] {
+  background-color: #005FFF
+}
+
+div.highlight .-Color[class*=-C28] {
+  color: #008700
+}
+
+div.highlight .-Color[class*=-BGC28] {
+  background-color: #008700
+}
+
+div.highlight .-Color[class*=-C29] {
+  color: #00875F
+}
+
+div.highlight .-Color[class*=-BGC29] {
+  background-color: #00875F
+}
+
+div.highlight .-Color[class*=-C30] {
+  color: #008787
+}
+
+div.highlight .-Color[class*=-BGC30] {
+  background-color: #008787
+}
+
+div.highlight .-Color[class*=-C31] {
+  color: #0087AF
+}
+
+div.highlight .-Color[class*=-BGC31] {
+  background-color: #0087AF
+}
+
+div.highlight .-Color[class*=-C32] {
+  color: #0087D7
+}
+
+div.highlight .-Color[class*=-BGC32] {
+  background-color: #0087D7
+}
+
+div.highlight .-Color[class*=-C33] {
+  color: #0087FF
+}
+
+div.highlight .-Color[class*=-BGC33] {
+  background-color: #0087FF
+}
+
+div.highlight .-Color[class*=-C34] {
+  color: #00AF00
+}
+
+div.highlight .-Color[class*=-BGC34] {
+  background-color: #00AF00
+}
+
+div.highlight .-Color[class*=-C35] {
+  color: #00AF5F
+}
+
+div.highlight .-Color[class*=-BGC35] {
+  background-color: #00AF5F
+}
+
+div.highlight .-Color[class*=-C36] {
+  color: #00AF87
+}
+
+div.highlight .-Color[class*=-BGC36] {
+  background-color: #00AF87
+}
+
+div.highlight .-Color[class*=-C37] {
+  color: #00AFAF
+}
+
+div.highlight .-Color[class*=-BGC37] {
+  background-color: #00AFAF
+}
+
+div.highlight .-Color[class*=-C38] {
+  color: #00AFD7
+}
+
+div.highlight .-Color[class*=-BGC38] {
+  background-color: #00AFD7
+}
+
+div.highlight .-Color[class*=-C39] {
+  color: #00AFFF
+}
+
+div.highlight .-Color[class*=-BGC39] {
+  background-color: #00AFFF
+}
+
+div.highlight .-Color[class*=-C40] {
+  color: #00D700
+}
+
+div.highlight .-Color[class*=-BGC40] {
+  background-color: #00D700
+}
+
+div.highlight .-Color[class*=-C41] {
+  color: #00D75F
+}
+
+div.highlight .-Color[class*=-BGC41] {
+  background-color: #00D75F
+}
+
+div.highlight .-Color[class*=-C42] {
+  color: #00D787
+}
+
+div.highlight .-Color[class*=-BGC42] {
+  background-color: #00D787
+}
+
+div.highlight .-Color[class*=-C43] {
+  color: #00D7AF
+}
+
+div.highlight .-Color[class*=-BGC43] {
+  background-color: #00D7AF
+}
+
+div.highlight .-Color[class*=-C44] {
+  color: #00D7D7
+}
+
+div.highlight .-Color[class*=-BGC44] {
+  background-color: #00D7D7
+}
+
+div.highlight .-Color[class*=-C45] {
+  color: #00D7FF
+}
+
+div.highlight .-Color[class*=-BGC45] {
+  background-color: #00D7FF
+}
+
+div.highlight .-Color[class*=-C46] {
+  color: #00FF00
+}
+
+div.highlight .-Color[class*=-BGC46] {
+  background-color: #00FF00
+}
+
+div.highlight .-Color[class*=-C47] {
+  color: #00FF5F
+}
+
+div.highlight .-Color[class*=-BGC47] {
+  background-color: #00FF5F
+}
+
+div.highlight .-Color[class*=-C48] {
+  color: #00FF87
+}
+
+div.highlight .-Color[class*=-BGC48] {
+  background-color: #00FF87
+}
+
+div.highlight .-Color[class*=-C49] {
+  color: #00FFAF
+}
+
+div.highlight .-Color[class*=-BGC49] {
+  background-color: #00FFAF
+}
+
+div.highlight .-Color[class*=-C50] {
+  color: #00FFD7
+}
+
+div.highlight .-Color[class*=-BGC50] {
+  background-color: #00FFD7
+}
+
+div.highlight .-Color[class*=-C51] {
+  color: #00FFFF
+}
+
+div.highlight .-Color[class*=-BGC51] {
+  background-color: #00FFFF
+}
+
+div.highlight .-Color[class*=-C52] {
+  color: #5F0000
+}
+
+div.highlight .-Color[class*=-BGC52] {
+  background-color: #5F0000
+}
+
+div.highlight .-Color[class*=-C53] {
+  color: #5F005F
+}
+
+div.highlight .-Color[class*=-BGC53] {
+  background-color: #5F005F
+}
+
+div.highlight .-Color[class*=-C54] {
+  color: #5F0087
+}
+
+div.highlight .-Color[class*=-BGC54] {
+  background-color: #5F0087
+}
+
+div.highlight .-Color[class*=-C55] {
+  color: #5F00AF
+}
+
+div.highlight .-Color[class*=-BGC55] {
+  background-color: #5F00AF
+}
+
+div.highlight .-Color[class*=-C56] {
+  color: #5F00D7
+}
+
+div.highlight .-Color[class*=-BGC56] {
+  background-color: #5F00D7
+}
+
+div.highlight .-Color[class*=-C57] {
+  color: #5F00FF
+}
+
+div.highlight .-Color[class*=-BGC57] {
+  background-color: #5F00FF
+}
+
+div.highlight .-Color[class*=-C58] {
+  color: #5F5F00
+}
+
+div.highlight .-Color[class*=-BGC58] {
+  background-color: #5F5F00
+}
+
+div.highlight .-Color[class*=-C59] {
+  color: #5F5F5F
+}
+
+div.highlight .-Color[class*=-BGC59] {
+  background-color: #5F5F5F
+}
+
+div.highlight .-Color[class*=-C60] {
+  color: #5F5F87
+}
+
+div.highlight .-Color[class*=-BGC60] {
+  background-color: #5F5F87
+}
+
+div.highlight .-Color[class*=-C61] {
+  color: #5F5FAF
+}
+
+div.highlight .-Color[class*=-BGC61] {
+  background-color: #5F5FAF
+}
+
+div.highlight .-Color[class*=-C62] {
+  color: #5F5FD7
+}
+
+div.highlight .-Color[class*=-BGC62] {
+  background-color: #5F5FD7
+}
+
+div.highlight .-Color[class*=-C63] {
+  color: #5F5FFF
+}
+
+div.highlight .-Color[class*=-BGC63] {
+  background-color: #5F5FFF
+}
+
+div.highlight .-Color[class*=-C64] {
+  color: #5F8700
+}
+
+div.highlight .-Color[class*=-BGC64] {
+  background-color: #5F8700
+}
+
+div.highlight .-Color[class*=-C65] {
+  color: #5F875F
+}
+
+div.highlight .-Color[class*=-BGC65] {
+  background-color: #5F875F
+}
+
+div.highlight .-Color[class*=-C66] {
+  color: #5F8787
+}
+
+div.highlight .-Color[class*=-BGC66] {
+  background-color: #5F8787
+}
+
+div.highlight .-Color[class*=-C67] {
+  color: #5F87AF
+}
+
+div.highlight .-Color[class*=-BGC67] {
+  background-color: #5F87AF
+}
+
+div.highlight .-Color[class*=-C68] {
+  color: #5F87D7
+}
+
+div.highlight .-Color[class*=-BGC68] {
+  background-color: #5F87D7
+}
+
+div.highlight .-Color[class*=-C69] {
+  color: #5F87FF
+}
+
+div.highlight .-Color[class*=-BGC69] {
+  background-color: #5F87FF
+}
+
+div.highlight .-Color[class*=-C70] {
+  color: #5FAF00
+}
+
+div.highlight .-Color[class*=-BGC70] {
+  background-color: #5FAF00
+}
+
+div.highlight .-Color[class*=-C71] {
+  color: #5FAF5F
+}
+
+div.highlight .-Color[class*=-BGC71] {
+  background-color: #5FAF5F
+}
+
+div.highlight .-Color[class*=-C72] {
+  color: #5FAF87
+}
+
+div.highlight .-Color[class*=-BGC72] {
+  background-color: #5FAF87
+}
+
+div.highlight .-Color[class*=-C73] {
+  color: #5FAFAF
+}
+
+div.highlight .-Color[class*=-BGC73] {
+  background-color: #5FAFAF
+}
+
+div.highlight .-Color[class*=-C74] {
+  color: #5FAFD7
+}
+
+div.highlight .-Color[class*=-BGC74] {
+  background-color: #5FAFD7
+}
+
+div.highlight .-Color[class*=-C75] {
+  color: #5FAFFF
+}
+
+div.highlight .-Color[class*=-BGC75] {
+  background-color: #5FAFFF
+}
+
+div.highlight .-Color[class*=-C76] {
+  color: #5FD700
+}
+
+div.highlight .-Color[class*=-BGC76] {
+  background-color: #5FD700
+}
+
+div.highlight .-Color[class*=-C77] {
+  color: #5FD75F
+}
+
+div.highlight .-Color[class*=-BGC77] {
+  background-color: #5FD75F
+}
+
+div.highlight .-Color[class*=-C78] {
+  color: #5FD787
+}
+
+div.highlight .-Color[class*=-BGC78] {
+  background-color: #5FD787
+}
+
+div.highlight .-Color[class*=-C79] {
+  color: #5FD7AF
+}
+
+div.highlight .-Color[class*=-BGC79] {
+  background-color: #5FD7AF
+}
+
+div.highlight .-Color[class*=-C80] {
+  color: #5FD7D7
+}
+
+div.highlight .-Color[class*=-BGC80] {
+  background-color: #5FD7D7
+}
+
+div.highlight .-Color[class*=-C81] {
+  color: #5FD7FF
+}
+
+div.highlight .-Color[class*=-BGC81] {
+  background-color: #5FD7FF
+}
+
+div.highlight .-Color[class*=-C82] {
+  color: #5FFF00
+}
+
+div.highlight .-Color[class*=-BGC82] {
+  background-color: #5FFF00
+}
+
+div.highlight .-Color[class*=-C83] {
+  color: #5FFF5F
+}
+
+div.highlight .-Color[class*=-BGC83] {
+  background-color: #5FFF5F
+}
+
+div.highlight .-Color[class*=-C84] {
+  color: #5FFF87
+}
+
+div.highlight .-Color[class*=-BGC84] {
+  background-color: #5FFF87
+}
+
+div.highlight .-Color[class*=-C85] {
+  color: #5FFFAF
+}
+
+div.highlight .-Color[class*=-BGC85] {
+  background-color: #5FFFAF
+}
+
+div.highlight .-Color[class*=-C86] {
+  color: #5FFFD7
+}
+
+div.highlight .-Color[class*=-BGC86] {
+  background-color: #5FFFD7
+}
+
+div.highlight .-Color[class*=-C87] {
+  color: #5FFFFF
+}
+
+div.highlight .-Color[class*=-BGC87] {
+  background-color: #5FFFFF
+}
+
+div.highlight .-Color[class*=-C88] {
+  color: #870000
+}
+
+div.highlight .-Color[class*=-BGC88] {
+  background-color: #870000
+}
+
+div.highlight .-Color[class*=-C89] {
+  color: #87005F
+}
+
+div.highlight .-Color[class*=-BGC89] {
+  background-color: #87005F
+}
+
+div.highlight .-Color[class*=-C90] {
+  color: #870087
+}
+
+div.highlight .-Color[class*=-BGC90] {
+  background-color: #870087
+}
+
+div.highlight .-Color[class*=-C91] {
+  color: #8700AF
+}
+
+div.highlight .-Color[class*=-BGC91] {
+  background-color: #8700AF
+}
+
+div.highlight .-Color[class*=-C92] {
+  color: #8700D7
+}
+
+div.highlight .-Color[class*=-BGC92] {
+  background-color: #8700D7
+}
+
+div.highlight .-Color[class*=-C93] {
+  color: #8700FF
+}
+
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
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+}
+
+div.highlight .-Color[class*=-BGC209] {
+  background-color: #FF875F
+}
+
+div.highlight .-Color[class*=-C210] {
+  color: #FF8787
+}
+
+div.highlight .-Color[class*=-BGC210] {
+  background-color: #FF8787
+}
+
+div.highlight .-Color[class*=-C211] {
+  color: #FF87AF
+}
+
+div.highlight .-Color[class*=-BGC211] {
+  background-color: #FF87AF
+}
+
+div.highlight .-Color[class*=-C212] {
+  color: #FF87D7
+}
+
+div.highlight .-Color[class*=-BGC212] {
+  background-color: #FF87D7
+}
+
+div.highlight .-Color[class*=-C213] {
+  color: #FF87FF
+}
+
+div.highlight .-Color[class*=-BGC213] {
+  background-color: #FF87FF
+}
+
+div.highlight .-Color[class*=-C214] {
+  color: #FFAF00
+}
+
+div.highlight .-Color[class*=-BGC214] {
+  background-color: #FFAF00
+}
+
+div.highlight .-Color[class*=-C215] {
+  color: #FFAF5F
+}
+
+div.highlight .-Color[class*=-BGC215] {
+  background-color: #FFAF5F
+}
+
+div.highlight .-Color[class*=-C216] {
+  color: #FFAF87
+}
+
+div.highlight .-Color[class*=-BGC216] {
+  background-color: #FFAF87
+}
+
+div.highlight .-Color[class*=-C217] {
+  color: #FFAFAF
+}
+
+div.highlight .-Color[class*=-BGC217] {
+  background-color: #FFAFAF
+}
+
+div.highlight .-Color[class*=-C218] {
+  color: #FFAFD7
+}
+
+div.highlight .-Color[class*=-BGC218] {
+  background-color: #FFAFD7
+}
+
+div.highlight .-Color[class*=-C219] {
+  color: #FFAFFF
+}
+
+div.highlight .-Color[class*=-BGC219] {
+  background-color: #FFAFFF
+}
+
+div.highlight .-Color[class*=-C220] {
+  color: #FFD700
+}
+
+div.highlight .-Color[class*=-BGC220] {
+  background-color: #FFD700
+}
+
+div.highlight .-Color[class*=-C221] {
+  color: #FFD75F
+}
+
+div.highlight .-Color[class*=-BGC221] {
+  background-color: #FFD75F
+}
+
+div.highlight .-Color[class*=-C222] {
+  color: #FFD787
+}
+
+div.highlight .-Color[class*=-BGC222] {
+  background-color: #FFD787
+}
+
+div.highlight .-Color[class*=-C223] {
+  color: #FFD7AF
+}
+
+div.highlight .-Color[class*=-BGC223] {
+  background-color: #FFD7AF
+}
+
+div.highlight .-Color[class*=-C224] {
+  color: #FFD7D7
+}
+
+div.highlight .-Color[class*=-BGC224] {
+  background-color: #FFD7D7
+}
+
+div.highlight .-Color[class*=-C225] {
+  color: #FFD7FF
+}
+
+div.highlight .-Color[class*=-BGC225] {
+  background-color: #FFD7FF
+}
+
+div.highlight .-Color[class*=-C226] {
+  color: #FFFF00
+}
+
+div.highlight .-Color[class*=-BGC226] {
+  background-color: #FFFF00
+}
+
+div.highlight .-Color[class*=-C227] {
+  color: #FFFF5F
+}
+
+div.highlight .-Color[class*=-BGC227] {
+  background-color: #FFFF5F
+}
+
+div.highlight .-Color[class*=-C228] {
+  color: #FFFF87
+}
+
+div.highlight .-Color[class*=-BGC228] {
+  background-color: #FFFF87
+}
+
+div.highlight .-Color[class*=-C229] {
+  color: #FFFFAF
+}
+
+div.highlight .-Color[class*=-BGC229] {
+  background-color: #FFFFAF
+}
+
+div.highlight .-Color[class*=-C230] {
+  color: #FFFFD7
+}
+
+div.highlight .-Color[class*=-BGC230] {
+  background-color: #FFFFD7
+}
+
+div.highlight .-Color[class*=-C231] {
+  color: #FFFFFF
+}
+
+div.highlight .-Color[class*=-BGC231] {
+  background-color: #FFFFFF
+}
+
+div.highlight .-Color[class*=-C232] {
+  color: #080808
+}
+
+div.highlight .-Color[class*=-BGC232] {
+  background-color: #080808
+}
+
+div.highlight .-Color[class*=-C233] {
+  color: #121212
+}
+
+div.highlight .-Color[class*=-BGC233] {
+  background-color: #121212
+}
+
+div.highlight .-Color[class*=-C234] {
+  color: #1C1C1C
+}
+
+div.highlight .-Color[class*=-BGC234] {
+  background-color: #1C1C1C
+}
+
+div.highlight .-Color[class*=-C235] {
+  color: #262626
+}
+
+div.highlight .-Color[class*=-BGC235] {
+  background-color: #262626
+}
+
+div.highlight .-Color[class*=-C236] {
+  color: #303030
+}
+
+div.highlight .-Color[class*=-BGC236] {
+  background-color: #303030
+}
+
+div.highlight .-Color[class*=-C237] {
+  color: #3A3A3A
+}
+
+div.highlight .-Color[class*=-BGC237] {
+  background-color: #3A3A3A
+}
+
+div.highlight .-Color[class*=-C238] {
+  color: #444444
+}
+
+div.highlight .-Color[class*=-BGC238] {
+  background-color: #444444
+}
+
+div.highlight .-Color[class*=-C239] {
+  color: #4E4E4E
+}
+
+div.highlight .-Color[class*=-BGC239] {
+  background-color: #4E4E4E
+}
+
+div.highlight .-Color[class*=-C240] {
+  color: #585858
+}
+
+div.highlight .-Color[class*=-BGC240] {
+  background-color: #585858
+}
+
+div.highlight .-Color[class*=-C241] {
+  color: #626262
+}
+
+div.highlight .-Color[class*=-BGC241] {
+  background-color: #626262
+}
+
+div.highlight .-Color[class*=-C242] {
+  color: #6C6C6C
+}
+
+div.highlight .-Color[class*=-BGC242] {
+  background-color: #6C6C6C
+}
+
+div.highlight .-Color[class*=-C243] {
+  color: #767676
+}
+
+div.highlight .-Color[class*=-BGC243] {
+  background-color: #767676
+}
+
+div.highlight .-Color[class*=-C244] {
+  color: #808080
+}
+
+div.highlight .-Color[class*=-BGC244] {
+  background-color: #808080
+}
+
+div.highlight .-Color[class*=-C245] {
+  color: #8A8A8A
+}
+
+div.highlight .-Color[class*=-BGC245] {
+  background-color: #8A8A8A
+}
+
+div.highlight .-Color[class*=-C246] {
+  color: #949494
+}
+
+div.highlight .-Color[class*=-BGC246] {
+  background-color: #949494
+}
+
+div.highlight .-Color[class*=-C247] {
+  color: #9E9E9E
+}
+
+div.highlight .-Color[class*=-BGC247] {
+  background-color: #9E9E9E
+}
+
+div.highlight .-Color[class*=-C248] {
+  color: #A8A8A8
+}
+
+div.highlight .-Color[class*=-BGC248] {
+  background-color: #A8A8A8
+}
+
+div.highlight .-Color[class*=-C249] {
+  color: #B2B2B2
+}
+
+div.highlight .-Color[class*=-BGC249] {
+  background-color: #B2B2B2
+}
+
+div.highlight .-Color[class*=-C250] {
+  color: #BCBCBC
+}
+
+div.highlight .-Color[class*=-BGC250] {
+  background-color: #BCBCBC
+}
+
+div.highlight .-Color[class*=-C251] {
+  color: #C6C6C6
+}
+
+div.highlight .-Color[class*=-BGC251] {
+  background-color: #C6C6C6
+}
+
+div.highlight .-Color[class*=-C252] {
+  color: #D0D0D0
+}
+
+div.highlight .-Color[class*=-BGC252] {
+  background-color: #D0D0D0
+}
+
+div.highlight .-Color[class*=-C253] {
+  color: #DADADA
+}
+
+div.highlight .-Color[class*=-BGC253] {
+  background-color: #DADADA
+}
+
+div.highlight .-Color[class*=-C254] {
+  color: #E4E4E4
+}
+
+div.highlight .-Color[class*=-BGC254] {
+  background-color: #E4E4E4
+}
+
+div.highlight .-Color[class*=-C255] {
+  color: #EEEEEE
+}
+
+div.highlight .-Color[class*=-BGC255] {
+  background-color: #EEEEEE
+}
diff --git a/_static/play-solid.svg b/_static/play-solid.svg
new file mode 100644
index 0000000..bcd81f7
--- /dev/null
+++ b/_static/play-solid.svg
@@ -0,0 +1 @@
+<svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="play" class="svg-inline--fa fa-play fa-w-14" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><path fill="currentColor" d="M424.4 214.7L72.4 6.6C43.8-10.3 0 6.1 0 47.9V464c0 37.5 40.7 60.1 72.4 41.3l352-208c31.4-18.5 31.5-64.1 0-82.6z"></path></svg>
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diff --git a/_static/plus.png b/_static/plus.png
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GIT binary patch
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diff --git a/_static/proof.css b/_static/proof.css
new file mode 100644
index 0000000..51e0c9f
--- /dev/null
+++ b/_static/proof.css
@@ -0,0 +1,196 @@
+/*********************************************
+* Variables *
+*********************************************/
+:root {
+  --note-title-color: rgba(68,138,255,.1);
+  --note-border-color: #007bff;
+  --warning-title-color: rgba(220,53,69,.1);
+  --warning-border-color: #dc3545;
+  --hint-title-color: rgba(255,193,7,.2);
+  --hint-border-color: #ffc107;
+  --caution-title-color: rgba(253,126,20,.1);
+  --caution-border-color: #fd7e14;
+  --grey-title-color: rgba(204,204,204,.2);
+  --grey-border-color: #ccc;
+}
+
+/*********************************************
+* Main body *
+*********************************************/
+
+/* Remove content box */
+div.proof p.admonition-title::before {
+	content: none;
+}
+
+/*********************************************
+* Proof *
+*********************************************/
+div#proof{
+	padding: .4rem .6rem .4rem 2rem !important;
+	border-color: var(--grey-border-color);
+	background-color: none;
+}
+
+/*********************************************
+* Theorem *
+*********************************************/
+div.theorem {
+	border-color: var(--note-border-color);
+	background-color: var(--note-title-color);
+}
+
+div.theorem p.admonition-title {
+	background-color: var(--note-title-color);
+}
+
+/*********************************************
+* Axiom *
+*********************************************/
+div.axiom {
+	border-color: var(--hint-border-color);
+	background-color: var(--hint-title-color);
+}
+
+div.axiom p.admonition-title {
+	background-color: var(--hint-title-color);
+}
+
+/*********************************************
+* Criterion *
+*********************************************/
+div.criterion {
+	border-color: var(--caution-border-color);
+	background-color: var(--caution-title-color);
+}
+
+div.criterion p.admonition-title {
+	background-color: var(--caution-title-color);
+}
+
+/*********************************************
+* Lemma *
+*********************************************/
+div.lemma {
+	border-color: var(--hint-border-color);
+	background-color: var(--hint-title-color);
+}
+
+div.lemma p.admonition-title {
+	background-color: var(--hint-title-color);
+}
+
+/*********************************************
+* Definition *
+*********************************************/
+div.definition {
+	border-color: var(--note-border-color);
+	background-color: var(--note-title-color);
+}
+
+div.definition p.admonition-title {
+	background-color: var(--note-title-color);
+}
+
+/*********************************************
+* Remark *
+*********************************************/
+div.remark {
+	border-color: var(--warning-border-color);
+	background-color: var(--warning-title-color);
+}
+
+div.remark p.admonition-title {
+	background-color: var(--warning-title-color);
+}
+
+/*********************************************
+* Conjecture *
+*********************************************/
+div.conjecture {
+	border-color: var(--hint-border-color);
+	background-color: var(--hint-title-color);
+}
+
+div.conjecture p.admonition-title {
+	background-color: var(--hint-title-color);
+}
+
+/*********************************************
+* Corollary *
+*********************************************/
+div.corollary {
+	border-color: var(--caution-border-color);
+	background-color: var(--caution-title-color);
+}
+
+div.corollary p.admonition-title {
+	background-color: var(--caution-title-color);
+}
+
+/*********************************************
+* Algorithm *
+*********************************************/
+div.algorithm {
+	border: none;
+	background-color: none;
+}
+
+div.algorithm p.admonition-title {
+	background-color: transparent;
+	border-top: .15rem solid var(--grey-border-color);
+	border-bottom: .15rem solid var(--grey-border-color);
+}
+
+div.algorithm div.section {
+	font-family: SFMono-Regular,Menlo,Monaco,Consolas,Liberation Mono,Courier New,monospace;
+	font-size: .85rem;
+}
+
+/*********************************************
+* Example *
+*********************************************/
+div.example {
+	border-color: var(--hint-border-color);
+	background-color: none;
+}
+
+div.example p.admonition-title {
+	background-color: transparent;
+}
+
+/*********************************************
+* Property *
+*********************************************/
+div.property {
+	border-color: var(--caution-border-color);
+	background-color: var(--caution-title-color);
+}
+
+div.property p.admonition-title {
+	background-color: var(--caution-title-color);
+}
+
+/*********************************************
+* Observation *
+*********************************************/
+div.observation {
+	border-color: var(--hint-border-color);
+	background-color: var(--hint-title-color);
+}
+
+div.observation p.admonition-title {
+	background-color: var(--hint-title-color);
+}
+
+/*********************************************
+* Proposition *
+*********************************************/
+div.proposition {
+	border-color: var(--note-border-color);
+	background-color: var(--note-title-color);
+}
+
+div.proposition p.admonition-title {
+	background-color: var(--note-title-color);
+}
diff --git a/_static/pygments.css b/_static/pygments.css
new file mode 100644
index 0000000..997797f
--- /dev/null
+++ b/_static/pygments.css
@@ -0,0 +1,152 @@
+html[data-theme="light"] .highlight pre { line-height: 125%; }
+html[data-theme="light"] .highlight td.linenos .normal { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+html[data-theme="light"] .highlight span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+html[data-theme="light"] .highlight td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+html[data-theme="light"] .highlight span.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+html[data-theme="light"] .highlight .hll { background-color: #7971292e }
+html[data-theme="light"] .highlight { background: #fefefe; color: #545454 }
+html[data-theme="light"] .highlight .c { color: #797129 } /* Comment */
+html[data-theme="light"] .highlight .err { color: #d91e18 } /* Error */
+html[data-theme="light"] .highlight .k { color: #7928a1 } /* Keyword */
+html[data-theme="light"] .highlight .l { color: #797129 } /* Literal */
+html[data-theme="light"] .highlight .n { color: #545454 } /* Name */
+html[data-theme="light"] .highlight .o { color: #008000 } /* Operator */
+html[data-theme="light"] .highlight .p { color: #545454 } /* Punctuation */
+html[data-theme="light"] .highlight .ch { color: #797129 } /* Comment.Hashbang */
+html[data-theme="light"] .highlight .cm { color: #797129 } /* Comment.Multiline */
+html[data-theme="light"] .highlight .cp { color: #797129 } /* Comment.Preproc */
+html[data-theme="light"] .highlight .cpf { color: #797129 } /* Comment.PreprocFile */
+html[data-theme="light"] .highlight .c1 { color: #797129 } /* Comment.Single */
+html[data-theme="light"] .highlight .cs { color: #797129 } /* Comment.Special */
+html[data-theme="light"] .highlight .gd { color: #007faa } /* Generic.Deleted */
+html[data-theme="light"] .highlight .ge { font-style: italic } /* Generic.Emph */
+html[data-theme="light"] .highlight .gh { color: #007faa } /* Generic.Heading */
+html[data-theme="light"] .highlight .gs { font-weight: bold } /* Generic.Strong */
+html[data-theme="light"] .highlight .gu { color: #007faa } /* Generic.Subheading */
+html[data-theme="light"] .highlight .kc { color: #7928a1 } /* Keyword.Constant */
+html[data-theme="light"] .highlight .kd { color: #7928a1 } /* Keyword.Declaration */
+html[data-theme="light"] .highlight .kn { color: #7928a1 } /* Keyword.Namespace */
+html[data-theme="light"] .highlight .kp { color: #7928a1 } /* Keyword.Pseudo */
+html[data-theme="light"] .highlight .kr { color: #7928a1 } /* Keyword.Reserved */
+html[data-theme="light"] .highlight .kt { color: #797129 } /* Keyword.Type */
+html[data-theme="light"] .highlight .ld { color: #797129 } /* Literal.Date */
+html[data-theme="light"] .highlight .m { color: #797129 } /* Literal.Number */
+html[data-theme="light"] .highlight .s { color: #008000 } /* Literal.String */
+html[data-theme="light"] .highlight .na { color: #797129 } /* Name.Attribute */
+html[data-theme="light"] .highlight .nb { color: #797129 } /* Name.Builtin */
+html[data-theme="light"] .highlight .nc { color: #007faa } /* Name.Class */
+html[data-theme="light"] .highlight .no { color: #007faa } /* Name.Constant */
+html[data-theme="light"] .highlight .nd { color: #797129 } /* Name.Decorator */
+html[data-theme="light"] .highlight .ni { color: #008000 } /* Name.Entity */
+html[data-theme="light"] .highlight .ne { color: #7928a1 } /* Name.Exception */
+html[data-theme="light"] .highlight .nf { color: #007faa } /* Name.Function */
+html[data-theme="light"] .highlight .nl { color: #797129 } /* Name.Label */
+html[data-theme="light"] .highlight .nn { color: #545454 } /* Name.Namespace */
+html[data-theme="light"] .highlight .nx { color: #545454 } /* Name.Other */
+html[data-theme="light"] .highlight .py { color: #007faa } /* Name.Property */
+html[data-theme="light"] .highlight .nt { color: #007faa } /* Name.Tag */
+html[data-theme="light"] .highlight .nv { color: #d91e18 } /* Name.Variable */
+html[data-theme="light"] .highlight .ow { color: #7928a1 } /* Operator.Word */
+html[data-theme="light"] .highlight .pm { color: #545454 } /* Punctuation.Marker */
+html[data-theme="light"] .highlight .w { color: #545454 } /* Text.Whitespace */
+html[data-theme="light"] .highlight .mb { color: #797129 } /* Literal.Number.Bin */
+html[data-theme="light"] .highlight .mf { color: #797129 } /* Literal.Number.Float */
+html[data-theme="light"] .highlight .mh { color: #797129 } /* Literal.Number.Hex */
+html[data-theme="light"] .highlight .mi { color: #797129 } /* Literal.Number.Integer */
+html[data-theme="light"] .highlight .mo { color: #797129 } /* Literal.Number.Oct */
+html[data-theme="light"] .highlight .sa { color: #008000 } /* Literal.String.Affix */
+html[data-theme="light"] .highlight .sb { color: #008000 } /* Literal.String.Backtick */
+html[data-theme="light"] .highlight .sc { color: #008000 } /* Literal.String.Char */
+html[data-theme="light"] .highlight .dl { color: #008000 } /* Literal.String.Delimiter */
+html[data-theme="light"] .highlight .sd { color: #008000 } /* Literal.String.Doc */
+html[data-theme="light"] .highlight .s2 { color: #008000 } /* Literal.String.Double */
+html[data-theme="light"] .highlight .se { color: #008000 } /* Literal.String.Escape */
+html[data-theme="light"] .highlight .sh { color: #008000 } /* Literal.String.Heredoc */
+html[data-theme="light"] .highlight .si { color: #008000 } /* Literal.String.Interpol */
+html[data-theme="light"] .highlight .sx { color: #008000 } /* Literal.String.Other */
+html[data-theme="light"] .highlight .sr { color: #d91e18 } /* Literal.String.Regex */
+html[data-theme="light"] .highlight .s1 { color: #008000 } /* Literal.String.Single */
+html[data-theme="light"] .highlight .ss { color: #007faa } /* Literal.String.Symbol */
+html[data-theme="light"] .highlight .bp { color: #797129 } /* Name.Builtin.Pseudo */
+html[data-theme="light"] .highlight .fm { color: #007faa } /* Name.Function.Magic */
+html[data-theme="light"] .highlight .vc { color: #d91e18 } /* Name.Variable.Class */
+html[data-theme="light"] .highlight .vg { color: #d91e18 } /* Name.Variable.Global */
+html[data-theme="light"] .highlight .vi { color: #d91e18 } /* Name.Variable.Instance */
+html[data-theme="light"] .highlight .vm { color: #797129 } /* Name.Variable.Magic */
+html[data-theme="light"] .highlight .il { color: #797129 } /* Literal.Number.Integer.Long */
+html[data-theme="dark"] .highlight pre { line-height: 125%; }
+html[data-theme="dark"] .highlight td.linenos .normal { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+html[data-theme="dark"] .highlight span.linenos { color: inherit; background-color: transparent; padding-left: 5px; padding-right: 5px; }
+html[data-theme="dark"] .highlight td.linenos .special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+html[data-theme="dark"] .highlight span.linenos.special { color: #000000; background-color: #ffffc0; padding-left: 5px; padding-right: 5px; }
+html[data-theme="dark"] .highlight .hll { background-color: #ffd9002e }
+html[data-theme="dark"] .highlight { background: #2b2b2b; color: #f8f8f2 }
+html[data-theme="dark"] .highlight .c { color: #ffd900 } /* Comment */
+html[data-theme="dark"] .highlight .err { color: #ffa07a } /* Error */
+html[data-theme="dark"] .highlight .k { color: #dcc6e0 } /* Keyword */
+html[data-theme="dark"] .highlight .l { color: #ffd900 } /* Literal */
+html[data-theme="dark"] .highlight .n { color: #f8f8f2 } /* Name */
+html[data-theme="dark"] .highlight .o { color: #abe338 } /* Operator */
+html[data-theme="dark"] .highlight .p { color: #f8f8f2 } /* Punctuation */
+html[data-theme="dark"] .highlight .ch { color: #ffd900 } /* Comment.Hashbang */
+html[data-theme="dark"] .highlight .cm { color: #ffd900 } /* Comment.Multiline */
+html[data-theme="dark"] .highlight .cp { color: #ffd900 } /* Comment.Preproc */
+html[data-theme="dark"] .highlight .cpf { color: #ffd900 } /* Comment.PreprocFile */
+html[data-theme="dark"] .highlight .c1 { color: #ffd900 } /* Comment.Single */
+html[data-theme="dark"] .highlight .cs { color: #ffd900 } /* Comment.Special */
+html[data-theme="dark"] .highlight .gd { color: #00e0e0 } /* Generic.Deleted */
+html[data-theme="dark"] .highlight .ge { font-style: italic } /* Generic.Emph */
+html[data-theme="dark"] .highlight .gh { color: #00e0e0 } /* Generic.Heading */
+html[data-theme="dark"] .highlight .gs { font-weight: bold } /* Generic.Strong */
+html[data-theme="dark"] .highlight .gu { color: #00e0e0 } /* Generic.Subheading */
+html[data-theme="dark"] .highlight .kc { color: #dcc6e0 } /* Keyword.Constant */
+html[data-theme="dark"] .highlight .kd { color: #dcc6e0 } /* Keyword.Declaration */
+html[data-theme="dark"] .highlight .kn { color: #dcc6e0 } /* Keyword.Namespace */
+html[data-theme="dark"] .highlight .kp { color: #dcc6e0 } /* Keyword.Pseudo */
+html[data-theme="dark"] .highlight .kr { color: #dcc6e0 } /* Keyword.Reserved */
+html[data-theme="dark"] .highlight .kt { color: #ffd900 } /* Keyword.Type */
+html[data-theme="dark"] .highlight .ld { color: #ffd900 } /* Literal.Date */
+html[data-theme="dark"] .highlight .m { color: #ffd900 } /* Literal.Number */
+html[data-theme="dark"] .highlight .s { color: #abe338 } /* Literal.String */
+html[data-theme="dark"] .highlight .na { color: #ffd900 } /* Name.Attribute */
+html[data-theme="dark"] .highlight .nb { color: #ffd900 } /* Name.Builtin */
+html[data-theme="dark"] .highlight .nc { color: #00e0e0 } /* Name.Class */
+html[data-theme="dark"] .highlight .no { color: #00e0e0 } /* Name.Constant */
+html[data-theme="dark"] .highlight .nd { color: #ffd900 } /* Name.Decorator */
+html[data-theme="dark"] .highlight .ni { color: #abe338 } /* Name.Entity */
+html[data-theme="dark"] .highlight .ne { color: #dcc6e0 } /* Name.Exception */
+html[data-theme="dark"] .highlight .nf { color: #00e0e0 } /* Name.Function */
+html[data-theme="dark"] .highlight .nl { color: #ffd900 } /* Name.Label */
+html[data-theme="dark"] .highlight .nn { color: #f8f8f2 } /* Name.Namespace */
+html[data-theme="dark"] .highlight .nx { color: #f8f8f2 } /* Name.Other */
+html[data-theme="dark"] .highlight .py { color: #00e0e0 } /* Name.Property */
+html[data-theme="dark"] .highlight .nt { color: #00e0e0 } /* Name.Tag */
+html[data-theme="dark"] .highlight .nv { color: #ffa07a } /* Name.Variable */
+html[data-theme="dark"] .highlight .ow { color: #dcc6e0 } /* Operator.Word */
+html[data-theme="dark"] .highlight .pm { color: #f8f8f2 } /* Punctuation.Marker */
+html[data-theme="dark"] .highlight .w { color: #f8f8f2 } /* Text.Whitespace */
+html[data-theme="dark"] .highlight .mb { color: #ffd900 } /* Literal.Number.Bin */
+html[data-theme="dark"] .highlight .mf { color: #ffd900 } /* Literal.Number.Float */
+html[data-theme="dark"] .highlight .mh { color: #ffd900 } /* Literal.Number.Hex */
+html[data-theme="dark"] .highlight .mi { color: #ffd900 } /* Literal.Number.Integer */
+html[data-theme="dark"] .highlight .mo { color: #ffd900 } /* Literal.Number.Oct */
+html[data-theme="dark"] .highlight .sa { color: #abe338 } /* Literal.String.Affix */
+html[data-theme="dark"] .highlight .sb { color: #abe338 } /* Literal.String.Backtick */
+html[data-theme="dark"] .highlight .sc { color: #abe338 } /* Literal.String.Char */
+html[data-theme="dark"] .highlight .dl { color: #abe338 } /* Literal.String.Delimiter */
+html[data-theme="dark"] .highlight .sd { color: #abe338 } /* Literal.String.Doc */
+html[data-theme="dark"] .highlight .s2 { color: #abe338 } /* Literal.String.Double */
+html[data-theme="dark"] .highlight .se { color: #abe338 } /* Literal.String.Escape */
+html[data-theme="dark"] .highlight .sh { color: #abe338 } /* Literal.String.Heredoc */
+html[data-theme="dark"] .highlight .si { color: #abe338 } /* Literal.String.Interpol */
+html[data-theme="dark"] .highlight .sx { color: #abe338 } /* Literal.String.Other */
+html[data-theme="dark"] .highlight .sr { color: #ffa07a } /* Literal.String.Regex */
+html[data-theme="dark"] .highlight .s1 { color: #abe338 } /* Literal.String.Single */
+html[data-theme="dark"] .highlight .ss { color: #00e0e0 } /* Literal.String.Symbol */
+html[data-theme="dark"] .highlight .bp { color: #ffd900 } /* Name.Builtin.Pseudo */
+html[data-theme="dark"] .highlight .fm { color: #00e0e0 } /* Name.Function.Magic */
+html[data-theme="dark"] .highlight .vc { color: #ffa07a } /* Name.Variable.Class */
+html[data-theme="dark"] .highlight .vg { color: #ffa07a } /* Name.Variable.Global */
+html[data-theme="dark"] .highlight .vi { color: #ffa07a } /* Name.Variable.Instance */
+html[data-theme="dark"] .highlight .vm { color: #ffd900 } /* Name.Variable.Magic */
+html[data-theme="dark"] .highlight .il { color: #ffd900 } /* Literal.Number.Integer.Long */
\ No newline at end of file
diff --git a/_static/sbt-webpack-macros.html b/_static/sbt-webpack-macros.html
new file mode 100644
index 0000000..6cbf559
--- /dev/null
+++ b/_static/sbt-webpack-macros.html
@@ -0,0 +1,11 @@
+<!--
+  All these macros are auto-generated and must **NOT** be edited by hand.
+  See the webpack.config.js file, to learn more about how this is generated.
+-->
+{% macro head_pre_bootstrap() %}
+  <link href="{{ pathto('_static/styles/sphinx-book-theme.css', 1) }}" rel="stylesheet">
+{% endmacro %}
+
+{% macro body_post() %}
+  <script src="{{ pathto('_static/scripts/sphinx-book-theme.js', 1) }}"></script>
+{% endmacro %}
diff --git a/_static/scripts/bootstrap.js b/_static/scripts/bootstrap.js
new file mode 100644
index 0000000..4e209b0
--- /dev/null
+++ b/_static/scripts/bootstrap.js
@@ -0,0 +1,3 @@
+/*! For license information please see bootstrap.js.LICENSE.txt */
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t=this._mergeConfigObj(t),t=this._configAfterMerge(t),this._typeCheckConfig(t),t}_configAfterMerge(t){return t}_mergeConfigObj(t,e){const i=Ft(e)?_e.getDataAttribute(e,"config"):{};return{...this.constructor.Default,..."object"==typeof i?i:{},...Ft(e)?_e.getDataAttributes(e):{},..."object"==typeof t?t:{}}}_typeCheckConfig(t,e=this.constructor.DefaultType){for(const[n,s]of Object.entries(e)){const e=t[n],o=Ft(e)?"element":null==(i=e)?`${i}`:Object.prototype.toString.call(i).match(/\s([a-z]+)/i)[1].toLowerCase();if(!new RegExp(s).test(o))throw new TypeError(`${this.constructor.NAME.toUpperCase()}: Option "${n}" provided type "${o}" but expected type "${s}".`)}var i}}class ve extends be{constructor(t,e){super(),(t=Ht(t))&&(this._element=t,this._config=this._getConfig(e),Nt.set(this._element,this.constructor.DATA_KEY,this))}dispose(){Nt.remove(this._element,this.constructor.DATA_KEY),fe.off(this._element,this.constructor.EVENT_KEY);for(const t of 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e?we.find(e):[]}},Ee=(t,e="hide")=>{const i=`click.dismiss${t.EVENT_KEY}`,n=t.NAME;fe.on(document,i,`[data-bs-dismiss="${n}"]`,(function(i){if(["A","AREA"].includes(this.tagName)&&i.preventDefault(),Wt(this))return;const s=we.getElementFromSelector(this)||this.closest(`.${n}`);t.getOrCreateInstance(s)[e]()}))},Ae=".bs.alert",Te=`close${Ae}`,Ce=`closed${Ae}`;class Oe extends ve{static get NAME(){return"alert"}close(){if(fe.trigger(this._element,Te).defaultPrevented)return;this._element.classList.remove("show");const t=this._element.classList.contains("fade");this._queueCallback((()=>this._destroyElement()),this._element,t)}_destroyElement(){this._element.remove(),fe.trigger(this._element,Ce),this.dispose()}static jQueryInterface(t){return this.each((function(){const e=Oe.getOrCreateInstance(this);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}Ee(Oe,"close"),Qt(Oe);const xe='[data-bs-toggle="button"]';class ke extends ve{static get NAME(){return"button"}toggle(){this._element.setAttribute("aria-pressed",this._element.classList.toggle("active"))}static jQueryInterface(t){return this.each((function(){const e=ke.getOrCreateInstance(this);"toggle"===t&&e[t]()}))}}fe.on(document,"click.bs.button.data-api",xe,(t=>{t.preventDefault();const e=t.target.closest(xe);ke.getOrCreateInstance(e).toggle()})),Qt(ke);const Le=".bs.swipe",Se=`touchstart${Le}`,De=`touchmove${Le}`,$e=`touchend${Le}`,Ie=`pointerdown${Le}`,Ne=`pointerup${Le}`,Pe={endCallback:null,leftCallback:null,rightCallback:null},Me={endCallback:"(function|null)",leftCallback:"(function|null)",rightCallback:"(function|null)"};class je extends be{constructor(t,e){super(),this._element=t,t&&je.isSupported()&&(this._config=this._getConfig(e),this._deltaX=0,this._supportPointerEvents=Boolean(window.PointerEvent),this._initEvents())}static get Default(){return Pe}static get DefaultType(){return Me}static get NAME(){return"swipe"}dispose(){fe.off(this._element,Le)}_start(t){this._supportPointerEvents?this._eventIsPointerPenTouch(t)&&(this._deltaX=t.clientX):this._deltaX=t.touches[0].clientX}_end(t){this._eventIsPointerPenTouch(t)&&(this._deltaX=t.clientX-this._deltaX),this._handleSwipe(),Xt(this._config.endCallback)}_move(t){this._deltaX=t.touches&&t.touches.length>1?0:t.touches[0].clientX-this._deltaX}_handleSwipe(){const t=Math.abs(this._deltaX);if(t<=40)return;const e=t/this._deltaX;this._deltaX=0,e&&Xt(e>0?this._config.rightCallback:this._config.leftCallback)}_initEvents(){this._supportPointerEvents?(fe.on(this._element,Ie,(t=>this._start(t))),fe.on(this._element,Ne,(t=>this._end(t))),this._element.classList.add("pointer-event")):(fe.on(this._element,Se,(t=>this._start(t))),fe.on(this._element,De,(t=>this._move(t))),fe.on(this._element,$e,(t=>this._end(t))))}_eventIsPointerPenTouch(t){return this._supportPointerEvents&&("pen"===t.pointerType||"touch"===t.pointerType)}static isSupported(){return"ontouchstart"in document.documentElement||navigator.maxTouchPoints>0}}const Fe=".bs.carousel",He=".data-api",Be="next",We="prev",ze="left",Re="right",qe=`slide${Fe}`,Ve=`slid${Fe}`,Ye=`keydown${Fe}`,Ke=`mouseenter${Fe}`,Qe=`mouseleave${Fe}`,Xe=`dragstart${Fe}`,Ue=`load${Fe}${He}`,Ge=`click${Fe}${He}`,Je="carousel",Ze="active",ti=".active",ei=".carousel-item",ii=ti+ei,ni={ArrowLeft:Re,ArrowRight:ze},si={interval:5e3,keyboard:!0,pause:"hover",ride:!1,touch:!0,wrap:!0},oi={interval:"(number|boolean)",keyboard:"boolean",pause:"(string|boolean)",ride:"(boolean|string)",touch:"boolean",wrap:"boolean"};class ri extends ve{constructor(t,e){super(t,e),this._interval=null,this._activeElement=null,this._isSliding=!1,this.touchTimeout=null,this._swipeHelper=null,this._indicatorsElement=we.findOne(".carousel-indicators",this._element),this._addEventListeners(),this._config.ride===Je&&this.cycle()}static get Default(){return si}static get DefaultType(){return oi}static get 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t.defaultInterval=t.interval,t}_addEventListeners(){this._config.keyboard&&fe.on(this._element,Ye,(t=>this._keydown(t))),"hover"===this._config.pause&&(fe.on(this._element,Ke,(()=>this.pause())),fe.on(this._element,Qe,(()=>this._maybeEnableCycle()))),this._config.touch&&je.isSupported()&&this._addTouchEventListeners()}_addTouchEventListeners(){for(const t of we.find(".carousel-item img",this._element))fe.on(t,Xe,(t=>t.preventDefault()));const t={leftCallback:()=>this._slide(this._directionToOrder(ze)),rightCallback:()=>this._slide(this._directionToOrder(Re)),endCallback:()=>{"hover"===this._config.pause&&(this.pause(),this.touchTimeout&&clearTimeout(this.touchTimeout),this.touchTimeout=setTimeout((()=>this._maybeEnableCycle()),500+this._config.interval))}};this._swipeHelper=new je(this._element,t)}_keydown(t){if(/input|textarea/i.test(t.target.tagName))return;const e=ni[t.key];e&&(t.preventDefault(),this._slide(this._directionToOrder(e)))}_getItemIndex(t){return 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l=n?"carousel-item-start":"carousel-item-end",c=n?"carousel-item-next":"carousel-item-prev";s.classList.add(c),qt(s),i.classList.add(l),s.classList.add(l),this._queueCallback((()=>{s.classList.remove(l,c),s.classList.add(Ze),i.classList.remove(Ze,c,l),this._isSliding=!1,r(Ve)}),i,this._isAnimated()),a&&this.cycle()}_isAnimated(){return this._element.classList.contains("slide")}_getActive(){return we.findOne(ii,this._element)}_getItems(){return we.find(ei,this._element)}_clearInterval(){this._interval&&(clearInterval(this._interval),this._interval=null)}_directionToOrder(t){return Kt()?t===ze?We:Be:t===ze?Be:We}_orderToDirection(t){return Kt()?t===We?ze:Re:t===We?Re:ze}static jQueryInterface(t){return this.each((function(){const e=ri.getOrCreateInstance(this,t);if("number"!=typeof t){if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t]()}}else e.to(t)}))}}fe.on(document,Ge,"[data-bs-slide], 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e=we.getSelectorFromElement(t),i=we.find(e).filter((t=>t===this._element));null!==e&&i.length&&this._triggerArray.push(t)}this._initializeChildren(),this._config.parent||this._addAriaAndCollapsedClass(this._triggerArray,this._isShown()),this._config.toggle&&this.toggle()}static get Default(){return bi}static get DefaultType(){return vi}static get NAME(){return"collapse"}toggle(){this._isShown()?this.hide():this.show()}show(){if(this._isTransitioning||this._isShown())return;let t=[];if(this._config.parent&&(t=this._getFirstLevelChildren(".collapse.show, .collapse.collapsing").filter((t=>t!==this._element)).map((t=>yi.getOrCreateInstance(t,{toggle:!1})))),t.length&&t[0]._isTransitioning)return;if(fe.trigger(this._element,li).defaultPrevented)return;for(const e of t)e.hide();const e=this._getDimension();this._element.classList.remove(pi),this._element.classList.add(mi),this._element.style[e]=0,this._addAriaAndCollapsedClass(this._triggerArray,!0),this._isTransitioning=!0;const i=`scroll${e[0].toUpperCase()+e.slice(1)}`;this._queueCallback((()=>{this._isTransitioning=!1,this._element.classList.remove(mi),this._element.classList.add(pi,fi),this._element.style[e]="",fe.trigger(this._element,ci)}),this._element,!0),this._element.style[e]=`${this._element[i]}px`}hide(){if(this._isTransitioning||!this._isShown())return;if(fe.trigger(this._element,hi).defaultPrevented)return;const t=this._getDimension();this._element.style[t]=`${this._element.getBoundingClientRect()[t]}px`,qt(this._element),this._element.classList.add(mi),this._element.classList.remove(pi,fi);for(const t of this._triggerArray){const e=we.getElementFromSelector(t);e&&!this._isShown(e)&&this._addAriaAndCollapsedClass([t],!1)}this._isTransitioning=!0,this._element.style[t]="",this._queueCallback((()=>{this._isTransitioning=!1,this._element.classList.remove(mi),this._element.classList.add(pi),fe.trigger(this._element,di)}),this._element,!0)}_isShown(t=this._element){return t.classList.contains(fi)}_configAfterMerge(t){return t.toggle=Boolean(t.toggle),t.parent=Ht(t.parent),t}_getDimension(){return this._element.classList.contains("collapse-horizontal")?"width":"height"}_initializeChildren(){if(!this._config.parent)return;const t=this._getFirstLevelChildren(_i);for(const e of t){const t=we.getElementFromSelector(e);t&&this._addAriaAndCollapsedClass([e],this._isShown(t))}}_getFirstLevelChildren(t){const e=we.find(gi,this._config.parent);return we.find(t,this._config.parent).filter((t=>!e.includes(t)))}_addAriaAndCollapsedClass(t,e){if(t.length)for(const i of t)i.classList.toggle("collapsed",!e),i.setAttribute("aria-expanded",e)}static jQueryInterface(t){const e={};return"string"==typeof t&&/show|hide/.test(t)&&(e.toggle=!1),this.each((function(){const i=yi.getOrCreateInstance(this,e);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t]()}}))}}fe.on(document,ui,_i,(function(t){("A"===t.target.tagName||t.delegateTarget&&"A"===t.delegateTarget.tagName)&&t.preventDefault();for(const t of we.getMultipleElementsFromSelector(this))yi.getOrCreateInstance(t,{toggle:!1}).toggle()})),Qt(yi);const wi="dropdown",Ei=".bs.dropdown",Ai=".data-api",Ti="ArrowUp",Ci="ArrowDown",Oi=`hide${Ei}`,xi=`hidden${Ei}`,ki=`show${Ei}`,Li=`shown${Ei}`,Si=`click${Ei}${Ai}`,Di=`keydown${Ei}${Ai}`,$i=`keyup${Ei}${Ai}`,Ii="show",Ni='[data-bs-toggle="dropdown"]:not(.disabled):not(:disabled)',Pi=`${Ni}.${Ii}`,Mi=".dropdown-menu",ji=Kt()?"top-end":"top-start",Fi=Kt()?"top-start":"top-end",Hi=Kt()?"bottom-end":"bottom-start",Bi=Kt()?"bottom-start":"bottom-end",Wi=Kt()?"left-start":"right-start",zi=Kt()?"right-start":"left-start",Ri={autoClose:!0,boundary:"clippingParents",display:"dynamic",offset:[0,2],popperConfig:null,reference:"toggle"},qi={autoClose:"(boolean|string)",boundary:"(string|element)",display:"string",offset:"(array|string|function)",popperConfig:"(null|object|function)",reference:"(string|element|object)"};class Vi extends ve{constructor(t,e){super(t,e),this._popper=null,this._parent=this._element.parentNode,this._menu=we.next(this._element,Mi)[0]||we.prev(this._element,Mi)[0]||we.findOne(Mi,this._parent),this._inNavbar=this._detectNavbar()}static get Default(){return Ri}static get DefaultType(){return qi}static get NAME(){return wi}toggle(){return this._isShown()?this.hide():this.show()}show(){if(Wt(this._element)||this._isShown())return;const t={relatedTarget:this._element};if(!fe.trigger(this._element,ki,t).defaultPrevented){if(this._createPopper(),"ontouchstart"in document.documentElement&&!this._parent.closest(".navbar-nav"))for(const t of[].concat(...document.body.children))fe.on(t,"mouseover",Rt);this._element.focus(),this._element.setAttribute("aria-expanded",!0),this._menu.classList.add(Ii),this._element.classList.add(Ii),fe.trigger(this._element,Li,t)}}hide(){if(Wt(this._element)||!this._isShown())return;const t={relatedTarget:this._element};this._completeHide(t)}dispose(){this._popper&&this._popper.destroy(),super.dispose()}update(){this._inNavbar=this._detectNavbar(),this._popper&&this._popper.update()}_completeHide(t){if(!fe.trigger(this._element,Oi,t).defaultPrevented){if("ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))fe.off(t,"mouseover",Rt);this._popper&&this._popper.destroy(),this._menu.classList.remove(Ii),this._element.classList.remove(Ii),this._element.setAttribute("aria-expanded","false"),_e.removeDataAttribute(this._menu,"popper"),fe.trigger(this._element,xi,t)}}_getConfig(t){if("object"==typeof(t=super._getConfig(t)).reference&&!Ft(t.reference)&&"function"!=typeof t.reference.getBoundingClientRect)throw new TypeError(`${wi.toUpperCase()}: Option "reference" provided type "object" without a required "getBoundingClientRect" method.`);return t}_createPopper(){if(void 0===e)throw new TypeError("Bootstrap's dropdowns require Popper 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t={placement:this._getPlacement(),modifiers:[{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"offset",options:{offset:this._getOffset()}}]};return(this._inNavbar||"static"===this._config.display)&&(_e.setDataAttribute(this._menu,"popper","static"),t.modifiers=[{name:"applyStyles",enabled:!1}]),{...t,...Xt(this._config.popperConfig,[t])}}_selectMenuItem({key:t,target:e}){const i=we.find(".dropdown-menu .dropdown-item:not(.disabled):not(:disabled)",this._menu).filter((t=>Bt(t)));i.length&&Gt(i,e,t===Ci,!i.includes(e)).focus()}static jQueryInterface(t){return this.each((function(){const e=Vi.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}static clearMenus(t){if(2===t.button||"keyup"===t.type&&"Tab"!==t.key)return;const e=we.find(Pi);for(const i of e){const e=Vi.getInstance(i);if(!e||!1===e._config.autoClose)continue;const n=t.composedPath(),s=n.includes(e._menu);if(n.includes(e._element)||"inside"===e._config.autoClose&&!s||"outside"===e._config.autoClose&&s)continue;if(e._menu.contains(t.target)&&("keyup"===t.type&&"Tab"===t.key||/input|select|option|textarea|form/i.test(t.target.tagName)))continue;const o={relatedTarget:e._element};"click"===t.type&&(o.clickEvent=t),e._completeHide(o)}}static dataApiKeydownHandler(t){const e=/input|textarea/i.test(t.target.tagName),i="Escape"===t.key,n=[Ti,Ci].includes(t.key);if(!n&&!i)return;if(e&&!i)return;t.preventDefault();const s=this.matches(Ni)?this:we.prev(this,Ni)[0]||we.next(this,Ni)[0]||we.findOne(Ni,t.delegateTarget.parentNode),o=Vi.getOrCreateInstance(s);if(n)return t.stopPropagation(),o.show(),void o._selectMenuItem(t);o._isShown()&&(t.stopPropagation(),o.hide(),s.focus())}}fe.on(document,Di,Ni,Vi.dataApiKeydownHandler),fe.on(document,Di,Mi,Vi.dataApiKeydownHandler),fe.on(document,Si,Vi.clearMenus),fe.on(document,$i,Vi.clearMenus),fe.on(document,Si,Ni,(function(t){t.preventDefault(),Vi.getOrCreateInstance(this).toggle()})),Qt(Vi);const Yi="backdrop",Ki="show",Qi=`mousedown.bs.${Yi}`,Xi={className:"modal-backdrop",clickCallback:null,isAnimated:!1,isVisible:!0,rootElement:"body"},Ui={className:"string",clickCallback:"(function|null)",isAnimated:"boolean",isVisible:"boolean",rootElement:"(element|string)"};class Gi extends be{constructor(t){super(),this._config=this._getConfig(t),this._isAppended=!1,this._element=null}static get Default(){return Xi}static get DefaultType(){return Ui}static get NAME(){return Yi}show(t){if(!this._config.isVisible)return void Xt(t);this._append();const e=this._getElement();this._config.isAnimated&&qt(e),e.classList.add(Ki),this._emulateAnimation((()=>{Xt(t)}))}hide(t){this._config.isVisible?(this._getElement().classList.remove(Ki),this._emulateAnimation((()=>{this.dispose(),Xt(t)}))):Xt(t)}dispose(){this._isAppended&&(fe.off(this._element,Qi),this._element.remove(),this._isAppended=!1)}_getElement(){if(!this._element){const t=document.createElement("div");t.className=this._config.className,this._config.isAnimated&&t.classList.add("fade"),this._element=t}return this._element}_configAfterMerge(t){return t.rootElement=Ht(t.rootElement),t}_append(){if(this._isAppended)return;const t=this._getElement();this._config.rootElement.append(t),fe.on(t,Qi,(()=>{Xt(this._config.clickCallback)})),this._isAppended=!0}_emulateAnimation(t){Ut(t,this._getElement(),this._config.isAnimated)}}const Ji=".bs.focustrap",Zi=`focusin${Ji}`,tn=`keydown.tab${Ji}`,en="backward",nn={autofocus:!0,trapElement:null},sn={autofocus:"boolean",trapElement:"element"};class on extends be{constructor(t){super(),this._config=this._getConfig(t),this._isActive=!1,this._lastTabNavDirection=null}static get Default(){return nn}static get DefaultType(){return sn}static get NAME(){return"focustrap"}activate(){this._isActive||(this._config.autofocus&&this._config.trapElement.focus(),fe.off(document,Ji),fe.on(document,Zi,(t=>this._handleFocusin(t))),fe.on(document,tn,(t=>this._handleKeydown(t))),this._isActive=!0)}deactivate(){this._isActive&&(this._isActive=!1,fe.off(document,Ji))}_handleFocusin(t){const{trapElement:e}=this._config;if(t.target===document||t.target===e||e.contains(t.target))return;const i=we.focusableChildren(e);0===i.length?e.focus():this._lastTabNavDirection===en?i[i.length-1].focus():i[0].focus()}_handleKeydown(t){"Tab"===t.key&&(this._lastTabNavDirection=t.shiftKey?en:"forward")}}const rn=".fixed-top, .fixed-bottom, .is-fixed, .sticky-top",an=".sticky-top",ln="padding-right",cn="margin-right";class hn{constructor(){this._element=document.body}getWidth(){const t=document.documentElement.clientWidth;return Math.abs(window.innerWidth-t)}hide(){const t=this.getWidth();this._disableOverFlow(),this._setElementAttributes(this._element,ln,(e=>e+t)),this._setElementAttributes(rn,ln,(e=>e+t)),this._setElementAttributes(an,cn,(e=>e-t))}reset(){this._resetElementAttributes(this._element,"overflow"),this._resetElementAttributes(this._element,ln),this._resetElementAttributes(rn,ln),this._resetElementAttributes(an,cn)}isOverflowing(){return this.getWidth()>0}_disableOverFlow(){this._saveInitialAttribute(this._element,"overflow"),this._element.style.overflow="hidden"}_setElementAttributes(t,e,i){const n=this.getWidth();this._applyManipulationCallback(t,(t=>{if(t!==this._element&&window.innerWidth>t.clientWidth+n)return;this._saveInitialAttribute(t,e);const s=window.getComputedStyle(t).getPropertyValue(e);t.style.setProperty(e,`${i(Number.parseFloat(s))}px`)}))}_saveInitialAttribute(t,e){const i=t.style.getPropertyValue(e);i&&_e.setDataAttribute(t,e,i)}_resetElementAttributes(t,e){this._applyManipulationCallback(t,(t=>{const i=_e.getDataAttribute(t,e);null!==i?(_e.removeDataAttribute(t,e),t.style.setProperty(e,i)):t.style.removeProperty(e)}))}_applyManipulationCallback(t,e){if(Ft(t))e(t);else for(const i of we.find(t,this._element))e(i)}}const dn=".bs.modal",un=`hide${dn}`,fn=`hidePrevented${dn}`,pn=`hidden${dn}`,mn=`show${dn}`,gn=`shown${dn}`,_n=`resize${dn}`,bn=`click.dismiss${dn}`,vn=`mousedown.dismiss${dn}`,yn=`keydown.dismiss${dn}`,wn=`click${dn}.data-api`,En="modal-open",An="show",Tn="modal-static",Cn={backdrop:!0,focus:!0,keyboard:!0},On={backdrop:"(boolean|string)",focus:"boolean",keyboard:"boolean"};class xn extends ve{constructor(t,e){super(t,e),this._dialog=we.findOne(".modal-dialog",this._element),this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._isShown=!1,this._isTransitioning=!1,this._scrollBar=new hn,this._addEventListeners()}static get Default(){return Cn}static get DefaultType(){return On}static get NAME(){return"modal"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||this._isTransitioning||fe.trigger(this._element,mn,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._isTransitioning=!0,this._scrollBar.hide(),document.body.classList.add(En),this._adjustDialog(),this._backdrop.show((()=>this._showElement(t))))}hide(){this._isShown&&!this._isTransitioning&&(fe.trigger(this._element,un).defaultPrevented||(this._isShown=!1,this._isTransitioning=!0,this._focustrap.deactivate(),this._element.classList.remove(An),this._queueCallback((()=>this._hideModal()),this._element,this._isAnimated())))}dispose(){fe.off(window,dn),fe.off(this._dialog,dn),this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}handleUpdate(){this._adjustDialog()}_initializeBackDrop(){return new Gi({isVisible:Boolean(this._config.backdrop),isAnimated:this._isAnimated()})}_initializeFocusTrap(){return new on({trapElement:this._element})}_showElement(t){document.body.contains(this._element)||document.body.append(this._element),this._element.style.display="block",this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.scrollTop=0;const e=we.findOne(".modal-body",this._dialog);e&&(e.scrollTop=0),qt(this._element),this._element.classList.add(An),this._queueCallback((()=>{this._config.focus&&this._focustrap.activate(),this._isTransitioning=!1,fe.trigger(this._element,gn,{relatedTarget:t})}),this._dialog,this._isAnimated())}_addEventListeners(){fe.on(this._element,yn,(t=>{"Escape"===t.key&&(this._config.keyboard?this.hide():this._triggerBackdropTransition())})),fe.on(window,_n,(()=>{this._isShown&&!this._isTransitioning&&this._adjustDialog()})),fe.on(this._element,vn,(t=>{fe.one(this._element,bn,(e=>{this._element===t.target&&this._element===e.target&&("static"!==this._config.backdrop?this._config.backdrop&&this.hide():this._triggerBackdropTransition())}))}))}_hideModal(){this._element.style.display="none",this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._isTransitioning=!1,this._backdrop.hide((()=>{document.body.classList.remove(En),this._resetAdjustments(),this._scrollBar.reset(),fe.trigger(this._element,pn)}))}_isAnimated(){return this._element.classList.contains("fade")}_triggerBackdropTransition(){if(fe.trigger(this._element,fn).defaultPrevented)return;const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._element.style.overflowY;"hidden"===e||this._element.classList.contains(Tn)||(t||(this._element.style.overflowY="hidden"),this._element.classList.add(Tn),this._queueCallback((()=>{this._element.classList.remove(Tn),this._queueCallback((()=>{this._element.style.overflowY=e}),this._dialog)}),this._dialog),this._element.focus())}_adjustDialog(){const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._scrollBar.getWidth(),i=e>0;if(i&&!t){const t=Kt()?"paddingLeft":"paddingRight";this._element.style[t]=`${e}px`}if(!i&&t){const t=Kt()?"paddingRight":"paddingLeft";this._element.style[t]=`${e}px`}}_resetAdjustments(){this._element.style.paddingLeft="",this._element.style.paddingRight=""}static jQueryInterface(t,e){return this.each((function(){const i=xn.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t](e)}}))}}fe.on(document,wn,'[data-bs-toggle="modal"]',(function(t){const e=we.getElementFromSelector(this);["A","AREA"].includes(this.tagName)&&t.preventDefault(),fe.one(e,mn,(t=>{t.defaultPrevented||fe.one(e,pn,(()=>{Bt(this)&&this.focus()}))}));const i=we.findOne(".modal.show");i&&xn.getInstance(i).hide(),xn.getOrCreateInstance(e).toggle(this)})),Ee(xn),Qt(xn);const kn=".bs.offcanvas",Ln=".data-api",Sn=`load${kn}${Ln}`,Dn="show",$n="showing",In="hiding",Nn=".offcanvas.show",Pn=`show${kn}`,Mn=`shown${kn}`,jn=`hide${kn}`,Fn=`hidePrevented${kn}`,Hn=`hidden${kn}`,Bn=`resize${kn}`,Wn=`click${kn}${Ln}`,zn=`keydown.dismiss${kn}`,Rn={backdrop:!0,keyboard:!0,scroll:!1},qn={backdrop:"(boolean|string)",keyboard:"boolean",scroll:"boolean"};class Vn extends ve{constructor(t,e){super(t,e),this._isShown=!1,this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._addEventListeners()}static get Default(){return Rn}static get DefaultType(){return qn}static get NAME(){return"offcanvas"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||fe.trigger(this._element,Pn,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._backdrop.show(),this._config.scroll||(new hn).hide(),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.classList.add($n),this._queueCallback((()=>{this._config.scroll&&!this._config.backdrop||this._focustrap.activate(),this._element.classList.add(Dn),this._element.classList.remove($n),fe.trigger(this._element,Mn,{relatedTarget:t})}),this._element,!0))}hide(){this._isShown&&(fe.trigger(this._element,jn).defaultPrevented||(this._focustrap.deactivate(),this._element.blur(),this._isShown=!1,this._element.classList.add(In),this._backdrop.hide(),this._queueCallback((()=>{this._element.classList.remove(Dn,In),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._config.scroll||(new hn).reset(),fe.trigger(this._element,Hn)}),this._element,!0)))}dispose(){this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}_initializeBackDrop(){const t=Boolean(this._config.backdrop);return new Gi({className:"offcanvas-backdrop",isVisible:t,isAnimated:!0,rootElement:this._element.parentNode,clickCallback:t?()=>{"static"!==this._config.backdrop?this.hide():fe.trigger(this._element,Fn)}:null})}_initializeFocusTrap(){return new on({trapElement:this._element})}_addEventListeners(){fe.on(this._element,zn,(t=>{"Escape"===t.key&&(this._config.keyboard?this.hide():fe.trigger(this._element,Fn))}))}static jQueryInterface(t){return this.each((function(){const e=Vn.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}fe.on(document,Wn,'[data-bs-toggle="offcanvas"]',(function(t){const e=we.getElementFromSelector(this);if(["A","AREA"].includes(this.tagName)&&t.preventDefault(),Wt(this))return;fe.one(e,Hn,(()=>{Bt(this)&&this.focus()}));const i=we.findOne(Nn);i&&i!==e&&Vn.getInstance(i).hide(),Vn.getOrCreateInstance(e).toggle(this)})),fe.on(window,Sn,(()=>{for(const t of we.find(Nn))Vn.getOrCreateInstance(t).show()})),fe.on(window,Bn,(()=>{for(const t of we.find("[aria-modal][class*=show][class*=offcanvas-]"))"fixed"!==getComputedStyle(t).position&&Vn.getOrCreateInstance(t).hide()})),Ee(Vn),Qt(Vn);const Yn={"*":["class","dir","id","lang","role",/^aria-[\w-]*$/i],a:["target","href","title","rel"],area:[],b:[],br:[],col:[],code:[],div:[],em:[],hr:[],h1:[],h2:[],h3:[],h4:[],h5:[],h6:[],i:[],img:["src","srcset","alt","title","width","height"],li:[],ol:[],p:[],pre:[],s:[],small:[],span:[],sub:[],sup:[],strong:[],u:[],ul:[]},Kn=new Set(["background","cite","href","itemtype","longdesc","poster","src","xlink:href"]),Qn=/^(?!javascript:)(?:[a-z0-9+.-]+:|[^&:/?#]*(?:[/?#]|$))/i,Xn=(t,e)=>{const i=t.nodeName.toLowerCase();return e.includes(i)?!Kn.has(i)||Boolean(Qn.test(t.nodeValue)):e.filter((t=>t instanceof RegExp)).some((t=>t.test(i)))},Un={allowList:Yn,content:{},extraClass:"",html:!1,sanitize:!0,sanitizeFn:null,template:"<div></div>"},Gn={allowList:"object",content:"object",extraClass:"(string|function)",html:"boolean",sanitize:"boolean",sanitizeFn:"(null|function)",template:"string"},Jn={entry:"(string|element|function|null)",selector:"(string|element)"};class Zn extends be{constructor(t){super(),this._config=this._getConfig(t)}static get Default(){return Un}static get DefaultType(){return Gn}static get NAME(){return"TemplateFactory"}getContent(){return Object.values(this._config.content).map((t=>this._resolvePossibleFunction(t))).filter(Boolean)}hasContent(){return this.getContent().length>0}changeContent(t){return this._checkContent(t),this._config.content={...this._config.content,...t},this}toHtml(){const t=document.createElement("div");t.innerHTML=this._maybeSanitize(this._config.template);for(const[e,i]of Object.entries(this._config.content))this._setContent(t,i,e);const e=t.children[0],i=this._resolvePossibleFunction(this._config.extraClass);return i&&e.classList.add(...i.split(" ")),e}_typeCheckConfig(t){super._typeCheckConfig(t),this._checkContent(t.content)}_checkContent(t){for(const[e,i]of Object.entries(t))super._typeCheckConfig({selector:e,entry:i},Jn)}_setContent(t,e,i){const n=we.findOne(i,t);n&&((e=this._resolvePossibleFunction(e))?Ft(e)?this._putElementInTemplate(Ht(e),n):this._config.html?n.innerHTML=this._maybeSanitize(e):n.textContent=e:n.remove())}_maybeSanitize(t){return this._config.sanitize?function(t,e,i){if(!t.length)return t;if(i&&"function"==typeof i)return i(t);const n=(new window.DOMParser).parseFromString(t,"text/html"),s=[].concat(...n.body.querySelectorAll("*"));for(const t of s){const i=t.nodeName.toLowerCase();if(!Object.keys(e).includes(i)){t.remove();continue}const n=[].concat(...t.attributes),s=[].concat(e["*"]||[],e[i]||[]);for(const e of n)Xn(e,s)||t.removeAttribute(e.nodeName)}return n.body.innerHTML}(t,this._config.allowList,this._config.sanitizeFn):t}_resolvePossibleFunction(t){return Xt(t,[this])}_putElementInTemplate(t,e){if(this._config.html)return e.innerHTML="",void e.append(t);e.textContent=t.textContent}}const ts=new Set(["sanitize","allowList","sanitizeFn"]),es="fade",is="show",ns=".modal",ss="hide.bs.modal",os="hover",rs="focus",as={AUTO:"auto",TOP:"top",RIGHT:Kt()?"left":"right",BOTTOM:"bottom",LEFT:Kt()?"right":"left"},ls={allowList:Yn,animation:!0,boundary:"clippingParents",container:!1,customClass:"",delay:0,fallbackPlacements:["top","right","bottom","left"],html:!1,offset:[0,6],placement:"top",popperConfig:null,sanitize:!0,sanitizeFn:null,selector:!1,template:'<div class="tooltip" role="tooltip"><div class="tooltip-arrow"></div><div class="tooltip-inner"></div></div>',title:"",trigger:"hover focus"},cs={allowList:"object",animation:"boolean",boundary:"(string|element)",container:"(string|element|boolean)",customClass:"(string|function)",delay:"(number|object)",fallbackPlacements:"array",html:"boolean",offset:"(array|string|function)",placement:"(string|function)",popperConfig:"(null|object|function)",sanitize:"boolean",sanitizeFn:"(null|function)",selector:"(string|boolean)",template:"string",title:"(string|element|function)",trigger:"string"};class hs extends ve{constructor(t,i){if(void 0===e)throw new TypeError("Bootstrap's tooltips require Popper (https://popper.js.org)");super(t,i),this._isEnabled=!0,this._timeout=0,this._isHovered=null,this._activeTrigger={},this._popper=null,this._templateFactory=null,this._newContent=null,this.tip=null,this._setListeners(),this._config.selector||this._fixTitle()}static get Default(){return ls}static get DefaultType(){return cs}static get NAME(){return"tooltip"}enable(){this._isEnabled=!0}disable(){this._isEnabled=!1}toggleEnabled(){this._isEnabled=!this._isEnabled}toggle(){this._isEnabled&&(this._activeTrigger.click=!this._activeTrigger.click,this._isShown()?this._leave():this._enter())}dispose(){clearTimeout(this._timeout),fe.off(this._element.closest(ns),ss,this._hideModalHandler),this._element.getAttribute("data-bs-original-title")&&this._element.setAttribute("title",this._element.getAttribute("data-bs-original-title")),this._disposePopper(),super.dispose()}show(){if("none"===this._element.style.display)throw new Error("Please use show on visible elements");if(!this._isWithContent()||!this._isEnabled)return;const t=fe.trigger(this._element,this.constructor.eventName("show")),e=(zt(this._element)||this._element.ownerDocument.documentElement).contains(this._element);if(t.defaultPrevented||!e)return;this._disposePopper();const i=this._getTipElement();this._element.setAttribute("aria-describedby",i.getAttribute("id"));const{container:n}=this._config;if(this._element.ownerDocument.documentElement.contains(this.tip)||(n.append(i),fe.trigger(this._element,this.constructor.eventName("inserted"))),this._popper=this._createPopper(i),i.classList.add(is),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))fe.on(t,"mouseover",Rt);this._queueCallback((()=>{fe.trigger(this._element,this.constructor.eventName("shown")),!1===this._isHovered&&this._leave(),this._isHovered=!1}),this.tip,this._isAnimated())}hide(){if(this._isShown()&&!fe.trigger(this._element,this.constructor.eventName("hide")).defaultPrevented){if(this._getTipElement().classList.remove(is),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))fe.off(t,"mouseover",Rt);this._activeTrigger.click=!1,this._activeTrigger[rs]=!1,this._activeTrigger[os]=!1,this._isHovered=null,this._queueCallback((()=>{this._isWithActiveTrigger()||(this._isHovered||this._disposePopper(),this._element.removeAttribute("aria-describedby"),fe.trigger(this._element,this.constructor.eventName("hidden")))}),this.tip,this._isAnimated())}}update(){this._popper&&this._popper.update()}_isWithContent(){return Boolean(this._getTitle())}_getTipElement(){return this.tip||(this.tip=this._createTipElement(this._newContent||this._getContentForTemplate())),this.tip}_createTipElement(t){const e=this._getTemplateFactory(t).toHtml();if(!e)return null;e.classList.remove(es,is),e.classList.add(`bs-${this.constructor.NAME}-auto`);const i=(t=>{do{t+=Math.floor(1e6*Math.random())}while(document.getElementById(t));return t})(this.constructor.NAME).toString();return e.setAttribute("id",i),this._isAnimated()&&e.classList.add(es),e}setContent(t){this._newContent=t,this._isShown()&&(this._disposePopper(),this.show())}_getTemplateFactory(t){return this._templateFactory?this._templateFactory.changeContent(t):this._templateFactory=new Zn({...this._config,content:t,extraClass:this._resolvePossibleFunction(this._config.customClass)}),this._templateFactory}_getContentForTemplate(){return{".tooltip-inner":this._getTitle()}}_getTitle(){return this._resolvePossibleFunction(this._config.title)||this._element.getAttribute("data-bs-original-title")}_initializeOnDelegatedTarget(t){return this.constructor.getOrCreateInstance(t.delegateTarget,this._getDelegateConfig())}_isAnimated(){return this._config.animation||this.tip&&this.tip.classList.contains(es)}_isShown(){return this.tip&&this.tip.classList.contains(is)}_createPopper(t){const e=Xt(this._config.placement,[this,t,this._element]),i=as[e.toUpperCase()];return Dt(this._element,t,this._getPopperConfig(i))}_getOffset(){const{offset:t}=this._config;return"string"==typeof t?t.split(",").map((t=>Number.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_resolvePossibleFunction(t){return Xt(t,[this._element])}_getPopperConfig(t){const e={placement:t,modifiers:[{name:"flip",options:{fallbackPlacements:this._config.fallbackPlacements}},{name:"offset",options:{offset:this._getOffset()}},{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"arrow",options:{element:`.${this.constructor.NAME}-arrow`}},{name:"preSetPlacement",enabled:!0,phase:"beforeMain",fn:t=>{this._getTipElement().setAttribute("data-popper-placement",t.state.placement)}}]};return{...e,...Xt(this._config.popperConfig,[e])}}_setListeners(){const t=this._config.trigger.split(" ");for(const e of t)if("click"===e)fe.on(this._element,this.constructor.eventName("click"),this._config.selector,(t=>{this._initializeOnDelegatedTarget(t).toggle()}));else if("manual"!==e){const t=e===os?this.constructor.eventName("mouseenter"):this.constructor.eventName("focusin"),i=e===os?this.constructor.eventName("mouseleave"):this.constructor.eventName("focusout");fe.on(this._element,t,this._config.selector,(t=>{const e=this._initializeOnDelegatedTarget(t);e._activeTrigger["focusin"===t.type?rs:os]=!0,e._enter()})),fe.on(this._element,i,this._config.selector,(t=>{const e=this._initializeOnDelegatedTarget(t);e._activeTrigger["focusout"===t.type?rs:os]=e._element.contains(t.relatedTarget),e._leave()}))}this._hideModalHandler=()=>{this._element&&this.hide()},fe.on(this._element.closest(ns),ss,this._hideModalHandler)}_fixTitle(){const t=this._element.getAttribute("title");t&&(this._element.getAttribute("aria-label")||this._element.textContent.trim()||this._element.setAttribute("aria-label",t),this._element.setAttribute("data-bs-original-title",t),this._element.removeAttribute("title"))}_enter(){this._isShown()||this._isHovered?this._isHovered=!0:(this._isHovered=!0,this._setTimeout((()=>{this._isHovered&&this.show()}),this._config.delay.show))}_leave(){this._isWithActiveTrigger()||(this._isHovered=!1,this._setTimeout((()=>{this._isHovered||this.hide()}),this._config.delay.hide))}_setTimeout(t,e){clearTimeout(this._timeout),this._timeout=setTimeout(t,e)}_isWithActiveTrigger(){return Object.values(this._activeTrigger).includes(!0)}_getConfig(t){const e=_e.getDataAttributes(this._element);for(const t of Object.keys(e))ts.has(t)&&delete e[t];return t={...e,..."object"==typeof t&&t?t:{}},t=this._mergeConfigObj(t),t=this._configAfterMerge(t),this._typeCheckConfig(t),t}_configAfterMerge(t){return t.container=!1===t.container?document.body:Ht(t.container),"number"==typeof t.delay&&(t.delay={show:t.delay,hide:t.delay}),"number"==typeof t.title&&(t.title=t.title.toString()),"number"==typeof t.content&&(t.content=t.content.toString()),t}_getDelegateConfig(){const t={};for(const[e,i]of Object.entries(this._config))this.constructor.Default[e]!==i&&(t[e]=i);return t.selector=!1,t.trigger="manual",t}_disposePopper(){this._popper&&(this._popper.destroy(),this._popper=null),this.tip&&(this.tip.remove(),this.tip=null)}static jQueryInterface(t){return this.each((function(){const e=hs.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}Qt(hs);const ds={...hs.Default,content:"",offset:[0,8],placement:"right",template:'<div class="popover" role="tooltip"><div class="popover-arrow"></div><h3 class="popover-header"></h3><div class="popover-body"></div></div>',trigger:"click"},us={...hs.DefaultType,content:"(null|string|element|function)"};class fs extends hs{static get Default(){return ds}static get DefaultType(){return us}static get NAME(){return"popover"}_isWithContent(){return this._getTitle()||this._getContent()}_getContentForTemplate(){return{".popover-header":this._getTitle(),".popover-body":this._getContent()}}_getContent(){return this._resolvePossibleFunction(this._config.content)}static jQueryInterface(t){return this.each((function(){const e=fs.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}Qt(fs);const ps=".bs.scrollspy",ms=`activate${ps}`,gs=`click${ps}`,_s=`load${ps}.data-api`,bs="active",vs="[href]",ys=".nav-link",ws=`${ys}, .nav-item > ${ys}, .list-group-item`,Es={offset:null,rootMargin:"0px 0px -25%",smoothScroll:!1,target:null,threshold:[.1,.5,1]},As={offset:"(number|null)",rootMargin:"string",smoothScroll:"boolean",target:"element",threshold:"array"};class Ts extends ve{constructor(t,e){super(t,e),this._targetLinks=new Map,this._observableSections=new Map,this._rootElement="visible"===getComputedStyle(this._element).overflowY?null:this._element,this._activeTarget=null,this._observer=null,this._previousScrollData={visibleEntryTop:0,parentScrollTop:0},this.refresh()}static get Default(){return Es}static get DefaultType(){return As}static get NAME(){return"scrollspy"}refresh(){this._initializeTargetsAndObservables(),this._maybeEnableSmoothScroll(),this._observer?this._observer.disconnect():this._observer=this._getNewObserver();for(const t of this._observableSections.values())this._observer.observe(t)}dispose(){this._observer.disconnect(),super.dispose()}_configAfterMerge(t){return t.target=Ht(t.target)||document.body,t.rootMargin=t.offset?`${t.offset}px 0px -30%`:t.rootMargin,"string"==typeof t.threshold&&(t.threshold=t.threshold.split(",").map((t=>Number.parseFloat(t)))),t}_maybeEnableSmoothScroll(){this._config.smoothScroll&&(fe.off(this._config.target,gs),fe.on(this._config.target,gs,vs,(t=>{const e=this._observableSections.get(t.target.hash);if(e){t.preventDefault();const i=this._rootElement||window,n=e.offsetTop-this._element.offsetTop;if(i.scrollTo)return void i.scrollTo({top:n,behavior:"smooth"});i.scrollTop=n}})))}_getNewObserver(){const t={root:this._rootElement,threshold:this._config.threshold,rootMargin:this._config.rootMargin};return new IntersectionObserver((t=>this._observerCallback(t)),t)}_observerCallback(t){const e=t=>this._targetLinks.get(`#${t.target.id}`),i=t=>{this._previousScrollData.visibleEntryTop=t.target.offsetTop,this._process(e(t))},n=(this._rootElement||document.documentElement).scrollTop,s=n>=this._previousScrollData.parentScrollTop;this._previousScrollData.parentScrollTop=n;for(const o of t){if(!o.isIntersecting){this._activeTarget=null,this._clearActiveClass(e(o));continue}const t=o.target.offsetTop>=this._previousScrollData.visibleEntryTop;if(s&&t){if(i(o),!n)return}else s||t||i(o)}}_initializeTargetsAndObservables(){this._targetLinks=new Map,this._observableSections=new Map;const t=we.find(vs,this._config.target);for(const e of t){if(!e.hash||Wt(e))continue;const t=we.findOne(decodeURI(e.hash),this._element);Bt(t)&&(this._targetLinks.set(decodeURI(e.hash),e),this._observableSections.set(e.hash,t))}}_process(t){this._activeTarget!==t&&(this._clearActiveClass(this._config.target),this._activeTarget=t,t.classList.add(bs),this._activateParents(t),fe.trigger(this._element,ms,{relatedTarget:t}))}_activateParents(t){if(t.classList.contains("dropdown-item"))we.findOne(".dropdown-toggle",t.closest(".dropdown")).classList.add(bs);else for(const e of we.parents(t,".nav, .list-group"))for(const t of we.prev(e,ws))t.classList.add(bs)}_clearActiveClass(t){t.classList.remove(bs);const e=we.find(`${vs}.${bs}`,t);for(const t of e)t.classList.remove(bs)}static jQueryInterface(t){return this.each((function(){const e=Ts.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t]()}}))}}fe.on(window,_s,(()=>{for(const t of we.find('[data-bs-spy="scroll"]'))Ts.getOrCreateInstance(t)})),Qt(Ts);const Cs=".bs.tab",Os=`hide${Cs}`,xs=`hidden${Cs}`,ks=`show${Cs}`,Ls=`shown${Cs}`,Ss=`click${Cs}`,Ds=`keydown${Cs}`,$s=`load${Cs}`,Is="ArrowLeft",Ns="ArrowRight",Ps="ArrowUp",Ms="ArrowDown",js="Home",Fs="End",Hs="active",Bs="fade",Ws="show",zs=".dropdown-toggle",Rs=`:not(${zs})`,qs='[data-bs-toggle="tab"], [data-bs-toggle="pill"], [data-bs-toggle="list"]',Vs=`.nav-link${Rs}, .list-group-item${Rs}, [role="tab"]${Rs}, ${qs}`,Ys=`.${Hs}[data-bs-toggle="tab"], .${Hs}[data-bs-toggle="pill"], .${Hs}[data-bs-toggle="list"]`;class Ks extends ve{constructor(t){super(t),this._parent=this._element.closest('.list-group, .nav, [role="tablist"]'),this._parent&&(this._setInitialAttributes(this._parent,this._getChildren()),fe.on(this._element,Ds,(t=>this._keydown(t))))}static get NAME(){return"tab"}show(){const t=this._element;if(this._elemIsActive(t))return;const e=this._getActiveElem(),i=e?fe.trigger(e,Os,{relatedTarget:t}):null;fe.trigger(t,ks,{relatedTarget:e}).defaultPrevented||i&&i.defaultPrevented||(this._deactivate(e,t),this._activate(t,e))}_activate(t,e){t&&(t.classList.add(Hs),this._activate(we.getElementFromSelector(t)),this._queueCallback((()=>{"tab"===t.getAttribute("role")?(t.removeAttribute("tabindex"),t.setAttribute("aria-selected",!0),this._toggleDropDown(t,!0),fe.trigger(t,Ls,{relatedTarget:e})):t.classList.add(Ws)}),t,t.classList.contains(Bs)))}_deactivate(t,e){t&&(t.classList.remove(Hs),t.blur(),this._deactivate(we.getElementFromSelector(t)),this._queueCallback((()=>{"tab"===t.getAttribute("role")?(t.setAttribute("aria-selected",!1),t.setAttribute("tabindex","-1"),this._toggleDropDown(t,!1),fe.trigger(t,xs,{relatedTarget:e})):t.classList.remove(Ws)}),t,t.classList.contains(Bs)))}_keydown(t){if(![Is,Ns,Ps,Ms,js,Fs].includes(t.key))return;t.stopPropagation(),t.preventDefault();const e=this._getChildren().filter((t=>!Wt(t)));let i;if([js,Fs].includes(t.key))i=e[t.key===js?0:e.length-1];else{const n=[Ns,Ms].includes(t.key);i=Gt(e,t.target,n,!0)}i&&(i.focus({preventScroll:!0}),Ks.getOrCreateInstance(i).show())}_getChildren(){return we.find(Vs,this._parent)}_getActiveElem(){return this._getChildren().find((t=>this._elemIsActive(t)))||null}_setInitialAttributes(t,e){this._setAttributeIfNotExists(t,"role","tablist");for(const t of e)this._setInitialAttributesOnChild(t)}_setInitialAttributesOnChild(t){t=this._getInnerElement(t);const e=this._elemIsActive(t),i=this._getOuterElement(t);t.setAttribute("aria-selected",e),i!==t&&this._setAttributeIfNotExists(i,"role","presentation"),e||t.setAttribute("tabindex","-1"),this._setAttributeIfNotExists(t,"role","tab"),this._setInitialAttributesOnTargetPanel(t)}_setInitialAttributesOnTargetPanel(t){const e=we.getElementFromSelector(t);e&&(this._setAttributeIfNotExists(e,"role","tabpanel"),t.id&&this._setAttributeIfNotExists(e,"aria-labelledby",`${t.id}`))}_toggleDropDown(t,e){const i=this._getOuterElement(t);if(!i.classList.contains("dropdown"))return;const n=(t,n)=>{const s=we.findOne(t,i);s&&s.classList.toggle(n,e)};n(zs,Hs),n(".dropdown-menu",Ws),i.setAttribute("aria-expanded",e)}_setAttributeIfNotExists(t,e,i){t.hasAttribute(e)||t.setAttribute(e,i)}_elemIsActive(t){return t.classList.contains(Hs)}_getInnerElement(t){return t.matches(Vs)?t:we.findOne(Vs,t)}_getOuterElement(t){return t.closest(".nav-item, .list-group-item")||t}static jQueryInterface(t){return this.each((function(){const e=Ks.getOrCreateInstance(this);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t]()}}))}}fe.on(document,Ss,qs,(function(t){["A","AREA"].includes(this.tagName)&&t.preventDefault(),Wt(this)||Ks.getOrCreateInstance(this).show()})),fe.on(window,$s,(()=>{for(const t of we.find(Ys))Ks.getOrCreateInstance(t)})),Qt(Ks);const Qs=".bs.toast",Xs=`mouseover${Qs}`,Us=`mouseout${Qs}`,Gs=`focusin${Qs}`,Js=`focusout${Qs}`,Zs=`hide${Qs}`,to=`hidden${Qs}`,eo=`show${Qs}`,io=`shown${Qs}`,no="hide",so="show",oo="showing",ro={animation:"boolean",autohide:"boolean",delay:"number"},ao={animation:!0,autohide:!0,delay:5e3};class lo extends ve{constructor(t,e){super(t,e),this._timeout=null,this._hasMouseInteraction=!1,this._hasKeyboardInteraction=!1,this._setListeners()}static get Default(){return ao}static get DefaultType(){return ro}static get NAME(){return"toast"}show(){fe.trigger(this._element,eo).defaultPrevented||(this._clearTimeout(),this._config.animation&&this._element.classList.add("fade"),this._element.classList.remove(no),qt(this._element),this._element.classList.add(so,oo),this._queueCallback((()=>{this._element.classList.remove(oo),fe.trigger(this._element,io),this._maybeScheduleHide()}),this._element,this._config.animation))}hide(){this.isShown()&&(fe.trigger(this._element,Zs).defaultPrevented||(this._element.classList.add(oo),this._queueCallback((()=>{this._element.classList.add(no),this._element.classList.remove(oo,so),fe.trigger(this._element,to)}),this._element,this._config.animation)))}dispose(){this._clearTimeout(),this.isShown()&&this._element.classList.remove(so),super.dispose()}isShown(){return this._element.classList.contains(so)}_maybeScheduleHide(){this._config.autohide&&(this._hasMouseInteraction||this._hasKeyboardInteraction||(this._timeout=setTimeout((()=>{this.hide()}),this._config.delay)))}_onInteraction(t,e){switch(t.type){case"mouseover":case"mouseout":this._hasMouseInteraction=e;break;case"focusin":case"focusout":this._hasKeyboardInteraction=e}if(e)return void this._clearTimeout();const i=t.relatedTarget;this._element===i||this._element.contains(i)||this._maybeScheduleHide()}_setListeners(){fe.on(this._element,Xs,(t=>this._onInteraction(t,!0))),fe.on(this._element,Us,(t=>this._onInteraction(t,!1))),fe.on(this._element,Gs,(t=>this._onInteraction(t,!0))),fe.on(this._element,Js,(t=>this._onInteraction(t,!1)))}_clearTimeout(){clearTimeout(this._timeout),this._timeout=null}static jQueryInterface(t){return this.each((function(){const e=lo.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}function co(t){"loading"!=document.readyState?t():document.addEventListener("DOMContentLoaded",t)}Ee(lo),Qt(lo),co((function(){[].slice.call(document.querySelectorAll('[data-bs-toggle="tooltip"]')).map((function(t){return new hs(t,{delay:{show:500,hide:100}})}))})),co((function(){document.getElementById("pst-back-to-top").addEventListener("click",(function(){document.body.scrollTop=0,document.documentElement.scrollTop=0}))})),co((function(){var t=document.getElementById("pst-back-to-top"),e=document.getElementsByClassName("bd-header")[0].getBoundingClientRect();window.addEventListener("scroll",(function(){this.oldScroll>this.scrollY&&this.scrollY>e.bottom?t.style.display="block":t.style.display="none",this.oldScroll=this.scrollY}))})),window.bootstrap=i})();
+//# sourceMappingURL=bootstrap.js.map
\ No newline at end of file
diff --git a/_static/scripts/bootstrap.js.LICENSE.txt b/_static/scripts/bootstrap.js.LICENSE.txt
new file mode 100644
index 0000000..10f979d
--- /dev/null
+++ b/_static/scripts/bootstrap.js.LICENSE.txt
@@ -0,0 +1,5 @@
+/*!
+  * Bootstrap v5.3.2 (https://getbootstrap.com/)
+  * Copyright 2011-2023 The Bootstrap Authors (https://github.com/twbs/bootstrap/graphs/contributors)
+  * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)
+  */
diff --git a/_static/scripts/bootstrap.js.map b/_static/scripts/bootstrap.js.map
new file mode 100644
index 0000000..64e212b
--- /dev/null
+++ b/_static/scripts/bootstrap.js.map
@@ -0,0 +1 @@
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(element.nodeName || '').toLowerCase() : null;\n}","export default function getWindow(node) {\n  if (node == null) {\n    return window;\n  }\n\n  if (node.toString() !== '[object Window]') {\n    var ownerDocument = node.ownerDocument;\n    return ownerDocument ? ownerDocument.defaultView || window : window;\n  }\n\n  return node;\n}","import getWindow from \"./getWindow.js\";\n\nfunction isElement(node) {\n  var OwnElement = getWindow(node).Element;\n  return node instanceof OwnElement || node instanceof Element;\n}\n\nfunction isHTMLElement(node) {\n  var OwnElement = getWindow(node).HTMLElement;\n  return node instanceof OwnElement || node instanceof HTMLElement;\n}\n\nfunction isShadowRoot(node) {\n  // IE 11 has no ShadowRoot\n  if (typeof ShadowRoot === 'undefined') {\n    return false;\n  }\n\n  var OwnElement = getWindow(node).ShadowRoot;\n  return node instanceof OwnElement || node instanceof ShadowRoot;\n}\n\nexport { isElement, isHTMLElement, isShadowRoot };","import getNodeName from \"../dom-utils/getNodeName.js\";\nimport { isHTMLElement } from \"../dom-utils/instanceOf.js\"; // This modifier takes the styles prepared by the `computeStyles` modifier\n// and applies them to the HTMLElements such as popper and arrow\n\nfunction applyStyles(_ref) {\n  var state = _ref.state;\n  Object.keys(state.elements).forEach(function (name) {\n    var style = state.styles[name] || {};\n    var attributes = state.attributes[name] || {};\n    var element = state.elements[name]; // arrow is optional + virtual elements\n\n    if (!isHTMLElement(element) || !getNodeName(element)) {\n      return;\n    } // Flow doesn't support to extend this property, but it's the most\n    // effective way to apply styles to an HTMLElement\n    // $FlowFixMe[cannot-write]\n\n\n    Object.assign(element.style, style);\n    Object.keys(attributes).forEach(function (name) {\n      var value = attributes[name];\n\n      if (value === false) {\n        element.removeAttribute(name);\n      } else {\n        element.setAttribute(name, value === true ? '' : value);\n      }\n    });\n  });\n}\n\nfunction effect(_ref2) {\n  var state = _ref2.state;\n  var initialStyles = {\n    popper: {\n      position: state.options.strategy,\n      left: '0',\n      top: '0',\n      margin: '0'\n    },\n    arrow: {\n      position: 'absolute'\n    },\n    reference: {}\n  };\n  Object.assign(state.elements.popper.style, initialStyles.popper);\n  state.styles = initialStyles;\n\n  if (state.elements.arrow) {\n    Object.assign(state.elements.arrow.style, initialStyles.arrow);\n  }\n\n  return function () {\n    Object.keys(state.elements).forEach(function (name) {\n      var element = state.elements[name];\n      var attributes = state.attributes[name] || {};\n      var styleProperties = Object.keys(state.styles.hasOwnProperty(name) ? state.styles[name] : initialStyles[name]); // Set all values to an empty string to unset them\n\n      var style = styleProperties.reduce(function (style, property) {\n        style[property] = '';\n        return style;\n      }, {}); // arrow is optional + virtual elements\n\n      if (!isHTMLElement(element) || !getNodeName(element)) {\n        return;\n      }\n\n      Object.assign(element.style, style);\n      Object.keys(attributes).forEach(function (attribute) {\n        element.removeAttribute(attribute);\n      });\n    });\n  };\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'applyStyles',\n  enabled: true,\n  phase: 'write',\n  fn: applyStyles,\n  effect: effect,\n  requires: ['computeStyles']\n};","import { auto } from \"../enums.js\";\nexport default function getBasePlacement(placement) {\n  return placement.split('-')[0];\n}","export var max = Math.max;\nexport var min = Math.min;\nexport var round = Math.round;","export default function getUAString() {\n  var uaData = navigator.userAgentData;\n\n  if (uaData != null && uaData.brands && Array.isArray(uaData.brands)) {\n    return uaData.brands.map(function (item) {\n      return item.brand + \"/\" + item.version;\n    }).join(' ');\n  }\n\n  return navigator.userAgent;\n}","import getUAString from \"../utils/userAgent.js\";\nexport default function isLayoutViewport() {\n  return !/^((?!chrome|android).)*safari/i.test(getUAString());\n}","import { isElement, isHTMLElement } from \"./instanceOf.js\";\nimport { round } from \"../utils/math.js\";\nimport getWindow from \"./getWindow.js\";\nimport isLayoutViewport from \"./isLayoutViewport.js\";\nexport default function getBoundingClientRect(element, includeScale, isFixedStrategy) {\n  if (includeScale === void 0) {\n    includeScale = false;\n  }\n\n  if (isFixedStrategy === void 0) {\n    isFixedStrategy = false;\n  }\n\n  var clientRect = element.getBoundingClientRect();\n  var scaleX = 1;\n  var scaleY = 1;\n\n  if (includeScale && isHTMLElement(element)) {\n    scaleX = element.offsetWidth > 0 ? round(clientRect.width) / element.offsetWidth || 1 : 1;\n    scaleY = element.offsetHeight > 0 ? round(clientRect.height) / element.offsetHeight || 1 : 1;\n  }\n\n  var _ref = isElement(element) ? getWindow(element) : window,\n      visualViewport = _ref.visualViewport;\n\n  var addVisualOffsets = !isLayoutViewport() && isFixedStrategy;\n  var x = (clientRect.left + (addVisualOffsets && visualViewport ? visualViewport.offsetLeft : 0)) / scaleX;\n  var y = (clientRect.top + (addVisualOffsets && visualViewport ? visualViewport.offsetTop : 0)) / scaleY;\n  var width = clientRect.width / scaleX;\n  var height = clientRect.height / scaleY;\n  return {\n    width: width,\n    height: height,\n    top: y,\n    right: x + width,\n    bottom: y + height,\n    left: x,\n    x: x,\n    y: y\n  };\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\"; // Returns the layout rect of an element relative to its offsetParent. Layout\n// means it doesn't take into account transforms.\n\nexport default function getLayoutRect(element) {\n  var clientRect = getBoundingClientRect(element); // Use the clientRect sizes if it's not been transformed.\n  // Fixes https://github.com/popperjs/popper-core/issues/1223\n\n  var width = element.offsetWidth;\n  var height = element.offsetHeight;\n\n  if (Math.abs(clientRect.width - width) <= 1) {\n    width = clientRect.width;\n  }\n\n  if (Math.abs(clientRect.height - height) <= 1) {\n    height = clientRect.height;\n  }\n\n  return {\n    x: element.offsetLeft,\n    y: element.offsetTop,\n    width: width,\n    height: height\n  };\n}","import { isShadowRoot } from \"./instanceOf.js\";\nexport default function contains(parent, child) {\n  var rootNode = child.getRootNode && child.getRootNode(); // First, attempt with faster native method\n\n  if (parent.contains(child)) {\n    return true;\n  } // then fallback to custom implementation with Shadow DOM support\n  else if (rootNode && isShadowRoot(rootNode)) {\n      var next = child;\n\n      do {\n        if (next && parent.isSameNode(next)) {\n          return true;\n        } // $FlowFixMe[prop-missing]: need a better way to handle this...\n\n\n        next = next.parentNode || next.host;\n      } while (next);\n    } // Give up, the result is false\n\n\n  return false;\n}","import getWindow from \"./getWindow.js\";\nexport default function getComputedStyle(element) {\n  return getWindow(element).getComputedStyle(element);\n}","import getNodeName from \"./getNodeName.js\";\nexport default function isTableElement(element) {\n  return ['table', 'td', 'th'].indexOf(getNodeName(element)) >= 0;\n}","import { isElement } from \"./instanceOf.js\";\nexport default function getDocumentElement(element) {\n  // $FlowFixMe[incompatible-return]: assume body is always available\n  return ((isElement(element) ? element.ownerDocument : // $FlowFixMe[prop-missing]\n  element.document) || window.document).documentElement;\n}","import getNodeName from \"./getNodeName.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport { isShadowRoot } from \"./instanceOf.js\";\nexport default function getParentNode(element) {\n  if (getNodeName(element) === 'html') {\n    return element;\n  }\n\n  return (// this is a quicker (but less type safe) way to save quite some bytes from the bundle\n    // $FlowFixMe[incompatible-return]\n    // $FlowFixMe[prop-missing]\n    element.assignedSlot || // step into the shadow DOM of the parent of a slotted node\n    element.parentNode || ( // DOM Element detected\n    isShadowRoot(element) ? element.host : null) || // ShadowRoot detected\n    // $FlowFixMe[incompatible-call]: HTMLElement is a Node\n    getDocumentElement(element) // fallback\n\n  );\n}","import getWindow from \"./getWindow.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport { isHTMLElement, isShadowRoot } from \"./instanceOf.js\";\nimport isTableElement from \"./isTableElement.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport getUAString from \"../utils/userAgent.js\";\n\nfunction getTrueOffsetParent(element) {\n  if (!isHTMLElement(element) || // https://github.com/popperjs/popper-core/issues/837\n  getComputedStyle(element).position === 'fixed') {\n    return null;\n  }\n\n  return element.offsetParent;\n} // `.offsetParent` reports `null` for fixed elements, while absolute elements\n// return the containing block\n\n\nfunction getContainingBlock(element) {\n  var isFirefox = /firefox/i.test(getUAString());\n  var isIE = /Trident/i.test(getUAString());\n\n  if (isIE && isHTMLElement(element)) {\n    // In IE 9, 10 and 11 fixed elements containing block is always established by the viewport\n    var elementCss = getComputedStyle(element);\n\n    if (elementCss.position === 'fixed') {\n      return null;\n    }\n  }\n\n  var currentNode = getParentNode(element);\n\n  if (isShadowRoot(currentNode)) {\n    currentNode = currentNode.host;\n  }\n\n  while (isHTMLElement(currentNode) && ['html', 'body'].indexOf(getNodeName(currentNode)) < 0) {\n    var css = getComputedStyle(currentNode); // This is non-exhaustive but covers the most common CSS properties that\n    // create a containing block.\n    // https://developer.mozilla.org/en-US/docs/Web/CSS/Containing_block#identifying_the_containing_block\n\n    if (css.transform !== 'none' || css.perspective !== 'none' || css.contain === 'paint' || ['transform', 'perspective'].indexOf(css.willChange) !== -1 || isFirefox && css.willChange === 'filter' || isFirefox && css.filter && css.filter !== 'none') {\n      return currentNode;\n    } else {\n      currentNode = currentNode.parentNode;\n    }\n  }\n\n  return null;\n} // Gets the closest ancestor positioned element. Handles some edge cases,\n// such as table ancestors and cross browser bugs.\n\n\nexport default function getOffsetParent(element) {\n  var window = getWindow(element);\n  var offsetParent = getTrueOffsetParent(element);\n\n  while (offsetParent && isTableElement(offsetParent) && getComputedStyle(offsetParent).position === 'static') {\n    offsetParent = getTrueOffsetParent(offsetParent);\n  }\n\n  if (offsetParent && (getNodeName(offsetParent) === 'html' || getNodeName(offsetParent) === 'body' && getComputedStyle(offsetParent).position === 'static')) {\n    return window;\n  }\n\n  return offsetParent || getContainingBlock(element) || window;\n}","export default function getMainAxisFromPlacement(placement) {\n  return ['top', 'bottom'].indexOf(placement) >= 0 ? 'x' : 'y';\n}","import { max as mathMax, min as mathMin } from \"./math.js\";\nexport function within(min, value, max) {\n  return mathMax(min, mathMin(value, max));\n}\nexport function withinMaxClamp(min, value, max) {\n  var v = within(min, value, max);\n  return v > max ? max : v;\n}","import getFreshSideObject from \"./getFreshSideObject.js\";\nexport default function mergePaddingObject(paddingObject) {\n  return Object.assign({}, getFreshSideObject(), paddingObject);\n}","export default function getFreshSideObject() {\n  return {\n    top: 0,\n    right: 0,\n    bottom: 0,\n    left: 0\n  };\n}","export default function expandToHashMap(value, keys) {\n  return keys.reduce(function (hashMap, key) {\n    hashMap[key] = value;\n    return hashMap;\n  }, {});\n}","import getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getLayoutRect from \"../dom-utils/getLayoutRect.js\";\nimport contains from \"../dom-utils/contains.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport getMainAxisFromPlacement from \"../utils/getMainAxisFromPlacement.js\";\nimport { within } from \"../utils/within.js\";\nimport mergePaddingObject from \"../utils/mergePaddingObject.js\";\nimport expandToHashMap from \"../utils/expandToHashMap.js\";\nimport { left, right, basePlacements, top, bottom } from \"../enums.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar toPaddingObject = function toPaddingObject(padding, state) {\n  padding = typeof padding === 'function' ? padding(Object.assign({}, state.rects, {\n    placement: state.placement\n  })) : padding;\n  return mergePaddingObject(typeof padding !== 'number' ? padding : expandToHashMap(padding, basePlacements));\n};\n\nfunction arrow(_ref) {\n  var _state$modifiersData$;\n\n  var state = _ref.state,\n      name = _ref.name,\n      options = _ref.options;\n  var arrowElement = state.elements.arrow;\n  var popperOffsets = state.modifiersData.popperOffsets;\n  var basePlacement = getBasePlacement(state.placement);\n  var axis = getMainAxisFromPlacement(basePlacement);\n  var isVertical = [left, right].indexOf(basePlacement) >= 0;\n  var len = isVertical ? 'height' : 'width';\n\n  if (!arrowElement || !popperOffsets) {\n    return;\n  }\n\n  var paddingObject = toPaddingObject(options.padding, state);\n  var arrowRect = getLayoutRect(arrowElement);\n  var minProp = axis === 'y' ? top : left;\n  var maxProp = axis === 'y' ? bottom : right;\n  var endDiff = state.rects.reference[len] + state.rects.reference[axis] - popperOffsets[axis] - state.rects.popper[len];\n  var startDiff = popperOffsets[axis] - state.rects.reference[axis];\n  var arrowOffsetParent = getOffsetParent(arrowElement);\n  var clientSize = arrowOffsetParent ? axis === 'y' ? arrowOffsetParent.clientHeight || 0 : arrowOffsetParent.clientWidth || 0 : 0;\n  var centerToReference = endDiff / 2 - startDiff / 2; // Make sure the arrow doesn't overflow the popper if the center point is\n  // outside of the popper bounds\n\n  var min = paddingObject[minProp];\n  var max = clientSize - arrowRect[len] - paddingObject[maxProp];\n  var center = clientSize / 2 - arrowRect[len] / 2 + centerToReference;\n  var offset = within(min, center, max); // Prevents breaking syntax highlighting...\n\n  var axisProp = axis;\n  state.modifiersData[name] = (_state$modifiersData$ = {}, _state$modifiersData$[axisProp] = offset, _state$modifiersData$.centerOffset = offset - center, _state$modifiersData$);\n}\n\nfunction effect(_ref2) {\n  var state = _ref2.state,\n      options = _ref2.options;\n  var _options$element = options.element,\n      arrowElement = _options$element === void 0 ? '[data-popper-arrow]' : _options$element;\n\n  if (arrowElement == null) {\n    return;\n  } // CSS selector\n\n\n  if (typeof arrowElement === 'string') {\n    arrowElement = state.elements.popper.querySelector(arrowElement);\n\n    if (!arrowElement) {\n      return;\n    }\n  }\n\n  if (!contains(state.elements.popper, arrowElement)) {\n    return;\n  }\n\n  state.elements.arrow = arrowElement;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'arrow',\n  enabled: true,\n  phase: 'main',\n  fn: arrow,\n  effect: effect,\n  requires: ['popperOffsets'],\n  requiresIfExists: ['preventOverflow']\n};","export default function getVariation(placement) {\n  return placement.split('-')[1];\n}","import { top, left, right, bottom, end } from \"../enums.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport getWindow from \"../dom-utils/getWindow.js\";\nimport getDocumentElement from \"../dom-utils/getDocumentElement.js\";\nimport getComputedStyle from \"../dom-utils/getComputedStyle.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getVariation from \"../utils/getVariation.js\";\nimport { round } from \"../utils/math.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar unsetSides = {\n  top: 'auto',\n  right: 'auto',\n  bottom: 'auto',\n  left: 'auto'\n}; // Round the offsets to the nearest suitable subpixel based on the DPR.\n// Zooming can change the DPR, but it seems to report a value that will\n// cleanly divide the values into the appropriate subpixels.\n\nfunction roundOffsetsByDPR(_ref, win) {\n  var x = _ref.x,\n      y = _ref.y;\n  var dpr = win.devicePixelRatio || 1;\n  return {\n    x: round(x * dpr) / dpr || 0,\n    y: round(y * dpr) / dpr || 0\n  };\n}\n\nexport function mapToStyles(_ref2) {\n  var _Object$assign2;\n\n  var popper = _ref2.popper,\n      popperRect = _ref2.popperRect,\n      placement = _ref2.placement,\n      variation = _ref2.variation,\n      offsets = _ref2.offsets,\n      position = _ref2.position,\n      gpuAcceleration = _ref2.gpuAcceleration,\n      adaptive = _ref2.adaptive,\n      roundOffsets = _ref2.roundOffsets,\n      isFixed = _ref2.isFixed;\n  var _offsets$x = offsets.x,\n      x = _offsets$x === void 0 ? 0 : _offsets$x,\n      _offsets$y = offsets.y,\n      y = _offsets$y === void 0 ? 0 : _offsets$y;\n\n  var _ref3 = typeof roundOffsets === 'function' ? roundOffsets({\n    x: x,\n    y: y\n  }) : {\n    x: x,\n    y: y\n  };\n\n  x = _ref3.x;\n  y = _ref3.y;\n  var hasX = offsets.hasOwnProperty('x');\n  var hasY = offsets.hasOwnProperty('y');\n  var sideX = left;\n  var sideY = top;\n  var win = window;\n\n  if (adaptive) {\n    var offsetParent = getOffsetParent(popper);\n    var heightProp = 'clientHeight';\n    var widthProp = 'clientWidth';\n\n    if (offsetParent === getWindow(popper)) {\n      offsetParent = getDocumentElement(popper);\n\n      if (getComputedStyle(offsetParent).position !== 'static' && position === 'absolute') {\n        heightProp = 'scrollHeight';\n        widthProp = 'scrollWidth';\n      }\n    } // $FlowFixMe[incompatible-cast]: force type refinement, we compare offsetParent with window above, but Flow doesn't detect it\n\n\n    offsetParent = offsetParent;\n\n    if (placement === top || (placement === left || placement === right) && variation === end) {\n      sideY = bottom;\n      var offsetY = isFixed && offsetParent === win && win.visualViewport ? win.visualViewport.height : // $FlowFixMe[prop-missing]\n      offsetParent[heightProp];\n      y -= offsetY - popperRect.height;\n      y *= gpuAcceleration ? 1 : -1;\n    }\n\n    if (placement === left || (placement === top || placement === bottom) && variation === end) {\n      sideX = right;\n      var offsetX = isFixed && offsetParent === win && win.visualViewport ? win.visualViewport.width : // $FlowFixMe[prop-missing]\n      offsetParent[widthProp];\n      x -= offsetX - popperRect.width;\n      x *= gpuAcceleration ? 1 : -1;\n    }\n  }\n\n  var commonStyles = Object.assign({\n    position: position\n  }, adaptive && unsetSides);\n\n  var _ref4 = roundOffsets === true ? roundOffsetsByDPR({\n    x: x,\n    y: y\n  }, getWindow(popper)) : {\n    x: x,\n    y: y\n  };\n\n  x = _ref4.x;\n  y = _ref4.y;\n\n  if (gpuAcceleration) {\n    var _Object$assign;\n\n    return Object.assign({}, commonStyles, (_Object$assign = {}, _Object$assign[sideY] = hasY ? '0' : '', _Object$assign[sideX] = hasX ? '0' : '', _Object$assign.transform = (win.devicePixelRatio || 1) <= 1 ? \"translate(\" + x + \"px, \" + y + \"px)\" : \"translate3d(\" + x + \"px, \" + y + \"px, 0)\", _Object$assign));\n  }\n\n  return Object.assign({}, commonStyles, (_Object$assign2 = {}, _Object$assign2[sideY] = hasY ? y + \"px\" : '', _Object$assign2[sideX] = hasX ? x + \"px\" : '', _Object$assign2.transform = '', _Object$assign2));\n}\n\nfunction computeStyles(_ref5) {\n  var state = _ref5.state,\n      options = _ref5.options;\n  var _options$gpuAccelerat = options.gpuAcceleration,\n      gpuAcceleration = _options$gpuAccelerat === void 0 ? true : _options$gpuAccelerat,\n      _options$adaptive = options.adaptive,\n      adaptive = _options$adaptive === void 0 ? true : _options$adaptive,\n      _options$roundOffsets = options.roundOffsets,\n      roundOffsets = _options$roundOffsets === void 0 ? true : _options$roundOffsets;\n  var commonStyles = {\n    placement: getBasePlacement(state.placement),\n    variation: getVariation(state.placement),\n    popper: state.elements.popper,\n    popperRect: state.rects.popper,\n    gpuAcceleration: gpuAcceleration,\n    isFixed: state.options.strategy === 'fixed'\n  };\n\n  if (state.modifiersData.popperOffsets != null) {\n    state.styles.popper = Object.assign({}, state.styles.popper, mapToStyles(Object.assign({}, commonStyles, {\n      offsets: state.modifiersData.popperOffsets,\n      position: state.options.strategy,\n      adaptive: adaptive,\n      roundOffsets: roundOffsets\n    })));\n  }\n\n  if (state.modifiersData.arrow != null) {\n    state.styles.arrow = Object.assign({}, state.styles.arrow, mapToStyles(Object.assign({}, commonStyles, {\n      offsets: state.modifiersData.arrow,\n      position: 'absolute',\n      adaptive: false,\n      roundOffsets: roundOffsets\n    })));\n  }\n\n  state.attributes.popper = Object.assign({}, state.attributes.popper, {\n    'data-popper-placement': state.placement\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'computeStyles',\n  enabled: true,\n  phase: 'beforeWrite',\n  fn: computeStyles,\n  data: {}\n};","import getWindow from \"../dom-utils/getWindow.js\"; // eslint-disable-next-line import/no-unused-modules\n\nvar passive = {\n  passive: true\n};\n\nfunction effect(_ref) {\n  var state = _ref.state,\n      instance = _ref.instance,\n      options = _ref.options;\n  var _options$scroll = options.scroll,\n      scroll = _options$scroll === void 0 ? true : _options$scroll,\n      _options$resize = options.resize,\n      resize = _options$resize === void 0 ? true : _options$resize;\n  var window = getWindow(state.elements.popper);\n  var scrollParents = [].concat(state.scrollParents.reference, state.scrollParents.popper);\n\n  if (scroll) {\n    scrollParents.forEach(function (scrollParent) {\n      scrollParent.addEventListener('scroll', instance.update, passive);\n    });\n  }\n\n  if (resize) {\n    window.addEventListener('resize', instance.update, passive);\n  }\n\n  return function () {\n    if (scroll) {\n      scrollParents.forEach(function (scrollParent) {\n        scrollParent.removeEventListener('scroll', instance.update, passive);\n      });\n    }\n\n    if (resize) {\n      window.removeEventListener('resize', instance.update, passive);\n    }\n  };\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'eventListeners',\n  enabled: true,\n  phase: 'write',\n  fn: function fn() {},\n  effect: effect,\n  data: {}\n};","var hash = {\n  left: 'right',\n  right: 'left',\n  bottom: 'top',\n  top: 'bottom'\n};\nexport default function getOppositePlacement(placement) {\n  return placement.replace(/left|right|bottom|top/g, function (matched) {\n    return hash[matched];\n  });\n}","var hash = {\n  start: 'end',\n  end: 'start'\n};\nexport default function getOppositeVariationPlacement(placement) {\n  return placement.replace(/start|end/g, function (matched) {\n    return hash[matched];\n  });\n}","import getWindow from \"./getWindow.js\";\nexport default function getWindowScroll(node) {\n  var win = getWindow(node);\n  var scrollLeft = win.pageXOffset;\n  var scrollTop = win.pageYOffset;\n  return {\n    scrollLeft: scrollLeft,\n    scrollTop: scrollTop\n  };\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getWindowScroll from \"./getWindowScroll.js\";\nexport default function getWindowScrollBarX(element) {\n  // If <html> has a CSS width greater than the viewport, then this will be\n  // incorrect for RTL.\n  // Popper 1 is broken in this case and never had a bug report so let's assume\n  // it's not an issue. I don't think anyone ever specifies width on <html>\n  // anyway.\n  // Browsers where the left scrollbar doesn't cause an issue report `0` for\n  // this (e.g. Edge 2019, IE11, Safari)\n  return getBoundingClientRect(getDocumentElement(element)).left + getWindowScroll(element).scrollLeft;\n}","import getComputedStyle from \"./getComputedStyle.js\";\nexport default function isScrollParent(element) {\n  // Firefox wants us to check `-x` and `-y` variations as well\n  var _getComputedStyle = getComputedStyle(element),\n      overflow = _getComputedStyle.overflow,\n      overflowX = _getComputedStyle.overflowX,\n      overflowY = _getComputedStyle.overflowY;\n\n  return /auto|scroll|overlay|hidden/.test(overflow + overflowY + overflowX);\n}","import getParentNode from \"./getParentNode.js\";\nimport isScrollParent from \"./isScrollParent.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nexport default function getScrollParent(node) {\n  if (['html', 'body', '#document'].indexOf(getNodeName(node)) >= 0) {\n    // $FlowFixMe[incompatible-return]: assume body is always available\n    return node.ownerDocument.body;\n  }\n\n  if (isHTMLElement(node) && isScrollParent(node)) {\n    return node;\n  }\n\n  return getScrollParent(getParentNode(node));\n}","import getScrollParent from \"./getScrollParent.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport getWindow from \"./getWindow.js\";\nimport isScrollParent from \"./isScrollParent.js\";\n/*\ngiven a DOM element, return the list of all scroll parents, up the list of ancesors\nuntil we get to the top window object. This list is what we attach scroll listeners\nto, because if any of these parent elements scroll, we'll need to re-calculate the\nreference element's position.\n*/\n\nexport default function listScrollParents(element, list) {\n  var _element$ownerDocumen;\n\n  if (list === void 0) {\n    list = [];\n  }\n\n  var scrollParent = getScrollParent(element);\n  var isBody = scrollParent === ((_element$ownerDocumen = element.ownerDocument) == null ? void 0 : _element$ownerDocumen.body);\n  var win = getWindow(scrollParent);\n  var target = isBody ? [win].concat(win.visualViewport || [], isScrollParent(scrollParent) ? scrollParent : []) : scrollParent;\n  var updatedList = list.concat(target);\n  return isBody ? updatedList : // $FlowFixMe[incompatible-call]: isBody tells us target will be an HTMLElement here\n  updatedList.concat(listScrollParents(getParentNode(target)));\n}","export default function rectToClientRect(rect) {\n  return Object.assign({}, rect, {\n    left: rect.x,\n    top: rect.y,\n    right: rect.x + rect.width,\n    bottom: rect.y + rect.height\n  });\n}","import { viewport } from \"../enums.js\";\nimport getViewportRect from \"./getViewportRect.js\";\nimport getDocumentRect from \"./getDocumentRect.js\";\nimport listScrollParents from \"./listScrollParents.js\";\nimport getOffsetParent from \"./getOffsetParent.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport { isElement, isHTMLElement } from \"./instanceOf.js\";\nimport getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getParentNode from \"./getParentNode.js\";\nimport contains from \"./contains.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport rectToClientRect from \"../utils/rectToClientRect.js\";\nimport { max, min } from \"../utils/math.js\";\n\nfunction getInnerBoundingClientRect(element, strategy) {\n  var rect = getBoundingClientRect(element, false, strategy === 'fixed');\n  rect.top = rect.top + element.clientTop;\n  rect.left = rect.left + element.clientLeft;\n  rect.bottom = rect.top + element.clientHeight;\n  rect.right = rect.left + element.clientWidth;\n  rect.width = element.clientWidth;\n  rect.height = element.clientHeight;\n  rect.x = rect.left;\n  rect.y = rect.top;\n  return rect;\n}\n\nfunction getClientRectFromMixedType(element, clippingParent, strategy) {\n  return clippingParent === viewport ? rectToClientRect(getViewportRect(element, strategy)) : isElement(clippingParent) ? getInnerBoundingClientRect(clippingParent, strategy) : rectToClientRect(getDocumentRect(getDocumentElement(element)));\n} // A \"clipping parent\" is an overflowable container with the characteristic of\n// clipping (or hiding) overflowing elements with a position different from\n// `initial`\n\n\nfunction getClippingParents(element) {\n  var clippingParents = listScrollParents(getParentNode(element));\n  var canEscapeClipping = ['absolute', 'fixed'].indexOf(getComputedStyle(element).position) >= 0;\n  var clipperElement = canEscapeClipping && isHTMLElement(element) ? getOffsetParent(element) : element;\n\n  if (!isElement(clipperElement)) {\n    return [];\n  } // $FlowFixMe[incompatible-return]: https://github.com/facebook/flow/issues/1414\n\n\n  return clippingParents.filter(function (clippingParent) {\n    return isElement(clippingParent) && contains(clippingParent, clipperElement) && getNodeName(clippingParent) !== 'body';\n  });\n} // Gets the maximum area that the element is visible in due to any number of\n// clipping parents\n\n\nexport default function getClippingRect(element, boundary, rootBoundary, strategy) {\n  var mainClippingParents = boundary === 'clippingParents' ? getClippingParents(element) : [].concat(boundary);\n  var clippingParents = [].concat(mainClippingParents, [rootBoundary]);\n  var firstClippingParent = clippingParents[0];\n  var clippingRect = clippingParents.reduce(function (accRect, clippingParent) {\n    var rect = getClientRectFromMixedType(element, clippingParent, strategy);\n    accRect.top = max(rect.top, accRect.top);\n    accRect.right = min(rect.right, accRect.right);\n    accRect.bottom = min(rect.bottom, accRect.bottom);\n    accRect.left = max(rect.left, accRect.left);\n    return accRect;\n  }, getClientRectFromMixedType(element, firstClippingParent, strategy));\n  clippingRect.width = clippingRect.right - clippingRect.left;\n  clippingRect.height = clippingRect.bottom - clippingRect.top;\n  clippingRect.x = clippingRect.left;\n  clippingRect.y = clippingRect.top;\n  return clippingRect;\n}","import getWindow from \"./getWindow.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport isLayoutViewport from \"./isLayoutViewport.js\";\nexport default function getViewportRect(element, strategy) {\n  var win = getWindow(element);\n  var html = getDocumentElement(element);\n  var visualViewport = win.visualViewport;\n  var width = html.clientWidth;\n  var height = html.clientHeight;\n  var x = 0;\n  var y = 0;\n\n  if (visualViewport) {\n    width = visualViewport.width;\n    height = visualViewport.height;\n    var layoutViewport = isLayoutViewport();\n\n    if (layoutViewport || !layoutViewport && strategy === 'fixed') {\n      x = visualViewport.offsetLeft;\n      y = visualViewport.offsetTop;\n    }\n  }\n\n  return {\n    width: width,\n    height: height,\n    x: x + getWindowScrollBarX(element),\n    y: y\n  };\n}","import getDocumentElement from \"./getDocumentElement.js\";\nimport getComputedStyle from \"./getComputedStyle.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport getWindowScroll from \"./getWindowScroll.js\";\nimport { max } from \"../utils/math.js\"; // Gets the entire size of the scrollable document area, even extending outside\n// of the `<html>` and `<body>` rect bounds if horizontally scrollable\n\nexport default function getDocumentRect(element) {\n  var _element$ownerDocumen;\n\n  var html = getDocumentElement(element);\n  var winScroll = getWindowScroll(element);\n  var body = (_element$ownerDocumen = element.ownerDocument) == null ? void 0 : _element$ownerDocumen.body;\n  var width = max(html.scrollWidth, html.clientWidth, body ? body.scrollWidth : 0, body ? body.clientWidth : 0);\n  var height = max(html.scrollHeight, html.clientHeight, body ? body.scrollHeight : 0, body ? body.clientHeight : 0);\n  var x = -winScroll.scrollLeft + getWindowScrollBarX(element);\n  var y = -winScroll.scrollTop;\n\n  if (getComputedStyle(body || html).direction === 'rtl') {\n    x += max(html.clientWidth, body ? body.clientWidth : 0) - width;\n  }\n\n  return {\n    width: width,\n    height: height,\n    x: x,\n    y: y\n  };\n}","import getBasePlacement from \"./getBasePlacement.js\";\nimport getVariation from \"./getVariation.js\";\nimport getMainAxisFromPlacement from \"./getMainAxisFromPlacement.js\";\nimport { top, right, bottom, left, start, end } from \"../enums.js\";\nexport default function computeOffsets(_ref) {\n  var reference = _ref.reference,\n      element = _ref.element,\n      placement = _ref.placement;\n  var basePlacement = placement ? getBasePlacement(placement) : null;\n  var variation = placement ? getVariation(placement) : null;\n  var commonX = reference.x + reference.width / 2 - element.width / 2;\n  var commonY = reference.y + reference.height / 2 - element.height / 2;\n  var offsets;\n\n  switch (basePlacement) {\n    case top:\n      offsets = {\n        x: commonX,\n        y: reference.y - element.height\n      };\n      break;\n\n    case bottom:\n      offsets = {\n        x: commonX,\n        y: reference.y + reference.height\n      };\n      break;\n\n    case right:\n      offsets = {\n        x: reference.x + reference.width,\n        y: commonY\n      };\n      break;\n\n    case left:\n      offsets = {\n        x: reference.x - element.width,\n        y: commonY\n      };\n      break;\n\n    default:\n      offsets = {\n        x: reference.x,\n        y: reference.y\n      };\n  }\n\n  var mainAxis = basePlacement ? getMainAxisFromPlacement(basePlacement) : null;\n\n  if (mainAxis != null) {\n    var len = mainAxis === 'y' ? 'height' : 'width';\n\n    switch (variation) {\n      case start:\n        offsets[mainAxis] = offsets[mainAxis] - (reference[len] / 2 - element[len] / 2);\n        break;\n\n      case end:\n        offsets[mainAxis] = offsets[mainAxis] + (reference[len] / 2 - element[len] / 2);\n        break;\n\n      default:\n    }\n  }\n\n  return offsets;\n}","import getClippingRect from \"../dom-utils/getClippingRect.js\";\nimport getDocumentElement from \"../dom-utils/getDocumentElement.js\";\nimport getBoundingClientRect from \"../dom-utils/getBoundingClientRect.js\";\nimport computeOffsets from \"./computeOffsets.js\";\nimport rectToClientRect from \"./rectToClientRect.js\";\nimport { clippingParents, reference, popper, bottom, top, right, basePlacements, viewport } from \"../enums.js\";\nimport { isElement } from \"../dom-utils/instanceOf.js\";\nimport mergePaddingObject from \"./mergePaddingObject.js\";\nimport expandToHashMap from \"./expandToHashMap.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport default function detectOverflow(state, options) {\n  if (options === void 0) {\n    options = {};\n  }\n\n  var _options = options,\n      _options$placement = _options.placement,\n      placement = _options$placement === void 0 ? state.placement : _options$placement,\n      _options$strategy = _options.strategy,\n      strategy = _options$strategy === void 0 ? state.strategy : _options$strategy,\n      _options$boundary = _options.boundary,\n      boundary = _options$boundary === void 0 ? clippingParents : _options$boundary,\n      _options$rootBoundary = _options.rootBoundary,\n      rootBoundary = _options$rootBoundary === void 0 ? viewport : _options$rootBoundary,\n      _options$elementConte = _options.elementContext,\n      elementContext = _options$elementConte === void 0 ? popper : _options$elementConte,\n      _options$altBoundary = _options.altBoundary,\n      altBoundary = _options$altBoundary === void 0 ? false : _options$altBoundary,\n      _options$padding = _options.padding,\n      padding = _options$padding === void 0 ? 0 : _options$padding;\n  var paddingObject = mergePaddingObject(typeof padding !== 'number' ? padding : expandToHashMap(padding, basePlacements));\n  var altContext = elementContext === popper ? reference : popper;\n  var popperRect = state.rects.popper;\n  var element = state.elements[altBoundary ? altContext : elementContext];\n  var clippingClientRect = getClippingRect(isElement(element) ? element : element.contextElement || getDocumentElement(state.elements.popper), boundary, rootBoundary, strategy);\n  var referenceClientRect = getBoundingClientRect(state.elements.reference);\n  var popperOffsets = computeOffsets({\n    reference: referenceClientRect,\n    element: popperRect,\n    strategy: 'absolute',\n    placement: placement\n  });\n  var popperClientRect = rectToClientRect(Object.assign({}, popperRect, popperOffsets));\n  var elementClientRect = elementContext === popper ? popperClientRect : referenceClientRect; // positive = overflowing the clipping rect\n  // 0 or negative = within the clipping rect\n\n  var overflowOffsets = {\n    top: clippingClientRect.top - elementClientRect.top + paddingObject.top,\n    bottom: elementClientRect.bottom - clippingClientRect.bottom + paddingObject.bottom,\n    left: clippingClientRect.left - elementClientRect.left + paddingObject.left,\n    right: elementClientRect.right - clippingClientRect.right + paddingObject.right\n  };\n  var offsetData = state.modifiersData.offset; // Offsets can be applied only to the popper element\n\n  if (elementContext === popper && offsetData) {\n    var offset = offsetData[placement];\n    Object.keys(overflowOffsets).forEach(function (key) {\n      var multiply = [right, bottom].indexOf(key) >= 0 ? 1 : -1;\n      var axis = [top, bottom].indexOf(key) >= 0 ? 'y' : 'x';\n      overflowOffsets[key] += offset[axis] * multiply;\n    });\n  }\n\n  return overflowOffsets;\n}","import getOppositePlacement from \"../utils/getOppositePlacement.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getOppositeVariationPlacement from \"../utils/getOppositeVariationPlacement.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\nimport computeAutoPlacement from \"../utils/computeAutoPlacement.js\";\nimport { bottom, top, start, right, left, auto } from \"../enums.js\";\nimport getVariation from \"../utils/getVariation.js\"; // eslint-disable-next-line import/no-unused-modules\n\nfunction getExpandedFallbackPlacements(placement) {\n  if (getBasePlacement(placement) === auto) {\n    return [];\n  }\n\n  var oppositePlacement = getOppositePlacement(placement);\n  return [getOppositeVariationPlacement(placement), oppositePlacement, getOppositeVariationPlacement(oppositePlacement)];\n}\n\nfunction flip(_ref) {\n  var state = _ref.state,\n      options = _ref.options,\n      name = _ref.name;\n\n  if (state.modifiersData[name]._skip) {\n    return;\n  }\n\n  var _options$mainAxis = options.mainAxis,\n      checkMainAxis = _options$mainAxis === void 0 ? true : _options$mainAxis,\n      _options$altAxis = options.altAxis,\n      checkAltAxis = _options$altAxis === void 0 ? true : _options$altAxis,\n      specifiedFallbackPlacements = options.fallbackPlacements,\n      padding = options.padding,\n      boundary = options.boundary,\n      rootBoundary = options.rootBoundary,\n      altBoundary = options.altBoundary,\n      _options$flipVariatio = options.flipVariations,\n      flipVariations = _options$flipVariatio === void 0 ? true : _options$flipVariatio,\n      allowedAutoPlacements = options.allowedAutoPlacements;\n  var preferredPlacement = state.options.placement;\n  var basePlacement = getBasePlacement(preferredPlacement);\n  var isBasePlacement = basePlacement === preferredPlacement;\n  var fallbackPlacements = specifiedFallbackPlacements || (isBasePlacement || !flipVariations ? [getOppositePlacement(preferredPlacement)] : getExpandedFallbackPlacements(preferredPlacement));\n  var placements = [preferredPlacement].concat(fallbackPlacements).reduce(function (acc, placement) {\n    return acc.concat(getBasePlacement(placement) === auto ? computeAutoPlacement(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      padding: padding,\n      flipVariations: flipVariations,\n      allowedAutoPlacements: allowedAutoPlacements\n    }) : placement);\n  }, []);\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var checksMap = new Map();\n  var makeFallbackChecks = true;\n  var firstFittingPlacement = placements[0];\n\n  for (var i = 0; i < placements.length; i++) {\n    var placement = placements[i];\n\n    var _basePlacement = getBasePlacement(placement);\n\n    var isStartVariation = getVariation(placement) === start;\n    var isVertical = [top, bottom].indexOf(_basePlacement) >= 0;\n    var len = isVertical ? 'width' : 'height';\n    var overflow = detectOverflow(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      altBoundary: altBoundary,\n      padding: padding\n    });\n    var mainVariationSide = isVertical ? isStartVariation ? right : left : isStartVariation ? bottom : top;\n\n    if (referenceRect[len] > popperRect[len]) {\n      mainVariationSide = getOppositePlacement(mainVariationSide);\n    }\n\n    var altVariationSide = getOppositePlacement(mainVariationSide);\n    var checks = [];\n\n    if (checkMainAxis) {\n      checks.push(overflow[_basePlacement] <= 0);\n    }\n\n    if (checkAltAxis) {\n      checks.push(overflow[mainVariationSide] <= 0, overflow[altVariationSide] <= 0);\n    }\n\n    if (checks.every(function (check) {\n      return check;\n    })) {\n      firstFittingPlacement = placement;\n      makeFallbackChecks = false;\n      break;\n    }\n\n    checksMap.set(placement, checks);\n  }\n\n  if (makeFallbackChecks) {\n    // `2` may be desired in some cases – research later\n    var numberOfChecks = flipVariations ? 3 : 1;\n\n    var _loop = function _loop(_i) {\n      var fittingPlacement = placements.find(function (placement) {\n        var checks = checksMap.get(placement);\n\n        if (checks) {\n          return checks.slice(0, _i).every(function (check) {\n            return check;\n          });\n        }\n      });\n\n      if (fittingPlacement) {\n        firstFittingPlacement = fittingPlacement;\n        return \"break\";\n      }\n    };\n\n    for (var _i = numberOfChecks; _i > 0; _i--) {\n      var _ret = _loop(_i);\n\n      if (_ret === \"break\") break;\n    }\n  }\n\n  if (state.placement !== firstFittingPlacement) {\n    state.modifiersData[name]._skip = true;\n    state.placement = firstFittingPlacement;\n    state.reset = true;\n  }\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'flip',\n  enabled: true,\n  phase: 'main',\n  fn: flip,\n  requiresIfExists: ['offset'],\n  data: {\n    _skip: false\n  }\n};","import getVariation from \"./getVariation.js\";\nimport { variationPlacements, basePlacements, placements as allPlacements } from \"../enums.js\";\nimport detectOverflow from \"./detectOverflow.js\";\nimport getBasePlacement from \"./getBasePlacement.js\";\nexport default function computeAutoPlacement(state, options) {\n  if (options === void 0) {\n    options = {};\n  }\n\n  var _options = options,\n      placement = _options.placement,\n      boundary = _options.boundary,\n      rootBoundary = _options.rootBoundary,\n      padding = _options.padding,\n      flipVariations = _options.flipVariations,\n      _options$allowedAutoP = _options.allowedAutoPlacements,\n      allowedAutoPlacements = _options$allowedAutoP === void 0 ? allPlacements : _options$allowedAutoP;\n  var variation = getVariation(placement);\n  var placements = variation ? flipVariations ? variationPlacements : variationPlacements.filter(function (placement) {\n    return getVariation(placement) === variation;\n  }) : basePlacements;\n  var allowedPlacements = placements.filter(function (placement) {\n    return allowedAutoPlacements.indexOf(placement) >= 0;\n  });\n\n  if (allowedPlacements.length === 0) {\n    allowedPlacements = placements;\n  } // $FlowFixMe[incompatible-type]: Flow seems to have problems with two array unions...\n\n\n  var overflows = allowedPlacements.reduce(function (acc, placement) {\n    acc[placement] = detectOverflow(state, {\n      placement: placement,\n      boundary: boundary,\n      rootBoundary: rootBoundary,\n      padding: padding\n    })[getBasePlacement(placement)];\n    return acc;\n  }, {});\n  return Object.keys(overflows).sort(function (a, b) {\n    return overflows[a] - overflows[b];\n  });\n}","import { top, bottom, left, right } from \"../enums.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\n\nfunction getSideOffsets(overflow, rect, preventedOffsets) {\n  if (preventedOffsets === void 0) {\n    preventedOffsets = {\n      x: 0,\n      y: 0\n    };\n  }\n\n  return {\n    top: overflow.top - rect.height - preventedOffsets.y,\n    right: overflow.right - rect.width + preventedOffsets.x,\n    bottom: overflow.bottom - rect.height + preventedOffsets.y,\n    left: overflow.left - rect.width - preventedOffsets.x\n  };\n}\n\nfunction isAnySideFullyClipped(overflow) {\n  return [top, right, bottom, left].some(function (side) {\n    return overflow[side] >= 0;\n  });\n}\n\nfunction hide(_ref) {\n  var state = _ref.state,\n      name = _ref.name;\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var preventedOffsets = state.modifiersData.preventOverflow;\n  var referenceOverflow = detectOverflow(state, {\n    elementContext: 'reference'\n  });\n  var popperAltOverflow = detectOverflow(state, {\n    altBoundary: true\n  });\n  var referenceClippingOffsets = getSideOffsets(referenceOverflow, referenceRect);\n  var popperEscapeOffsets = getSideOffsets(popperAltOverflow, popperRect, preventedOffsets);\n  var isReferenceHidden = isAnySideFullyClipped(referenceClippingOffsets);\n  var hasPopperEscaped = isAnySideFullyClipped(popperEscapeOffsets);\n  state.modifiersData[name] = {\n    referenceClippingOffsets: referenceClippingOffsets,\n    popperEscapeOffsets: popperEscapeOffsets,\n    isReferenceHidden: isReferenceHidden,\n    hasPopperEscaped: hasPopperEscaped\n  };\n  state.attributes.popper = Object.assign({}, state.attributes.popper, {\n    'data-popper-reference-hidden': isReferenceHidden,\n    'data-popper-escaped': hasPopperEscaped\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'hide',\n  enabled: true,\n  phase: 'main',\n  requiresIfExists: ['preventOverflow'],\n  fn: hide\n};","import getBasePlacement from \"../utils/getBasePlacement.js\";\nimport { top, left, right, placements } from \"../enums.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport function distanceAndSkiddingToXY(placement, rects, offset) {\n  var basePlacement = getBasePlacement(placement);\n  var invertDistance = [left, top].indexOf(basePlacement) >= 0 ? -1 : 1;\n\n  var _ref = typeof offset === 'function' ? offset(Object.assign({}, rects, {\n    placement: placement\n  })) : offset,\n      skidding = _ref[0],\n      distance = _ref[1];\n\n  skidding = skidding || 0;\n  distance = (distance || 0) * invertDistance;\n  return [left, right].indexOf(basePlacement) >= 0 ? {\n    x: distance,\n    y: skidding\n  } : {\n    x: skidding,\n    y: distance\n  };\n}\n\nfunction offset(_ref2) {\n  var state = _ref2.state,\n      options = _ref2.options,\n      name = _ref2.name;\n  var _options$offset = options.offset,\n      offset = _options$offset === void 0 ? [0, 0] : _options$offset;\n  var data = placements.reduce(function (acc, placement) {\n    acc[placement] = distanceAndSkiddingToXY(placement, state.rects, offset);\n    return acc;\n  }, {});\n  var _data$state$placement = data[state.placement],\n      x = _data$state$placement.x,\n      y = _data$state$placement.y;\n\n  if (state.modifiersData.popperOffsets != null) {\n    state.modifiersData.popperOffsets.x += x;\n    state.modifiersData.popperOffsets.y += y;\n  }\n\n  state.modifiersData[name] = data;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'offset',\n  enabled: true,\n  phase: 'main',\n  requires: ['popperOffsets'],\n  fn: offset\n};","import computeOffsets from \"../utils/computeOffsets.js\";\n\nfunction popperOffsets(_ref) {\n  var state = _ref.state,\n      name = _ref.name;\n  // Offsets are the actual position the popper needs to have to be\n  // properly positioned near its reference element\n  // This is the most basic placement, and will be adjusted by\n  // the modifiers in the next step\n  state.modifiersData[name] = computeOffsets({\n    reference: state.rects.reference,\n    element: state.rects.popper,\n    strategy: 'absolute',\n    placement: state.placement\n  });\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'popperOffsets',\n  enabled: true,\n  phase: 'read',\n  fn: popperOffsets,\n  data: {}\n};","import { top, left, right, bottom, start } from \"../enums.js\";\nimport getBasePlacement from \"../utils/getBasePlacement.js\";\nimport getMainAxisFromPlacement from \"../utils/getMainAxisFromPlacement.js\";\nimport getAltAxis from \"../utils/getAltAxis.js\";\nimport { within, withinMaxClamp } from \"../utils/within.js\";\nimport getLayoutRect from \"../dom-utils/getLayoutRect.js\";\nimport getOffsetParent from \"../dom-utils/getOffsetParent.js\";\nimport detectOverflow from \"../utils/detectOverflow.js\";\nimport getVariation from \"../utils/getVariation.js\";\nimport getFreshSideObject from \"../utils/getFreshSideObject.js\";\nimport { min as mathMin, max as mathMax } from \"../utils/math.js\";\n\nfunction preventOverflow(_ref) {\n  var state = _ref.state,\n      options = _ref.options,\n      name = _ref.name;\n  var _options$mainAxis = options.mainAxis,\n      checkMainAxis = _options$mainAxis === void 0 ? true : _options$mainAxis,\n      _options$altAxis = options.altAxis,\n      checkAltAxis = _options$altAxis === void 0 ? false : _options$altAxis,\n      boundary = options.boundary,\n      rootBoundary = options.rootBoundary,\n      altBoundary = options.altBoundary,\n      padding = options.padding,\n      _options$tether = options.tether,\n      tether = _options$tether === void 0 ? true : _options$tether,\n      _options$tetherOffset = options.tetherOffset,\n      tetherOffset = _options$tetherOffset === void 0 ? 0 : _options$tetherOffset;\n  var overflow = detectOverflow(state, {\n    boundary: boundary,\n    rootBoundary: rootBoundary,\n    padding: padding,\n    altBoundary: altBoundary\n  });\n  var basePlacement = getBasePlacement(state.placement);\n  var variation = getVariation(state.placement);\n  var isBasePlacement = !variation;\n  var mainAxis = getMainAxisFromPlacement(basePlacement);\n  var altAxis = getAltAxis(mainAxis);\n  var popperOffsets = state.modifiersData.popperOffsets;\n  var referenceRect = state.rects.reference;\n  var popperRect = state.rects.popper;\n  var tetherOffsetValue = typeof tetherOffset === 'function' ? tetherOffset(Object.assign({}, state.rects, {\n    placement: state.placement\n  })) : tetherOffset;\n  var normalizedTetherOffsetValue = typeof tetherOffsetValue === 'number' ? {\n    mainAxis: tetherOffsetValue,\n    altAxis: tetherOffsetValue\n  } : Object.assign({\n    mainAxis: 0,\n    altAxis: 0\n  }, tetherOffsetValue);\n  var offsetModifierState = state.modifiersData.offset ? state.modifiersData.offset[state.placement] : null;\n  var data = {\n    x: 0,\n    y: 0\n  };\n\n  if (!popperOffsets) {\n    return;\n  }\n\n  if (checkMainAxis) {\n    var _offsetModifierState$;\n\n    var mainSide = mainAxis === 'y' ? top : left;\n    var altSide = mainAxis === 'y' ? bottom : right;\n    var len = mainAxis === 'y' ? 'height' : 'width';\n    var offset = popperOffsets[mainAxis];\n    var min = offset + overflow[mainSide];\n    var max = offset - overflow[altSide];\n    var additive = tether ? -popperRect[len] / 2 : 0;\n    var minLen = variation === start ? referenceRect[len] : popperRect[len];\n    var maxLen = variation === start ? -popperRect[len] : -referenceRect[len]; // We need to include the arrow in the calculation so the arrow doesn't go\n    // outside the reference bounds\n\n    var arrowElement = state.elements.arrow;\n    var arrowRect = tether && arrowElement ? getLayoutRect(arrowElement) : {\n      width: 0,\n      height: 0\n    };\n    var arrowPaddingObject = state.modifiersData['arrow#persistent'] ? state.modifiersData['arrow#persistent'].padding : getFreshSideObject();\n    var arrowPaddingMin = arrowPaddingObject[mainSide];\n    var arrowPaddingMax = arrowPaddingObject[altSide]; // If the reference length is smaller than the arrow length, we don't want\n    // to include its full size in the calculation. If the reference is small\n    // and near the edge of a boundary, the popper can overflow even if the\n    // reference is not overflowing as well (e.g. virtual elements with no\n    // width or height)\n\n    var arrowLen = within(0, referenceRect[len], arrowRect[len]);\n    var minOffset = isBasePlacement ? referenceRect[len] / 2 - additive - arrowLen - arrowPaddingMin - normalizedTetherOffsetValue.mainAxis : minLen - arrowLen - arrowPaddingMin - normalizedTetherOffsetValue.mainAxis;\n    var maxOffset = isBasePlacement ? -referenceRect[len] / 2 + additive + arrowLen + arrowPaddingMax + normalizedTetherOffsetValue.mainAxis : maxLen + arrowLen + arrowPaddingMax + normalizedTetherOffsetValue.mainAxis;\n    var arrowOffsetParent = state.elements.arrow && getOffsetParent(state.elements.arrow);\n    var clientOffset = arrowOffsetParent ? mainAxis === 'y' ? arrowOffsetParent.clientTop || 0 : arrowOffsetParent.clientLeft || 0 : 0;\n    var offsetModifierValue = (_offsetModifierState$ = offsetModifierState == null ? void 0 : offsetModifierState[mainAxis]) != null ? _offsetModifierState$ : 0;\n    var tetherMin = offset + minOffset - offsetModifierValue - clientOffset;\n    var tetherMax = offset + maxOffset - offsetModifierValue;\n    var preventedOffset = within(tether ? mathMin(min, tetherMin) : min, offset, tether ? mathMax(max, tetherMax) : max);\n    popperOffsets[mainAxis] = preventedOffset;\n    data[mainAxis] = preventedOffset - offset;\n  }\n\n  if (checkAltAxis) {\n    var _offsetModifierState$2;\n\n    var _mainSide = mainAxis === 'x' ? top : left;\n\n    var _altSide = mainAxis === 'x' ? bottom : right;\n\n    var _offset = popperOffsets[altAxis];\n\n    var _len = altAxis === 'y' ? 'height' : 'width';\n\n    var _min = _offset + overflow[_mainSide];\n\n    var _max = _offset - overflow[_altSide];\n\n    var isOriginSide = [top, left].indexOf(basePlacement) !== -1;\n\n    var _offsetModifierValue = (_offsetModifierState$2 = offsetModifierState == null ? void 0 : offsetModifierState[altAxis]) != null ? _offsetModifierState$2 : 0;\n\n    var _tetherMin = isOriginSide ? _min : _offset - referenceRect[_len] - popperRect[_len] - _offsetModifierValue + normalizedTetherOffsetValue.altAxis;\n\n    var _tetherMax = isOriginSide ? _offset + referenceRect[_len] + popperRect[_len] - _offsetModifierValue - normalizedTetherOffsetValue.altAxis : _max;\n\n    var _preventedOffset = tether && isOriginSide ? withinMaxClamp(_tetherMin, _offset, _tetherMax) : within(tether ? _tetherMin : _min, _offset, tether ? _tetherMax : _max);\n\n    popperOffsets[altAxis] = _preventedOffset;\n    data[altAxis] = _preventedOffset - _offset;\n  }\n\n  state.modifiersData[name] = data;\n} // eslint-disable-next-line import/no-unused-modules\n\n\nexport default {\n  name: 'preventOverflow',\n  enabled: true,\n  phase: 'main',\n  fn: preventOverflow,\n  requiresIfExists: ['offset']\n};","export default function getAltAxis(axis) {\n  return axis === 'x' ? 'y' : 'x';\n}","import getBoundingClientRect from \"./getBoundingClientRect.js\";\nimport getNodeScroll from \"./getNodeScroll.js\";\nimport getNodeName from \"./getNodeName.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nimport getWindowScrollBarX from \"./getWindowScrollBarX.js\";\nimport getDocumentElement from \"./getDocumentElement.js\";\nimport isScrollParent from \"./isScrollParent.js\";\nimport { round } from \"../utils/math.js\";\n\nfunction isElementScaled(element) {\n  var rect = element.getBoundingClientRect();\n  var scaleX = round(rect.width) / element.offsetWidth || 1;\n  var scaleY = round(rect.height) / element.offsetHeight || 1;\n  return scaleX !== 1 || scaleY !== 1;\n} // Returns the composite rect of an element relative to its offsetParent.\n// Composite means it takes into account transforms as well as layout.\n\n\nexport default function getCompositeRect(elementOrVirtualElement, offsetParent, isFixed) {\n  if (isFixed === void 0) {\n    isFixed = false;\n  }\n\n  var isOffsetParentAnElement = isHTMLElement(offsetParent);\n  var offsetParentIsScaled = isHTMLElement(offsetParent) && isElementScaled(offsetParent);\n  var documentElement = getDocumentElement(offsetParent);\n  var rect = getBoundingClientRect(elementOrVirtualElement, offsetParentIsScaled, isFixed);\n  var scroll = {\n    scrollLeft: 0,\n    scrollTop: 0\n  };\n  var offsets = {\n    x: 0,\n    y: 0\n  };\n\n  if (isOffsetParentAnElement || !isOffsetParentAnElement && !isFixed) {\n    if (getNodeName(offsetParent) !== 'body' || // https://github.com/popperjs/popper-core/issues/1078\n    isScrollParent(documentElement)) {\n      scroll = getNodeScroll(offsetParent);\n    }\n\n    if (isHTMLElement(offsetParent)) {\n      offsets = getBoundingClientRect(offsetParent, true);\n      offsets.x += offsetParent.clientLeft;\n      offsets.y += offsetParent.clientTop;\n    } else if (documentElement) {\n      offsets.x = getWindowScrollBarX(documentElement);\n    }\n  }\n\n  return {\n    x: rect.left + scroll.scrollLeft - offsets.x,\n    y: rect.top + scroll.scrollTop - offsets.y,\n    width: rect.width,\n    height: rect.height\n  };\n}","import getWindowScroll from \"./getWindowScroll.js\";\nimport getWindow from \"./getWindow.js\";\nimport { isHTMLElement } from \"./instanceOf.js\";\nimport getHTMLElementScroll from \"./getHTMLElementScroll.js\";\nexport default function getNodeScroll(node) {\n  if (node === getWindow(node) || !isHTMLElement(node)) {\n    return getWindowScroll(node);\n  } else {\n    return getHTMLElementScroll(node);\n  }\n}","export default function getHTMLElementScroll(element) {\n  return {\n    scrollLeft: element.scrollLeft,\n    scrollTop: element.scrollTop\n  };\n}","import { modifierPhases } from \"../enums.js\"; // source: https://stackoverflow.com/questions/49875255\n\nfunction order(modifiers) {\n  var map = new Map();\n  var visited = new Set();\n  var result = [];\n  modifiers.forEach(function (modifier) {\n    map.set(modifier.name, modifier);\n  }); // On visiting object, check for its dependencies and visit them recursively\n\n  function sort(modifier) {\n    visited.add(modifier.name);\n    var requires = [].concat(modifier.requires || [], modifier.requiresIfExists || []);\n    requires.forEach(function (dep) {\n      if (!visited.has(dep)) {\n        var depModifier = map.get(dep);\n\n        if (depModifier) {\n          sort(depModifier);\n        }\n      }\n    });\n    result.push(modifier);\n  }\n\n  modifiers.forEach(function (modifier) {\n    if (!visited.has(modifier.name)) {\n      // check for visited object\n      sort(modifier);\n    }\n  });\n  return result;\n}\n\nexport default function orderModifiers(modifiers) {\n  // order based on dependencies\n  var orderedModifiers = order(modifiers); // order based on phase\n\n  return modifierPhases.reduce(function (acc, phase) {\n    return acc.concat(orderedModifiers.filter(function (modifier) {\n      return modifier.phase === phase;\n    }));\n  }, []);\n}","import getCompositeRect from \"./dom-utils/getCompositeRect.js\";\nimport getLayoutRect from \"./dom-utils/getLayoutRect.js\";\nimport listScrollParents from \"./dom-utils/listScrollParents.js\";\nimport getOffsetParent from \"./dom-utils/getOffsetParent.js\";\nimport orderModifiers from \"./utils/orderModifiers.js\";\nimport debounce from \"./utils/debounce.js\";\nimport mergeByName from \"./utils/mergeByName.js\";\nimport detectOverflow from \"./utils/detectOverflow.js\";\nimport { isElement } from \"./dom-utils/instanceOf.js\";\nvar DEFAULT_OPTIONS = {\n  placement: 'bottom',\n  modifiers: [],\n  strategy: 'absolute'\n};\n\nfunction areValidElements() {\n  for (var _len = arguments.length, args = new Array(_len), _key = 0; _key < _len; _key++) {\n    args[_key] = arguments[_key];\n  }\n\n  return !args.some(function (element) {\n    return !(element && typeof element.getBoundingClientRect === 'function');\n  });\n}\n\nexport function popperGenerator(generatorOptions) {\n  if (generatorOptions === void 0) {\n    generatorOptions = {};\n  }\n\n  var _generatorOptions = generatorOptions,\n      _generatorOptions$def = _generatorOptions.defaultModifiers,\n      defaultModifiers = _generatorOptions$def === void 0 ? [] : _generatorOptions$def,\n      _generatorOptions$def2 = _generatorOptions.defaultOptions,\n      defaultOptions = _generatorOptions$def2 === void 0 ? DEFAULT_OPTIONS : _generatorOptions$def2;\n  return function createPopper(reference, popper, options) {\n    if (options === void 0) {\n      options = defaultOptions;\n    }\n\n    var state = {\n      placement: 'bottom',\n      orderedModifiers: [],\n      options: Object.assign({}, DEFAULT_OPTIONS, defaultOptions),\n      modifiersData: {},\n      elements: {\n        reference: reference,\n        popper: popper\n      },\n      attributes: {},\n      styles: {}\n    };\n    var effectCleanupFns = [];\n    var isDestroyed = false;\n    var instance = {\n      state: state,\n      setOptions: function setOptions(setOptionsAction) {\n        var options = typeof setOptionsAction === 'function' ? setOptionsAction(state.options) : setOptionsAction;\n        cleanupModifierEffects();\n        state.options = Object.assign({}, defaultOptions, state.options, options);\n        state.scrollParents = {\n          reference: isElement(reference) ? listScrollParents(reference) : reference.contextElement ? listScrollParents(reference.contextElement) : [],\n          popper: listScrollParents(popper)\n        }; // Orders the modifiers based on their dependencies and `phase`\n        // properties\n\n        var orderedModifiers = orderModifiers(mergeByName([].concat(defaultModifiers, state.options.modifiers))); // Strip out disabled modifiers\n\n        state.orderedModifiers = orderedModifiers.filter(function (m) {\n          return m.enabled;\n        });\n        runModifierEffects();\n        return instance.update();\n      },\n      // Sync update – it will always be executed, even if not necessary. This\n      // is useful for low frequency updates where sync behavior simplifies the\n      // logic.\n      // For high frequency updates (e.g. `resize` and `scroll` events), always\n      // prefer the async Popper#update method\n      forceUpdate: function forceUpdate() {\n        if (isDestroyed) {\n          return;\n        }\n\n        var _state$elements = state.elements,\n            reference = _state$elements.reference,\n            popper = _state$elements.popper; // Don't proceed if `reference` or `popper` are not valid elements\n        // anymore\n\n        if (!areValidElements(reference, popper)) {\n          return;\n        } // Store the reference and popper rects to be read by modifiers\n\n\n        state.rects = {\n          reference: getCompositeRect(reference, getOffsetParent(popper), state.options.strategy === 'fixed'),\n          popper: getLayoutRect(popper)\n        }; // Modifiers have the ability to reset the current update cycle. The\n        // most common use case for this is the `flip` modifier changing the\n        // placement, which then needs to re-run all the modifiers, because the\n        // logic was previously ran for the previous placement and is therefore\n        // stale/incorrect\n\n        state.reset = false;\n        state.placement = state.options.placement; // On each update cycle, the `modifiersData` property for each modifier\n        // is filled with the initial data specified by the modifier. This means\n        // it doesn't persist and is fresh on each update.\n        // To ensure persistent data, use `${name}#persistent`\n\n        state.orderedModifiers.forEach(function (modifier) {\n          return state.modifiersData[modifier.name] = Object.assign({}, modifier.data);\n        });\n\n        for (var index = 0; index < state.orderedModifiers.length; index++) {\n          if (state.reset === true) {\n            state.reset = false;\n            index = -1;\n            continue;\n          }\n\n          var _state$orderedModifie = state.orderedModifiers[index],\n              fn = _state$orderedModifie.fn,\n              _state$orderedModifie2 = _state$orderedModifie.options,\n              _options = _state$orderedModifie2 === void 0 ? {} : _state$orderedModifie2,\n              name = _state$orderedModifie.name;\n\n          if (typeof fn === 'function') {\n            state = fn({\n              state: state,\n              options: _options,\n              name: name,\n              instance: instance\n            }) || state;\n          }\n        }\n      },\n      // Async and optimistically optimized update – it will not be executed if\n      // not necessary (debounced to run at most once-per-tick)\n      update: debounce(function () {\n        return new Promise(function (resolve) {\n          instance.forceUpdate();\n          resolve(state);\n        });\n      }),\n      destroy: function destroy() {\n        cleanupModifierEffects();\n        isDestroyed = true;\n      }\n    };\n\n    if (!areValidElements(reference, popper)) {\n      return instance;\n    }\n\n    instance.setOptions(options).then(function (state) {\n      if (!isDestroyed && options.onFirstUpdate) {\n        options.onFirstUpdate(state);\n      }\n    }); // Modifiers have the ability to execute arbitrary code before the first\n    // update cycle runs. They will be executed in the same order as the update\n    // cycle. This is useful when a modifier adds some persistent data that\n    // other modifiers need to use, but the modifier is run after the dependent\n    // one.\n\n    function runModifierEffects() {\n      state.orderedModifiers.forEach(function (_ref) {\n        var name = _ref.name,\n            _ref$options = _ref.options,\n            options = _ref$options === void 0 ? {} : _ref$options,\n            effect = _ref.effect;\n\n        if (typeof effect === 'function') {\n          var cleanupFn = effect({\n            state: state,\n            name: name,\n            instance: instance,\n            options: options\n          });\n\n          var noopFn = function noopFn() {};\n\n          effectCleanupFns.push(cleanupFn || noopFn);\n        }\n      });\n    }\n\n    function cleanupModifierEffects() {\n      effectCleanupFns.forEach(function (fn) {\n        return fn();\n      });\n      effectCleanupFns = [];\n    }\n\n    return instance;\n  };\n}\nexport var createPopper = /*#__PURE__*/popperGenerator(); // eslint-disable-next-line import/no-unused-modules\n\nexport { detectOverflow };","export default function debounce(fn) {\n  var pending;\n  return function () {\n    if (!pending) {\n      pending = new Promise(function (resolve) {\n        Promise.resolve().then(function () {\n          pending = undefined;\n          resolve(fn());\n        });\n      });\n    }\n\n    return pending;\n  };\n}","export default function mergeByName(modifiers) {\n  var merged = modifiers.reduce(function (merged, current) {\n    var existing = merged[current.name];\n    merged[current.name] = existing ? Object.assign({}, existing, current, {\n      options: Object.assign({}, existing.options, current.options),\n      data: Object.assign({}, existing.data, current.data)\n    }) : current;\n    return merged;\n  }, {}); // IE11 does not support Object.values\n\n  return Object.keys(merged).map(function (key) {\n    return merged[key];\n  });\n}","import { popperGenerator, detectOverflow } from \"./createPopper.js\";\nimport eventListeners from \"./modifiers/eventListeners.js\";\nimport popperOffsets from \"./modifiers/popperOffsets.js\";\nimport computeStyles from \"./modifiers/computeStyles.js\";\nimport applyStyles from \"./modifiers/applyStyles.js\";\nimport offset from \"./modifiers/offset.js\";\nimport flip from \"./modifiers/flip.js\";\nimport preventOverflow from \"./modifiers/preventOverflow.js\";\nimport arrow from \"./modifiers/arrow.js\";\nimport hide from \"./modifiers/hide.js\";\nvar defaultModifiers = [eventListeners, popperOffsets, computeStyles, applyStyles, offset, flip, preventOverflow, arrow, hide];\nvar createPopper = /*#__PURE__*/popperGenerator({\n  defaultModifiers: defaultModifiers\n}); // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper, popperGenerator, defaultModifiers, detectOverflow }; // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper as createPopperLite } from \"./popper-lite.js\"; // eslint-disable-next-line import/no-unused-modules\n\nexport * from \"./modifiers/index.js\";","import { popperGenerator, detectOverflow } from \"./createPopper.js\";\nimport eventListeners from \"./modifiers/eventListeners.js\";\nimport popperOffsets from \"./modifiers/popperOffsets.js\";\nimport computeStyles from \"./modifiers/computeStyles.js\";\nimport applyStyles from \"./modifiers/applyStyles.js\";\nvar defaultModifiers = [eventListeners, popperOffsets, computeStyles, applyStyles];\nvar createPopper = /*#__PURE__*/popperGenerator({\n  defaultModifiers: defaultModifiers\n}); // eslint-disable-next-line import/no-unused-modules\n\nexport { createPopper, popperGenerator, defaultModifiers, detectOverflow };","/*!\n  * Bootstrap v5.3.2 (https://getbootstrap.com/)\n  * Copyright 2011-2023 The Bootstrap Authors (https://github.com/twbs/bootstrap/graphs/contributors)\n  * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n  */\nimport * as Popper from '@popperjs/core';\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/data.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n/**\n * Constants\n */\n\nconst elementMap = new Map();\nconst Data = {\n  set(element, key, instance) {\n    if (!elementMap.has(element)) {\n      elementMap.set(element, new Map());\n    }\n    const instanceMap = elementMap.get(element);\n\n    // make it clear we only want one instance per element\n    // can be removed later when multiple key/instances are fine to be used\n    if (!instanceMap.has(key) && instanceMap.size !== 0) {\n      // eslint-disable-next-line no-console\n      console.error(`Bootstrap doesn't allow more than one instance per element. Bound instance: ${Array.from(instanceMap.keys())[0]}.`);\n      return;\n    }\n    instanceMap.set(key, instance);\n  },\n  get(element, key) {\n    if (elementMap.has(element)) {\n      return elementMap.get(element).get(key) || null;\n    }\n    return null;\n  },\n  remove(element, key) {\n    if (!elementMap.has(element)) {\n      return;\n    }\n    const instanceMap = elementMap.get(element);\n    instanceMap.delete(key);\n\n    // free up element references if there are no instances left for an element\n    if (instanceMap.size === 0) {\n      elementMap.delete(element);\n    }\n  }\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/index.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nconst MAX_UID = 1000000;\nconst MILLISECONDS_MULTIPLIER = 1000;\nconst TRANSITION_END = 'transitionend';\n\n/**\n * Properly escape IDs selectors to handle weird IDs\n * @param {string} selector\n * @returns {string}\n */\nconst parseSelector = selector => {\n  if (selector && window.CSS && window.CSS.escape) {\n    // document.querySelector needs escaping to handle IDs (html5+) containing for instance /\n    selector = selector.replace(/#([^\\s\"#']+)/g, (match, id) => `#${CSS.escape(id)}`);\n  }\n  return selector;\n};\n\n// Shout-out Angus Croll (https://goo.gl/pxwQGp)\nconst toType = object => {\n  if (object === null || object === undefined) {\n    return `${object}`;\n  }\n  return Object.prototype.toString.call(object).match(/\\s([a-z]+)/i)[1].toLowerCase();\n};\n\n/**\n * Public Util API\n */\n\nconst getUID = prefix => {\n  do {\n    prefix += Math.floor(Math.random() * MAX_UID);\n  } while (document.getElementById(prefix));\n  return prefix;\n};\nconst getTransitionDurationFromElement = element => {\n  if (!element) {\n    return 0;\n  }\n\n  // Get transition-duration of the element\n  let {\n    transitionDuration,\n    transitionDelay\n  } = window.getComputedStyle(element);\n  const floatTransitionDuration = Number.parseFloat(transitionDuration);\n  const floatTransitionDelay = Number.parseFloat(transitionDelay);\n\n  // Return 0 if element or transition duration is not found\n  if (!floatTransitionDuration && !floatTransitionDelay) {\n    return 0;\n  }\n\n  // If multiple durations are defined, take the first\n  transitionDuration = transitionDuration.split(',')[0];\n  transitionDelay = transitionDelay.split(',')[0];\n  return (Number.parseFloat(transitionDuration) + Number.parseFloat(transitionDelay)) * MILLISECONDS_MULTIPLIER;\n};\nconst triggerTransitionEnd = element => {\n  element.dispatchEvent(new Event(TRANSITION_END));\n};\nconst isElement = object => {\n  if (!object || typeof object !== 'object') {\n    return false;\n  }\n  if (typeof object.jquery !== 'undefined') {\n    object = object[0];\n  }\n  return typeof object.nodeType !== 'undefined';\n};\nconst getElement = object => {\n  // it's a jQuery object or a node element\n  if (isElement(object)) {\n    return object.jquery ? object[0] : object;\n  }\n  if (typeof object === 'string' && object.length > 0) {\n    return document.querySelector(parseSelector(object));\n  }\n  return null;\n};\nconst isVisible = element => {\n  if (!isElement(element) || element.getClientRects().length === 0) {\n    return false;\n  }\n  const elementIsVisible = getComputedStyle(element).getPropertyValue('visibility') === 'visible';\n  // Handle `details` element as its content may falsie appear visible when it is closed\n  const closedDetails = element.closest('details:not([open])');\n  if (!closedDetails) {\n    return elementIsVisible;\n  }\n  if (closedDetails !== element) {\n    const summary = element.closest('summary');\n    if (summary && summary.parentNode !== closedDetails) {\n      return false;\n    }\n    if (summary === null) {\n      return false;\n    }\n  }\n  return elementIsVisible;\n};\nconst isDisabled = element => {\n  if (!element || element.nodeType !== Node.ELEMENT_NODE) {\n    return true;\n  }\n  if (element.classList.contains('disabled')) {\n    return true;\n  }\n  if (typeof element.disabled !== 'undefined') {\n    return element.disabled;\n  }\n  return element.hasAttribute('disabled') && element.getAttribute('disabled') !== 'false';\n};\nconst findShadowRoot = element => {\n  if (!document.documentElement.attachShadow) {\n    return null;\n  }\n\n  // Can find the shadow root otherwise it'll return the document\n  if (typeof element.getRootNode === 'function') {\n    const root = element.getRootNode();\n    return root instanceof ShadowRoot ? root : null;\n  }\n  if (element instanceof ShadowRoot) {\n    return element;\n  }\n\n  // when we don't find a shadow root\n  if (!element.parentNode) {\n    return null;\n  }\n  return findShadowRoot(element.parentNode);\n};\nconst noop = () => {};\n\n/**\n * Trick to restart an element's animation\n *\n * @param {HTMLElement} element\n * @return void\n *\n * @see https://www.charistheo.io/blog/2021/02/restart-a-css-animation-with-javascript/#restarting-a-css-animation\n */\nconst reflow = element => {\n  element.offsetHeight; // eslint-disable-line no-unused-expressions\n};\n\nconst getjQuery = () => {\n  if (window.jQuery && !document.body.hasAttribute('data-bs-no-jquery')) {\n    return window.jQuery;\n  }\n  return null;\n};\nconst DOMContentLoadedCallbacks = [];\nconst onDOMContentLoaded = callback => {\n  if (document.readyState === 'loading') {\n    // add listener on the first call when the document is in loading state\n    if (!DOMContentLoadedCallbacks.length) {\n      document.addEventListener('DOMContentLoaded', () => {\n        for (const callback of DOMContentLoadedCallbacks) {\n          callback();\n        }\n      });\n    }\n    DOMContentLoadedCallbacks.push(callback);\n  } else {\n    callback();\n  }\n};\nconst isRTL = () => document.documentElement.dir === 'rtl';\nconst defineJQueryPlugin = plugin => {\n  onDOMContentLoaded(() => {\n    const $ = getjQuery();\n    /* istanbul ignore if */\n    if ($) {\n      const name = plugin.NAME;\n      const JQUERY_NO_CONFLICT = $.fn[name];\n      $.fn[name] = plugin.jQueryInterface;\n      $.fn[name].Constructor = plugin;\n      $.fn[name].noConflict = () => {\n        $.fn[name] = JQUERY_NO_CONFLICT;\n        return plugin.jQueryInterface;\n      };\n    }\n  });\n};\nconst execute = (possibleCallback, args = [], defaultValue = possibleCallback) => {\n  return typeof possibleCallback === 'function' ? possibleCallback(...args) : defaultValue;\n};\nconst executeAfterTransition = (callback, transitionElement, waitForTransition = true) => {\n  if (!waitForTransition) {\n    execute(callback);\n    return;\n  }\n  const durationPadding = 5;\n  const emulatedDuration = getTransitionDurationFromElement(transitionElement) + durationPadding;\n  let called = false;\n  const handler = ({\n    target\n  }) => {\n    if (target !== transitionElement) {\n      return;\n    }\n    called = true;\n    transitionElement.removeEventListener(TRANSITION_END, handler);\n    execute(callback);\n  };\n  transitionElement.addEventListener(TRANSITION_END, handler);\n  setTimeout(() => {\n    if (!called) {\n      triggerTransitionEnd(transitionElement);\n    }\n  }, emulatedDuration);\n};\n\n/**\n * Return the previous/next element of a list.\n *\n * @param {array} list    The list of elements\n * @param activeElement   The active element\n * @param shouldGetNext   Choose to get next or previous element\n * @param isCycleAllowed\n * @return {Element|elem} The proper element\n */\nconst getNextActiveElement = (list, activeElement, shouldGetNext, isCycleAllowed) => {\n  const listLength = list.length;\n  let index = list.indexOf(activeElement);\n\n  // if the element does not exist in the list return an element\n  // depending on the direction and if cycle is allowed\n  if (index === -1) {\n    return !shouldGetNext && isCycleAllowed ? list[listLength - 1] : list[0];\n  }\n  index += shouldGetNext ? 1 : -1;\n  if (isCycleAllowed) {\n    index = (index + listLength) % listLength;\n  }\n  return list[Math.max(0, Math.min(index, listLength - 1))];\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/event-handler.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst namespaceRegex = /[^.]*(?=\\..*)\\.|.*/;\nconst stripNameRegex = /\\..*/;\nconst stripUidRegex = /::\\d+$/;\nconst eventRegistry = {}; // Events storage\nlet uidEvent = 1;\nconst customEvents = {\n  mouseenter: 'mouseover',\n  mouseleave: 'mouseout'\n};\nconst nativeEvents = new Set(['click', 'dblclick', 'mouseup', 'mousedown', 'contextmenu', 'mousewheel', 'DOMMouseScroll', 'mouseover', 'mouseout', 'mousemove', 'selectstart', 'selectend', 'keydown', 'keypress', 'keyup', 'orientationchange', 'touchstart', 'touchmove', 'touchend', 'touchcancel', 'pointerdown', 'pointermove', 'pointerup', 'pointerleave', 'pointercancel', 'gesturestart', 'gesturechange', 'gestureend', 'focus', 'blur', 'change', 'reset', 'select', 'submit', 'focusin', 'focusout', 'load', 'unload', 'beforeunload', 'resize', 'move', 'DOMContentLoaded', 'readystatechange', 'error', 'abort', 'scroll']);\n\n/**\n * Private methods\n */\n\nfunction makeEventUid(element, uid) {\n  return uid && `${uid}::${uidEvent++}` || element.uidEvent || uidEvent++;\n}\nfunction getElementEvents(element) {\n  const uid = makeEventUid(element);\n  element.uidEvent = uid;\n  eventRegistry[uid] = eventRegistry[uid] || {};\n  return eventRegistry[uid];\n}\nfunction bootstrapHandler(element, fn) {\n  return function handler(event) {\n    hydrateObj(event, {\n      delegateTarget: element\n    });\n    if (handler.oneOff) {\n      EventHandler.off(element, event.type, fn);\n    }\n    return fn.apply(element, [event]);\n  };\n}\nfunction bootstrapDelegationHandler(element, selector, fn) {\n  return function handler(event) {\n    const domElements = element.querySelectorAll(selector);\n    for (let {\n      target\n    } = event; target && target !== this; target = target.parentNode) {\n      for (const domElement of domElements) {\n        if (domElement !== target) {\n          continue;\n        }\n        hydrateObj(event, {\n          delegateTarget: target\n        });\n        if (handler.oneOff) {\n          EventHandler.off(element, event.type, selector, fn);\n        }\n        return fn.apply(target, [event]);\n      }\n    }\n  };\n}\nfunction findHandler(events, callable, delegationSelector = null) {\n  return Object.values(events).find(event => event.callable === callable && event.delegationSelector === delegationSelector);\n}\nfunction normalizeParameters(originalTypeEvent, handler, delegationFunction) {\n  const isDelegated = typeof handler === 'string';\n  // TODO: tooltip passes `false` instead of selector, so we need to check\n  const callable = isDelegated ? delegationFunction : handler || delegationFunction;\n  let typeEvent = getTypeEvent(originalTypeEvent);\n  if (!nativeEvents.has(typeEvent)) {\n    typeEvent = originalTypeEvent;\n  }\n  return [isDelegated, callable, typeEvent];\n}\nfunction addHandler(element, originalTypeEvent, handler, delegationFunction, oneOff) {\n  if (typeof originalTypeEvent !== 'string' || !element) {\n    return;\n  }\n  let [isDelegated, callable, typeEvent] = normalizeParameters(originalTypeEvent, handler, delegationFunction);\n\n  // in case of mouseenter or mouseleave wrap the handler within a function that checks for its DOM position\n  // this prevents the handler from being dispatched the same way as mouseover or mouseout does\n  if (originalTypeEvent in customEvents) {\n    const wrapFunction = fn => {\n      return function (event) {\n        if (!event.relatedTarget || event.relatedTarget !== event.delegateTarget && !event.delegateTarget.contains(event.relatedTarget)) {\n          return fn.call(this, event);\n        }\n      };\n    };\n    callable = wrapFunction(callable);\n  }\n  const events = getElementEvents(element);\n  const handlers = events[typeEvent] || (events[typeEvent] = {});\n  const previousFunction = findHandler(handlers, callable, isDelegated ? handler : null);\n  if (previousFunction) {\n    previousFunction.oneOff = previousFunction.oneOff && oneOff;\n    return;\n  }\n  const uid = makeEventUid(callable, originalTypeEvent.replace(namespaceRegex, ''));\n  const fn = isDelegated ? bootstrapDelegationHandler(element, handler, callable) : bootstrapHandler(element, callable);\n  fn.delegationSelector = isDelegated ? handler : null;\n  fn.callable = callable;\n  fn.oneOff = oneOff;\n  fn.uidEvent = uid;\n  handlers[uid] = fn;\n  element.addEventListener(typeEvent, fn, isDelegated);\n}\nfunction removeHandler(element, events, typeEvent, handler, delegationSelector) {\n  const fn = findHandler(events[typeEvent], handler, delegationSelector);\n  if (!fn) {\n    return;\n  }\n  element.removeEventListener(typeEvent, fn, Boolean(delegationSelector));\n  delete events[typeEvent][fn.uidEvent];\n}\nfunction removeNamespacedHandlers(element, events, typeEvent, namespace) {\n  const storeElementEvent = events[typeEvent] || {};\n  for (const [handlerKey, event] of Object.entries(storeElementEvent)) {\n    if (handlerKey.includes(namespace)) {\n      removeHandler(element, events, typeEvent, event.callable, event.delegationSelector);\n    }\n  }\n}\nfunction getTypeEvent(event) {\n  // allow to get the native events from namespaced events ('click.bs.button' --> 'click')\n  event = event.replace(stripNameRegex, '');\n  return customEvents[event] || event;\n}\nconst EventHandler = {\n  on(element, event, handler, delegationFunction) {\n    addHandler(element, event, handler, delegationFunction, false);\n  },\n  one(element, event, handler, delegationFunction) {\n    addHandler(element, event, handler, delegationFunction, true);\n  },\n  off(element, originalTypeEvent, handler, delegationFunction) {\n    if (typeof originalTypeEvent !== 'string' || !element) {\n      return;\n    }\n    const [isDelegated, callable, typeEvent] = normalizeParameters(originalTypeEvent, handler, delegationFunction);\n    const inNamespace = typeEvent !== originalTypeEvent;\n    const events = getElementEvents(element);\n    const storeElementEvent = events[typeEvent] || {};\n    const isNamespace = originalTypeEvent.startsWith('.');\n    if (typeof callable !== 'undefined') {\n      // Simplest case: handler is passed, remove that listener ONLY.\n      if (!Object.keys(storeElementEvent).length) {\n        return;\n      }\n      removeHandler(element, events, typeEvent, callable, isDelegated ? handler : null);\n      return;\n    }\n    if (isNamespace) {\n      for (const elementEvent of Object.keys(events)) {\n        removeNamespacedHandlers(element, events, elementEvent, originalTypeEvent.slice(1));\n      }\n    }\n    for (const [keyHandlers, event] of Object.entries(storeElementEvent)) {\n      const handlerKey = keyHandlers.replace(stripUidRegex, '');\n      if (!inNamespace || originalTypeEvent.includes(handlerKey)) {\n        removeHandler(element, events, typeEvent, event.callable, event.delegationSelector);\n      }\n    }\n  },\n  trigger(element, event, args) {\n    if (typeof event !== 'string' || !element) {\n      return null;\n    }\n    const $ = getjQuery();\n    const typeEvent = getTypeEvent(event);\n    const inNamespace = event !== typeEvent;\n    let jQueryEvent = null;\n    let bubbles = true;\n    let nativeDispatch = true;\n    let defaultPrevented = false;\n    if (inNamespace && $) {\n      jQueryEvent = $.Event(event, args);\n      $(element).trigger(jQueryEvent);\n      bubbles = !jQueryEvent.isPropagationStopped();\n      nativeDispatch = !jQueryEvent.isImmediatePropagationStopped();\n      defaultPrevented = jQueryEvent.isDefaultPrevented();\n    }\n    const evt = hydrateObj(new Event(event, {\n      bubbles,\n      cancelable: true\n    }), args);\n    if (defaultPrevented) {\n      evt.preventDefault();\n    }\n    if (nativeDispatch) {\n      element.dispatchEvent(evt);\n    }\n    if (evt.defaultPrevented && jQueryEvent) {\n      jQueryEvent.preventDefault();\n    }\n    return evt;\n  }\n};\nfunction hydrateObj(obj, meta = {}) {\n  for (const [key, value] of Object.entries(meta)) {\n    try {\n      obj[key] = value;\n    } catch (_unused) {\n      Object.defineProperty(obj, key, {\n        configurable: true,\n        get() {\n          return value;\n        }\n      });\n    }\n  }\n  return obj;\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/manipulator.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nfunction normalizeData(value) {\n  if (value === 'true') {\n    return true;\n  }\n  if (value === 'false') {\n    return false;\n  }\n  if (value === Number(value).toString()) {\n    return Number(value);\n  }\n  if (value === '' || value === 'null') {\n    return null;\n  }\n  if (typeof value !== 'string') {\n    return value;\n  }\n  try {\n    return JSON.parse(decodeURIComponent(value));\n  } catch (_unused) {\n    return value;\n  }\n}\nfunction normalizeDataKey(key) {\n  return key.replace(/[A-Z]/g, chr => `-${chr.toLowerCase()}`);\n}\nconst Manipulator = {\n  setDataAttribute(element, key, value) {\n    element.setAttribute(`data-bs-${normalizeDataKey(key)}`, value);\n  },\n  removeDataAttribute(element, key) {\n    element.removeAttribute(`data-bs-${normalizeDataKey(key)}`);\n  },\n  getDataAttributes(element) {\n    if (!element) {\n      return {};\n    }\n    const attributes = {};\n    const bsKeys = Object.keys(element.dataset).filter(key => key.startsWith('bs') && !key.startsWith('bsConfig'));\n    for (const key of bsKeys) {\n      let pureKey = key.replace(/^bs/, '');\n      pureKey = pureKey.charAt(0).toLowerCase() + pureKey.slice(1, pureKey.length);\n      attributes[pureKey] = normalizeData(element.dataset[key]);\n    }\n    return attributes;\n  },\n  getDataAttribute(element, key) {\n    return normalizeData(element.getAttribute(`data-bs-${normalizeDataKey(key)}`));\n  }\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/config.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Class definition\n */\n\nclass Config {\n  // Getters\n  static get Default() {\n    return {};\n  }\n  static get DefaultType() {\n    return {};\n  }\n  static get NAME() {\n    throw new Error('You have to implement the static method \"NAME\", for each component!');\n  }\n  _getConfig(config) {\n    config = this._mergeConfigObj(config);\n    config = this._configAfterMerge(config);\n    this._typeCheckConfig(config);\n    return config;\n  }\n  _configAfterMerge(config) {\n    return config;\n  }\n  _mergeConfigObj(config, element) {\n    const jsonConfig = isElement(element) ? Manipulator.getDataAttribute(element, 'config') : {}; // try to parse\n\n    return {\n      ...this.constructor.Default,\n      ...(typeof jsonConfig === 'object' ? jsonConfig : {}),\n      ...(isElement(element) ? Manipulator.getDataAttributes(element) : {}),\n      ...(typeof config === 'object' ? config : {})\n    };\n  }\n  _typeCheckConfig(config, configTypes = this.constructor.DefaultType) {\n    for (const [property, expectedTypes] of Object.entries(configTypes)) {\n      const value = config[property];\n      const valueType = isElement(value) ? 'element' : toType(value);\n      if (!new RegExp(expectedTypes).test(valueType)) {\n        throw new TypeError(`${this.constructor.NAME.toUpperCase()}: Option \"${property}\" provided type \"${valueType}\" but expected type \"${expectedTypes}\".`);\n      }\n    }\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap base-component.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst VERSION = '5.3.2';\n\n/**\n * Class definition\n */\n\nclass BaseComponent extends Config {\n  constructor(element, config) {\n    super();\n    element = getElement(element);\n    if (!element) {\n      return;\n    }\n    this._element = element;\n    this._config = this._getConfig(config);\n    Data.set(this._element, this.constructor.DATA_KEY, this);\n  }\n\n  // Public\n  dispose() {\n    Data.remove(this._element, this.constructor.DATA_KEY);\n    EventHandler.off(this._element, this.constructor.EVENT_KEY);\n    for (const propertyName of Object.getOwnPropertyNames(this)) {\n      this[propertyName] = null;\n    }\n  }\n  _queueCallback(callback, element, isAnimated = true) {\n    executeAfterTransition(callback, element, isAnimated);\n  }\n  _getConfig(config) {\n    config = this._mergeConfigObj(config, this._element);\n    config = this._configAfterMerge(config);\n    this._typeCheckConfig(config);\n    return config;\n  }\n\n  // Static\n  static getInstance(element) {\n    return Data.get(getElement(element), this.DATA_KEY);\n  }\n  static getOrCreateInstance(element, config = {}) {\n    return this.getInstance(element) || new this(element, typeof config === 'object' ? config : null);\n  }\n  static get VERSION() {\n    return VERSION;\n  }\n  static get DATA_KEY() {\n    return `bs.${this.NAME}`;\n  }\n  static get EVENT_KEY() {\n    return `.${this.DATA_KEY}`;\n  }\n  static eventName(name) {\n    return `${name}${this.EVENT_KEY}`;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dom/selector-engine.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nconst getSelector = element => {\n  let selector = element.getAttribute('data-bs-target');\n  if (!selector || selector === '#') {\n    let hrefAttribute = element.getAttribute('href');\n\n    // The only valid content that could double as a selector are IDs or classes,\n    // so everything starting with `#` or `.`. If a \"real\" URL is used as the selector,\n    // `document.querySelector` will rightfully complain it is invalid.\n    // See https://github.com/twbs/bootstrap/issues/32273\n    if (!hrefAttribute || !hrefAttribute.includes('#') && !hrefAttribute.startsWith('.')) {\n      return null;\n    }\n\n    // Just in case some CMS puts out a full URL with the anchor appended\n    if (hrefAttribute.includes('#') && !hrefAttribute.startsWith('#')) {\n      hrefAttribute = `#${hrefAttribute.split('#')[1]}`;\n    }\n    selector = hrefAttribute && hrefAttribute !== '#' ? parseSelector(hrefAttribute.trim()) : null;\n  }\n  return selector;\n};\nconst SelectorEngine = {\n  find(selector, element = document.documentElement) {\n    return [].concat(...Element.prototype.querySelectorAll.call(element, selector));\n  },\n  findOne(selector, element = document.documentElement) {\n    return Element.prototype.querySelector.call(element, selector);\n  },\n  children(element, selector) {\n    return [].concat(...element.children).filter(child => child.matches(selector));\n  },\n  parents(element, selector) {\n    const parents = [];\n    let ancestor = element.parentNode.closest(selector);\n    while (ancestor) {\n      parents.push(ancestor);\n      ancestor = ancestor.parentNode.closest(selector);\n    }\n    return parents;\n  },\n  prev(element, selector) {\n    let previous = element.previousElementSibling;\n    while (previous) {\n      if (previous.matches(selector)) {\n        return [previous];\n      }\n      previous = previous.previousElementSibling;\n    }\n    return [];\n  },\n  // TODO: this is now unused; remove later along with prev()\n  next(element, selector) {\n    let next = element.nextElementSibling;\n    while (next) {\n      if (next.matches(selector)) {\n        return [next];\n      }\n      next = next.nextElementSibling;\n    }\n    return [];\n  },\n  focusableChildren(element) {\n    const focusables = ['a', 'button', 'input', 'textarea', 'select', 'details', '[tabindex]', '[contenteditable=\"true\"]'].map(selector => `${selector}:not([tabindex^=\"-\"])`).join(',');\n    return this.find(focusables, element).filter(el => !isDisabled(el) && isVisible(el));\n  },\n  getSelectorFromElement(element) {\n    const selector = getSelector(element);\n    if (selector) {\n      return SelectorEngine.findOne(selector) ? selector : null;\n    }\n    return null;\n  },\n  getElementFromSelector(element) {\n    const selector = getSelector(element);\n    return selector ? SelectorEngine.findOne(selector) : null;\n  },\n  getMultipleElementsFromSelector(element) {\n    const selector = getSelector(element);\n    return selector ? SelectorEngine.find(selector) : [];\n  }\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/component-functions.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\nconst enableDismissTrigger = (component, method = 'hide') => {\n  const clickEvent = `click.dismiss${component.EVENT_KEY}`;\n  const name = component.NAME;\n  EventHandler.on(document, clickEvent, `[data-bs-dismiss=\"${name}\"]`, function (event) {\n    if (['A', 'AREA'].includes(this.tagName)) {\n      event.preventDefault();\n    }\n    if (isDisabled(this)) {\n      return;\n    }\n    const target = SelectorEngine.getElementFromSelector(this) || this.closest(`.${name}`);\n    const instance = component.getOrCreateInstance(target);\n\n    // Method argument is left, for Alert and only, as it doesn't implement the 'hide' method\n    instance[method]();\n  });\n};\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap alert.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$f = 'alert';\nconst DATA_KEY$a = 'bs.alert';\nconst EVENT_KEY$b = `.${DATA_KEY$a}`;\nconst EVENT_CLOSE = `close${EVENT_KEY$b}`;\nconst EVENT_CLOSED = `closed${EVENT_KEY$b}`;\nconst CLASS_NAME_FADE$5 = 'fade';\nconst CLASS_NAME_SHOW$8 = 'show';\n\n/**\n * Class definition\n */\n\nclass Alert extends BaseComponent {\n  // Getters\n  static get NAME() {\n    return NAME$f;\n  }\n\n  // Public\n  close() {\n    const closeEvent = EventHandler.trigger(this._element, EVENT_CLOSE);\n    if (closeEvent.defaultPrevented) {\n      return;\n    }\n    this._element.classList.remove(CLASS_NAME_SHOW$8);\n    const isAnimated = this._element.classList.contains(CLASS_NAME_FADE$5);\n    this._queueCallback(() => this._destroyElement(), this._element, isAnimated);\n  }\n\n  // Private\n  _destroyElement() {\n    this._element.remove();\n    EventHandler.trigger(this._element, EVENT_CLOSED);\n    this.dispose();\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Alert.getOrCreateInstance(this);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config](this);\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nenableDismissTrigger(Alert, 'close');\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Alert);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap button.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$e = 'button';\nconst DATA_KEY$9 = 'bs.button';\nconst EVENT_KEY$a = `.${DATA_KEY$9}`;\nconst DATA_API_KEY$6 = '.data-api';\nconst CLASS_NAME_ACTIVE$3 = 'active';\nconst SELECTOR_DATA_TOGGLE$5 = '[data-bs-toggle=\"button\"]';\nconst EVENT_CLICK_DATA_API$6 = `click${EVENT_KEY$a}${DATA_API_KEY$6}`;\n\n/**\n * Class definition\n */\n\nclass Button extends BaseComponent {\n  // Getters\n  static get NAME() {\n    return NAME$e;\n  }\n\n  // Public\n  toggle() {\n    // Toggle class and sync the `aria-pressed` attribute with the return value of the `.toggle()` method\n    this._element.setAttribute('aria-pressed', this._element.classList.toggle(CLASS_NAME_ACTIVE$3));\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Button.getOrCreateInstance(this);\n      if (config === 'toggle') {\n        data[config]();\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$6, SELECTOR_DATA_TOGGLE$5, event => {\n  event.preventDefault();\n  const button = event.target.closest(SELECTOR_DATA_TOGGLE$5);\n  const data = Button.getOrCreateInstance(button);\n  data.toggle();\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Button);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/swipe.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$d = 'swipe';\nconst EVENT_KEY$9 = '.bs.swipe';\nconst EVENT_TOUCHSTART = `touchstart${EVENT_KEY$9}`;\nconst EVENT_TOUCHMOVE = `touchmove${EVENT_KEY$9}`;\nconst EVENT_TOUCHEND = `touchend${EVENT_KEY$9}`;\nconst EVENT_POINTERDOWN = `pointerdown${EVENT_KEY$9}`;\nconst EVENT_POINTERUP = `pointerup${EVENT_KEY$9}`;\nconst POINTER_TYPE_TOUCH = 'touch';\nconst POINTER_TYPE_PEN = 'pen';\nconst CLASS_NAME_POINTER_EVENT = 'pointer-event';\nconst SWIPE_THRESHOLD = 40;\nconst Default$c = {\n  endCallback: null,\n  leftCallback: null,\n  rightCallback: null\n};\nconst DefaultType$c = {\n  endCallback: '(function|null)',\n  leftCallback: '(function|null)',\n  rightCallback: '(function|null)'\n};\n\n/**\n * Class definition\n */\n\nclass Swipe extends Config {\n  constructor(element, config) {\n    super();\n    this._element = element;\n    if (!element || !Swipe.isSupported()) {\n      return;\n    }\n    this._config = this._getConfig(config);\n    this._deltaX = 0;\n    this._supportPointerEvents = Boolean(window.PointerEvent);\n    this._initEvents();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$c;\n  }\n  static get DefaultType() {\n    return DefaultType$c;\n  }\n  static get NAME() {\n    return NAME$d;\n  }\n\n  // Public\n  dispose() {\n    EventHandler.off(this._element, EVENT_KEY$9);\n  }\n\n  // Private\n  _start(event) {\n    if (!this._supportPointerEvents) {\n      this._deltaX = event.touches[0].clientX;\n      return;\n    }\n    if (this._eventIsPointerPenTouch(event)) {\n      this._deltaX = event.clientX;\n    }\n  }\n  _end(event) {\n    if (this._eventIsPointerPenTouch(event)) {\n      this._deltaX = event.clientX - this._deltaX;\n    }\n    this._handleSwipe();\n    execute(this._config.endCallback);\n  }\n  _move(event) {\n    this._deltaX = event.touches && event.touches.length > 1 ? 0 : event.touches[0].clientX - this._deltaX;\n  }\n  _handleSwipe() {\n    const absDeltaX = Math.abs(this._deltaX);\n    if (absDeltaX <= SWIPE_THRESHOLD) {\n      return;\n    }\n    const direction = absDeltaX / this._deltaX;\n    this._deltaX = 0;\n    if (!direction) {\n      return;\n    }\n    execute(direction > 0 ? this._config.rightCallback : this._config.leftCallback);\n  }\n  _initEvents() {\n    if (this._supportPointerEvents) {\n      EventHandler.on(this._element, EVENT_POINTERDOWN, event => this._start(event));\n      EventHandler.on(this._element, EVENT_POINTERUP, event => this._end(event));\n      this._element.classList.add(CLASS_NAME_POINTER_EVENT);\n    } else {\n      EventHandler.on(this._element, EVENT_TOUCHSTART, event => this._start(event));\n      EventHandler.on(this._element, EVENT_TOUCHMOVE, event => this._move(event));\n      EventHandler.on(this._element, EVENT_TOUCHEND, event => this._end(event));\n    }\n  }\n  _eventIsPointerPenTouch(event) {\n    return this._supportPointerEvents && (event.pointerType === POINTER_TYPE_PEN || event.pointerType === POINTER_TYPE_TOUCH);\n  }\n\n  // Static\n  static isSupported() {\n    return 'ontouchstart' in document.documentElement || navigator.maxTouchPoints > 0;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap carousel.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$c = 'carousel';\nconst DATA_KEY$8 = 'bs.carousel';\nconst EVENT_KEY$8 = `.${DATA_KEY$8}`;\nconst DATA_API_KEY$5 = '.data-api';\nconst ARROW_LEFT_KEY$1 = 'ArrowLeft';\nconst ARROW_RIGHT_KEY$1 = 'ArrowRight';\nconst TOUCHEVENT_COMPAT_WAIT = 500; // Time for mouse compat events to fire after touch\n\nconst ORDER_NEXT = 'next';\nconst ORDER_PREV = 'prev';\nconst DIRECTION_LEFT = 'left';\nconst DIRECTION_RIGHT = 'right';\nconst EVENT_SLIDE = `slide${EVENT_KEY$8}`;\nconst EVENT_SLID = `slid${EVENT_KEY$8}`;\nconst EVENT_KEYDOWN$1 = `keydown${EVENT_KEY$8}`;\nconst EVENT_MOUSEENTER$1 = `mouseenter${EVENT_KEY$8}`;\nconst EVENT_MOUSELEAVE$1 = `mouseleave${EVENT_KEY$8}`;\nconst EVENT_DRAG_START = `dragstart${EVENT_KEY$8}`;\nconst EVENT_LOAD_DATA_API$3 = `load${EVENT_KEY$8}${DATA_API_KEY$5}`;\nconst EVENT_CLICK_DATA_API$5 = `click${EVENT_KEY$8}${DATA_API_KEY$5}`;\nconst CLASS_NAME_CAROUSEL = 'carousel';\nconst CLASS_NAME_ACTIVE$2 = 'active';\nconst CLASS_NAME_SLIDE = 'slide';\nconst CLASS_NAME_END = 'carousel-item-end';\nconst CLASS_NAME_START = 'carousel-item-start';\nconst CLASS_NAME_NEXT = 'carousel-item-next';\nconst CLASS_NAME_PREV = 'carousel-item-prev';\nconst SELECTOR_ACTIVE = '.active';\nconst SELECTOR_ITEM = '.carousel-item';\nconst SELECTOR_ACTIVE_ITEM = SELECTOR_ACTIVE + SELECTOR_ITEM;\nconst SELECTOR_ITEM_IMG = '.carousel-item img';\nconst SELECTOR_INDICATORS = '.carousel-indicators';\nconst SELECTOR_DATA_SLIDE = '[data-bs-slide], [data-bs-slide-to]';\nconst SELECTOR_DATA_RIDE = '[data-bs-ride=\"carousel\"]';\nconst KEY_TO_DIRECTION = {\n  [ARROW_LEFT_KEY$1]: DIRECTION_RIGHT,\n  [ARROW_RIGHT_KEY$1]: DIRECTION_LEFT\n};\nconst Default$b = {\n  interval: 5000,\n  keyboard: true,\n  pause: 'hover',\n  ride: false,\n  touch: true,\n  wrap: true\n};\nconst DefaultType$b = {\n  interval: '(number|boolean)',\n  // TODO:v6 remove boolean support\n  keyboard: 'boolean',\n  pause: '(string|boolean)',\n  ride: '(boolean|string)',\n  touch: 'boolean',\n  wrap: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Carousel extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._interval = null;\n    this._activeElement = null;\n    this._isSliding = false;\n    this.touchTimeout = null;\n    this._swipeHelper = null;\n    this._indicatorsElement = SelectorEngine.findOne(SELECTOR_INDICATORS, this._element);\n    this._addEventListeners();\n    if (this._config.ride === CLASS_NAME_CAROUSEL) {\n      this.cycle();\n    }\n  }\n\n  // Getters\n  static get Default() {\n    return Default$b;\n  }\n  static get DefaultType() {\n    return DefaultType$b;\n  }\n  static get NAME() {\n    return NAME$c;\n  }\n\n  // Public\n  next() {\n    this._slide(ORDER_NEXT);\n  }\n  nextWhenVisible() {\n    // FIXME TODO use `document.visibilityState`\n    // Don't call next when the page isn't visible\n    // or the carousel or its parent isn't visible\n    if (!document.hidden && isVisible(this._element)) {\n      this.next();\n    }\n  }\n  prev() {\n    this._slide(ORDER_PREV);\n  }\n  pause() {\n    if (this._isSliding) {\n      triggerTransitionEnd(this._element);\n    }\n    this._clearInterval();\n  }\n  cycle() {\n    this._clearInterval();\n    this._updateInterval();\n    this._interval = setInterval(() => this.nextWhenVisible(), this._config.interval);\n  }\n  _maybeEnableCycle() {\n    if (!this._config.ride) {\n      return;\n    }\n    if (this._isSliding) {\n      EventHandler.one(this._element, EVENT_SLID, () => this.cycle());\n      return;\n    }\n    this.cycle();\n  }\n  to(index) {\n    const items = this._getItems();\n    if (index > items.length - 1 || index < 0) {\n      return;\n    }\n    if (this._isSliding) {\n      EventHandler.one(this._element, EVENT_SLID, () => this.to(index));\n      return;\n    }\n    const activeIndex = this._getItemIndex(this._getActive());\n    if (activeIndex === index) {\n      return;\n    }\n    const order = index > activeIndex ? ORDER_NEXT : ORDER_PREV;\n    this._slide(order, items[index]);\n  }\n  dispose() {\n    if (this._swipeHelper) {\n      this._swipeHelper.dispose();\n    }\n    super.dispose();\n  }\n\n  // Private\n  _configAfterMerge(config) {\n    config.defaultInterval = config.interval;\n    return config;\n  }\n  _addEventListeners() {\n    if (this._config.keyboard) {\n      EventHandler.on(this._element, EVENT_KEYDOWN$1, event => this._keydown(event));\n    }\n    if (this._config.pause === 'hover') {\n      EventHandler.on(this._element, EVENT_MOUSEENTER$1, () => this.pause());\n      EventHandler.on(this._element, EVENT_MOUSELEAVE$1, () => this._maybeEnableCycle());\n    }\n    if (this._config.touch && Swipe.isSupported()) {\n      this._addTouchEventListeners();\n    }\n  }\n  _addTouchEventListeners() {\n    for (const img of SelectorEngine.find(SELECTOR_ITEM_IMG, this._element)) {\n      EventHandler.on(img, EVENT_DRAG_START, event => event.preventDefault());\n    }\n    const endCallBack = () => {\n      if (this._config.pause !== 'hover') {\n        return;\n      }\n\n      // If it's a touch-enabled device, mouseenter/leave are fired as\n      // part of the mouse compatibility events on first tap - the carousel\n      // would stop cycling until user tapped out of it;\n      // here, we listen for touchend, explicitly pause the carousel\n      // (as if it's the second time we tap on it, mouseenter compat event\n      // is NOT fired) and after a timeout (to allow for mouse compatibility\n      // events to fire) we explicitly restart cycling\n\n      this.pause();\n      if (this.touchTimeout) {\n        clearTimeout(this.touchTimeout);\n      }\n      this.touchTimeout = setTimeout(() => this._maybeEnableCycle(), TOUCHEVENT_COMPAT_WAIT + this._config.interval);\n    };\n    const swipeConfig = {\n      leftCallback: () => this._slide(this._directionToOrder(DIRECTION_LEFT)),\n      rightCallback: () => this._slide(this._directionToOrder(DIRECTION_RIGHT)),\n      endCallback: endCallBack\n    };\n    this._swipeHelper = new Swipe(this._element, swipeConfig);\n  }\n  _keydown(event) {\n    if (/input|textarea/i.test(event.target.tagName)) {\n      return;\n    }\n    const direction = KEY_TO_DIRECTION[event.key];\n    if (direction) {\n      event.preventDefault();\n      this._slide(this._directionToOrder(direction));\n    }\n  }\n  _getItemIndex(element) {\n    return this._getItems().indexOf(element);\n  }\n  _setActiveIndicatorElement(index) {\n    if (!this._indicatorsElement) {\n      return;\n    }\n    const activeIndicator = SelectorEngine.findOne(SELECTOR_ACTIVE, this._indicatorsElement);\n    activeIndicator.classList.remove(CLASS_NAME_ACTIVE$2);\n    activeIndicator.removeAttribute('aria-current');\n    const newActiveIndicator = SelectorEngine.findOne(`[data-bs-slide-to=\"${index}\"]`, this._indicatorsElement);\n    if (newActiveIndicator) {\n      newActiveIndicator.classList.add(CLASS_NAME_ACTIVE$2);\n      newActiveIndicator.setAttribute('aria-current', 'true');\n    }\n  }\n  _updateInterval() {\n    const element = this._activeElement || this._getActive();\n    if (!element) {\n      return;\n    }\n    const elementInterval = Number.parseInt(element.getAttribute('data-bs-interval'), 10);\n    this._config.interval = elementInterval || this._config.defaultInterval;\n  }\n  _slide(order, element = null) {\n    if (this._isSliding) {\n      return;\n    }\n    const activeElement = this._getActive();\n    const isNext = order === ORDER_NEXT;\n    const nextElement = element || getNextActiveElement(this._getItems(), activeElement, isNext, this._config.wrap);\n    if (nextElement === activeElement) {\n      return;\n    }\n    const nextElementIndex = this._getItemIndex(nextElement);\n    const triggerEvent = eventName => {\n      return EventHandler.trigger(this._element, eventName, {\n        relatedTarget: nextElement,\n        direction: this._orderToDirection(order),\n        from: this._getItemIndex(activeElement),\n        to: nextElementIndex\n      });\n    };\n    const slideEvent = triggerEvent(EVENT_SLIDE);\n    if (slideEvent.defaultPrevented) {\n      return;\n    }\n    if (!activeElement || !nextElement) {\n      // Some weirdness is happening, so we bail\n      // TODO: change tests that use empty divs to avoid this check\n      return;\n    }\n    const isCycling = Boolean(this._interval);\n    this.pause();\n    this._isSliding = true;\n    this._setActiveIndicatorElement(nextElementIndex);\n    this._activeElement = nextElement;\n    const directionalClassName = isNext ? CLASS_NAME_START : CLASS_NAME_END;\n    const orderClassName = isNext ? CLASS_NAME_NEXT : CLASS_NAME_PREV;\n    nextElement.classList.add(orderClassName);\n    reflow(nextElement);\n    activeElement.classList.add(directionalClassName);\n    nextElement.classList.add(directionalClassName);\n    const completeCallBack = () => {\n      nextElement.classList.remove(directionalClassName, orderClassName);\n      nextElement.classList.add(CLASS_NAME_ACTIVE$2);\n      activeElement.classList.remove(CLASS_NAME_ACTIVE$2, orderClassName, directionalClassName);\n      this._isSliding = false;\n      triggerEvent(EVENT_SLID);\n    };\n    this._queueCallback(completeCallBack, activeElement, this._isAnimated());\n    if (isCycling) {\n      this.cycle();\n    }\n  }\n  _isAnimated() {\n    return this._element.classList.contains(CLASS_NAME_SLIDE);\n  }\n  _getActive() {\n    return SelectorEngine.findOne(SELECTOR_ACTIVE_ITEM, this._element);\n  }\n  _getItems() {\n    return SelectorEngine.find(SELECTOR_ITEM, this._element);\n  }\n  _clearInterval() {\n    if (this._interval) {\n      clearInterval(this._interval);\n      this._interval = null;\n    }\n  }\n  _directionToOrder(direction) {\n    if (isRTL()) {\n      return direction === DIRECTION_LEFT ? ORDER_PREV : ORDER_NEXT;\n    }\n    return direction === DIRECTION_LEFT ? ORDER_NEXT : ORDER_PREV;\n  }\n  _orderToDirection(order) {\n    if (isRTL()) {\n      return order === ORDER_PREV ? DIRECTION_LEFT : DIRECTION_RIGHT;\n    }\n    return order === ORDER_PREV ? DIRECTION_RIGHT : DIRECTION_LEFT;\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Carousel.getOrCreateInstance(this, config);\n      if (typeof config === 'number') {\n        data.to(config);\n        return;\n      }\n      if (typeof config === 'string') {\n        if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n        data[config]();\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$5, SELECTOR_DATA_SLIDE, function (event) {\n  const target = SelectorEngine.getElementFromSelector(this);\n  if (!target || !target.classList.contains(CLASS_NAME_CAROUSEL)) {\n    return;\n  }\n  event.preventDefault();\n  const carousel = Carousel.getOrCreateInstance(target);\n  const slideIndex = this.getAttribute('data-bs-slide-to');\n  if (slideIndex) {\n    carousel.to(slideIndex);\n    carousel._maybeEnableCycle();\n    return;\n  }\n  if (Manipulator.getDataAttribute(this, 'slide') === 'next') {\n    carousel.next();\n    carousel._maybeEnableCycle();\n    return;\n  }\n  carousel.prev();\n  carousel._maybeEnableCycle();\n});\nEventHandler.on(window, EVENT_LOAD_DATA_API$3, () => {\n  const carousels = SelectorEngine.find(SELECTOR_DATA_RIDE);\n  for (const carousel of carousels) {\n    Carousel.getOrCreateInstance(carousel);\n  }\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Carousel);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap collapse.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$b = 'collapse';\nconst DATA_KEY$7 = 'bs.collapse';\nconst EVENT_KEY$7 = `.${DATA_KEY$7}`;\nconst DATA_API_KEY$4 = '.data-api';\nconst EVENT_SHOW$6 = `show${EVENT_KEY$7}`;\nconst EVENT_SHOWN$6 = `shown${EVENT_KEY$7}`;\nconst EVENT_HIDE$6 = `hide${EVENT_KEY$7}`;\nconst EVENT_HIDDEN$6 = `hidden${EVENT_KEY$7}`;\nconst EVENT_CLICK_DATA_API$4 = `click${EVENT_KEY$7}${DATA_API_KEY$4}`;\nconst CLASS_NAME_SHOW$7 = 'show';\nconst CLASS_NAME_COLLAPSE = 'collapse';\nconst CLASS_NAME_COLLAPSING = 'collapsing';\nconst CLASS_NAME_COLLAPSED = 'collapsed';\nconst CLASS_NAME_DEEPER_CHILDREN = `:scope .${CLASS_NAME_COLLAPSE} .${CLASS_NAME_COLLAPSE}`;\nconst CLASS_NAME_HORIZONTAL = 'collapse-horizontal';\nconst WIDTH = 'width';\nconst HEIGHT = 'height';\nconst SELECTOR_ACTIVES = '.collapse.show, .collapse.collapsing';\nconst SELECTOR_DATA_TOGGLE$4 = '[data-bs-toggle=\"collapse\"]';\nconst Default$a = {\n  parent: null,\n  toggle: true\n};\nconst DefaultType$a = {\n  parent: '(null|element)',\n  toggle: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Collapse extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._isTransitioning = false;\n    this._triggerArray = [];\n    const toggleList = SelectorEngine.find(SELECTOR_DATA_TOGGLE$4);\n    for (const elem of toggleList) {\n      const selector = SelectorEngine.getSelectorFromElement(elem);\n      const filterElement = SelectorEngine.find(selector).filter(foundElement => foundElement === this._element);\n      if (selector !== null && filterElement.length) {\n        this._triggerArray.push(elem);\n      }\n    }\n    this._initializeChildren();\n    if (!this._config.parent) {\n      this._addAriaAndCollapsedClass(this._triggerArray, this._isShown());\n    }\n    if (this._config.toggle) {\n      this.toggle();\n    }\n  }\n\n  // Getters\n  static get Default() {\n    return Default$a;\n  }\n  static get DefaultType() {\n    return DefaultType$a;\n  }\n  static get NAME() {\n    return NAME$b;\n  }\n\n  // Public\n  toggle() {\n    if (this._isShown()) {\n      this.hide();\n    } else {\n      this.show();\n    }\n  }\n  show() {\n    if (this._isTransitioning || this._isShown()) {\n      return;\n    }\n    let activeChildren = [];\n\n    // find active children\n    if (this._config.parent) {\n      activeChildren = this._getFirstLevelChildren(SELECTOR_ACTIVES).filter(element => element !== this._element).map(element => Collapse.getOrCreateInstance(element, {\n        toggle: false\n      }));\n    }\n    if (activeChildren.length && activeChildren[0]._isTransitioning) {\n      return;\n    }\n    const startEvent = EventHandler.trigger(this._element, EVENT_SHOW$6);\n    if (startEvent.defaultPrevented) {\n      return;\n    }\n    for (const activeInstance of activeChildren) {\n      activeInstance.hide();\n    }\n    const dimension = this._getDimension();\n    this._element.classList.remove(CLASS_NAME_COLLAPSE);\n    this._element.classList.add(CLASS_NAME_COLLAPSING);\n    this._element.style[dimension] = 0;\n    this._addAriaAndCollapsedClass(this._triggerArray, true);\n    this._isTransitioning = true;\n    const complete = () => {\n      this._isTransitioning = false;\n      this._element.classList.remove(CLASS_NAME_COLLAPSING);\n      this._element.classList.add(CLASS_NAME_COLLAPSE, CLASS_NAME_SHOW$7);\n      this._element.style[dimension] = '';\n      EventHandler.trigger(this._element, EVENT_SHOWN$6);\n    };\n    const capitalizedDimension = dimension[0].toUpperCase() + dimension.slice(1);\n    const scrollSize = `scroll${capitalizedDimension}`;\n    this._queueCallback(complete, this._element, true);\n    this._element.style[dimension] = `${this._element[scrollSize]}px`;\n  }\n  hide() {\n    if (this._isTransitioning || !this._isShown()) {\n      return;\n    }\n    const startEvent = EventHandler.trigger(this._element, EVENT_HIDE$6);\n    if (startEvent.defaultPrevented) {\n      return;\n    }\n    const dimension = this._getDimension();\n    this._element.style[dimension] = `${this._element.getBoundingClientRect()[dimension]}px`;\n    reflow(this._element);\n    this._element.classList.add(CLASS_NAME_COLLAPSING);\n    this._element.classList.remove(CLASS_NAME_COLLAPSE, CLASS_NAME_SHOW$7);\n    for (const trigger of this._triggerArray) {\n      const element = SelectorEngine.getElementFromSelector(trigger);\n      if (element && !this._isShown(element)) {\n        this._addAriaAndCollapsedClass([trigger], false);\n      }\n    }\n    this._isTransitioning = true;\n    const complete = () => {\n      this._isTransitioning = false;\n      this._element.classList.remove(CLASS_NAME_COLLAPSING);\n      this._element.classList.add(CLASS_NAME_COLLAPSE);\n      EventHandler.trigger(this._element, EVENT_HIDDEN$6);\n    };\n    this._element.style[dimension] = '';\n    this._queueCallback(complete, this._element, true);\n  }\n  _isShown(element = this._element) {\n    return element.classList.contains(CLASS_NAME_SHOW$7);\n  }\n\n  // Private\n  _configAfterMerge(config) {\n    config.toggle = Boolean(config.toggle); // Coerce string values\n    config.parent = getElement(config.parent);\n    return config;\n  }\n  _getDimension() {\n    return this._element.classList.contains(CLASS_NAME_HORIZONTAL) ? WIDTH : HEIGHT;\n  }\n  _initializeChildren() {\n    if (!this._config.parent) {\n      return;\n    }\n    const children = this._getFirstLevelChildren(SELECTOR_DATA_TOGGLE$4);\n    for (const element of children) {\n      const selected = SelectorEngine.getElementFromSelector(element);\n      if (selected) {\n        this._addAriaAndCollapsedClass([element], this._isShown(selected));\n      }\n    }\n  }\n  _getFirstLevelChildren(selector) {\n    const children = SelectorEngine.find(CLASS_NAME_DEEPER_CHILDREN, this._config.parent);\n    // remove children if greater depth\n    return SelectorEngine.find(selector, this._config.parent).filter(element => !children.includes(element));\n  }\n  _addAriaAndCollapsedClass(triggerArray, isOpen) {\n    if (!triggerArray.length) {\n      return;\n    }\n    for (const element of triggerArray) {\n      element.classList.toggle(CLASS_NAME_COLLAPSED, !isOpen);\n      element.setAttribute('aria-expanded', isOpen);\n    }\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    const _config = {};\n    if (typeof config === 'string' && /show|hide/.test(config)) {\n      _config.toggle = false;\n    }\n    return this.each(function () {\n      const data = Collapse.getOrCreateInstance(this, _config);\n      if (typeof config === 'string') {\n        if (typeof data[config] === 'undefined') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n        data[config]();\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$4, SELECTOR_DATA_TOGGLE$4, function (event) {\n  // preventDefault only for <a> elements (which change the URL) not inside the collapsible element\n  if (event.target.tagName === 'A' || event.delegateTarget && event.delegateTarget.tagName === 'A') {\n    event.preventDefault();\n  }\n  for (const element of SelectorEngine.getMultipleElementsFromSelector(this)) {\n    Collapse.getOrCreateInstance(element, {\n      toggle: false\n    }).toggle();\n  }\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Collapse);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap dropdown.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$a = 'dropdown';\nconst DATA_KEY$6 = 'bs.dropdown';\nconst EVENT_KEY$6 = `.${DATA_KEY$6}`;\nconst DATA_API_KEY$3 = '.data-api';\nconst ESCAPE_KEY$2 = 'Escape';\nconst TAB_KEY$1 = 'Tab';\nconst ARROW_UP_KEY$1 = 'ArrowUp';\nconst ARROW_DOWN_KEY$1 = 'ArrowDown';\nconst RIGHT_MOUSE_BUTTON = 2; // MouseEvent.button value for the secondary button, usually the right button\n\nconst EVENT_HIDE$5 = `hide${EVENT_KEY$6}`;\nconst EVENT_HIDDEN$5 = `hidden${EVENT_KEY$6}`;\nconst EVENT_SHOW$5 = `show${EVENT_KEY$6}`;\nconst EVENT_SHOWN$5 = `shown${EVENT_KEY$6}`;\nconst EVENT_CLICK_DATA_API$3 = `click${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst EVENT_KEYDOWN_DATA_API = `keydown${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst EVENT_KEYUP_DATA_API = `keyup${EVENT_KEY$6}${DATA_API_KEY$3}`;\nconst CLASS_NAME_SHOW$6 = 'show';\nconst CLASS_NAME_DROPUP = 'dropup';\nconst CLASS_NAME_DROPEND = 'dropend';\nconst CLASS_NAME_DROPSTART = 'dropstart';\nconst CLASS_NAME_DROPUP_CENTER = 'dropup-center';\nconst CLASS_NAME_DROPDOWN_CENTER = 'dropdown-center';\nconst SELECTOR_DATA_TOGGLE$3 = '[data-bs-toggle=\"dropdown\"]:not(.disabled):not(:disabled)';\nconst SELECTOR_DATA_TOGGLE_SHOWN = `${SELECTOR_DATA_TOGGLE$3}.${CLASS_NAME_SHOW$6}`;\nconst SELECTOR_MENU = '.dropdown-menu';\nconst SELECTOR_NAVBAR = '.navbar';\nconst SELECTOR_NAVBAR_NAV = '.navbar-nav';\nconst SELECTOR_VISIBLE_ITEMS = '.dropdown-menu .dropdown-item:not(.disabled):not(:disabled)';\nconst PLACEMENT_TOP = isRTL() ? 'top-end' : 'top-start';\nconst PLACEMENT_TOPEND = isRTL() ? 'top-start' : 'top-end';\nconst PLACEMENT_BOTTOM = isRTL() ? 'bottom-end' : 'bottom-start';\nconst PLACEMENT_BOTTOMEND = isRTL() ? 'bottom-start' : 'bottom-end';\nconst PLACEMENT_RIGHT = isRTL() ? 'left-start' : 'right-start';\nconst PLACEMENT_LEFT = isRTL() ? 'right-start' : 'left-start';\nconst PLACEMENT_TOPCENTER = 'top';\nconst PLACEMENT_BOTTOMCENTER = 'bottom';\nconst Default$9 = {\n  autoClose: true,\n  boundary: 'clippingParents',\n  display: 'dynamic',\n  offset: [0, 2],\n  popperConfig: null,\n  reference: 'toggle'\n};\nconst DefaultType$9 = {\n  autoClose: '(boolean|string)',\n  boundary: '(string|element)',\n  display: 'string',\n  offset: '(array|string|function)',\n  popperConfig: '(null|object|function)',\n  reference: '(string|element|object)'\n};\n\n/**\n * Class definition\n */\n\nclass Dropdown extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._popper = null;\n    this._parent = this._element.parentNode; // dropdown wrapper\n    // TODO: v6 revert #37011 & change markup https://getbootstrap.com/docs/5.3/forms/input-group/\n    this._menu = SelectorEngine.next(this._element, SELECTOR_MENU)[0] || SelectorEngine.prev(this._element, SELECTOR_MENU)[0] || SelectorEngine.findOne(SELECTOR_MENU, this._parent);\n    this._inNavbar = this._detectNavbar();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$9;\n  }\n  static get DefaultType() {\n    return DefaultType$9;\n  }\n  static get NAME() {\n    return NAME$a;\n  }\n\n  // Public\n  toggle() {\n    return this._isShown() ? this.hide() : this.show();\n  }\n  show() {\n    if (isDisabled(this._element) || this._isShown()) {\n      return;\n    }\n    const relatedTarget = {\n      relatedTarget: this._element\n    };\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$5, relatedTarget);\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._createPopper();\n\n    // If this is a touch-enabled device we add extra\n    // empty mouseover listeners to the body's immediate children;\n    // only needed because of broken event delegation on iOS\n    // https://www.quirksmode.org/blog/archives/2014/02/mouse_event_bub.html\n    if ('ontouchstart' in document.documentElement && !this._parent.closest(SELECTOR_NAVBAR_NAV)) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.on(element, 'mouseover', noop);\n      }\n    }\n    this._element.focus();\n    this._element.setAttribute('aria-expanded', true);\n    this._menu.classList.add(CLASS_NAME_SHOW$6);\n    this._element.classList.add(CLASS_NAME_SHOW$6);\n    EventHandler.trigger(this._element, EVENT_SHOWN$5, relatedTarget);\n  }\n  hide() {\n    if (isDisabled(this._element) || !this._isShown()) {\n      return;\n    }\n    const relatedTarget = {\n      relatedTarget: this._element\n    };\n    this._completeHide(relatedTarget);\n  }\n  dispose() {\n    if (this._popper) {\n      this._popper.destroy();\n    }\n    super.dispose();\n  }\n  update() {\n    this._inNavbar = this._detectNavbar();\n    if (this._popper) {\n      this._popper.update();\n    }\n  }\n\n  // Private\n  _completeHide(relatedTarget) {\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$5, relatedTarget);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n\n    // If this is a touch-enabled device we remove the extra\n    // empty mouseover listeners we added for iOS support\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.off(element, 'mouseover', noop);\n      }\n    }\n    if (this._popper) {\n      this._popper.destroy();\n    }\n    this._menu.classList.remove(CLASS_NAME_SHOW$6);\n    this._element.classList.remove(CLASS_NAME_SHOW$6);\n    this._element.setAttribute('aria-expanded', 'false');\n    Manipulator.removeDataAttribute(this._menu, 'popper');\n    EventHandler.trigger(this._element, EVENT_HIDDEN$5, relatedTarget);\n  }\n  _getConfig(config) {\n    config = super._getConfig(config);\n    if (typeof config.reference === 'object' && !isElement(config.reference) && typeof config.reference.getBoundingClientRect !== 'function') {\n      // Popper virtual elements require a getBoundingClientRect method\n      throw new TypeError(`${NAME$a.toUpperCase()}: Option \"reference\" provided type \"object\" without a required \"getBoundingClientRect\" method.`);\n    }\n    return config;\n  }\n  _createPopper() {\n    if (typeof Popper === 'undefined') {\n      throw new TypeError('Bootstrap\\'s dropdowns require Popper (https://popper.js.org)');\n    }\n    let referenceElement = this._element;\n    if (this._config.reference === 'parent') {\n      referenceElement = this._parent;\n    } else if (isElement(this._config.reference)) {\n      referenceElement = getElement(this._config.reference);\n    } else if (typeof this._config.reference === 'object') {\n      referenceElement = this._config.reference;\n    }\n    const popperConfig = this._getPopperConfig();\n    this._popper = Popper.createPopper(referenceElement, this._menu, popperConfig);\n  }\n  _isShown() {\n    return this._menu.classList.contains(CLASS_NAME_SHOW$6);\n  }\n  _getPlacement() {\n    const parentDropdown = this._parent;\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPEND)) {\n      return PLACEMENT_RIGHT;\n    }\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPSTART)) {\n      return PLACEMENT_LEFT;\n    }\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPUP_CENTER)) {\n      return PLACEMENT_TOPCENTER;\n    }\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPDOWN_CENTER)) {\n      return PLACEMENT_BOTTOMCENTER;\n    }\n\n    // We need to trim the value because custom properties can also include spaces\n    const isEnd = getComputedStyle(this._menu).getPropertyValue('--bs-position').trim() === 'end';\n    if (parentDropdown.classList.contains(CLASS_NAME_DROPUP)) {\n      return isEnd ? PLACEMENT_TOPEND : PLACEMENT_TOP;\n    }\n    return isEnd ? PLACEMENT_BOTTOMEND : PLACEMENT_BOTTOM;\n  }\n  _detectNavbar() {\n    return this._element.closest(SELECTOR_NAVBAR) !== null;\n  }\n  _getOffset() {\n    const {\n      offset\n    } = this._config;\n    if (typeof offset === 'string') {\n      return offset.split(',').map(value => Number.parseInt(value, 10));\n    }\n    if (typeof offset === 'function') {\n      return popperData => offset(popperData, this._element);\n    }\n    return offset;\n  }\n  _getPopperConfig() {\n    const defaultBsPopperConfig = {\n      placement: this._getPlacement(),\n      modifiers: [{\n        name: 'preventOverflow',\n        options: {\n          boundary: this._config.boundary\n        }\n      }, {\n        name: 'offset',\n        options: {\n          offset: this._getOffset()\n        }\n      }]\n    };\n\n    // Disable Popper if we have a static display or Dropdown is in Navbar\n    if (this._inNavbar || this._config.display === 'static') {\n      Manipulator.setDataAttribute(this._menu, 'popper', 'static'); // TODO: v6 remove\n      defaultBsPopperConfig.modifiers = [{\n        name: 'applyStyles',\n        enabled: false\n      }];\n    }\n    return {\n      ...defaultBsPopperConfig,\n      ...execute(this._config.popperConfig, [defaultBsPopperConfig])\n    };\n  }\n  _selectMenuItem({\n    key,\n    target\n  }) {\n    const items = SelectorEngine.find(SELECTOR_VISIBLE_ITEMS, this._menu).filter(element => isVisible(element));\n    if (!items.length) {\n      return;\n    }\n\n    // if target isn't included in items (e.g. when expanding the dropdown)\n    // allow cycling to get the last item in case key equals ARROW_UP_KEY\n    getNextActiveElement(items, target, key === ARROW_DOWN_KEY$1, !items.includes(target)).focus();\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Dropdown.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n  static clearMenus(event) {\n    if (event.button === RIGHT_MOUSE_BUTTON || event.type === 'keyup' && event.key !== TAB_KEY$1) {\n      return;\n    }\n    const openToggles = SelectorEngine.find(SELECTOR_DATA_TOGGLE_SHOWN);\n    for (const toggle of openToggles) {\n      const context = Dropdown.getInstance(toggle);\n      if (!context || context._config.autoClose === false) {\n        continue;\n      }\n      const composedPath = event.composedPath();\n      const isMenuTarget = composedPath.includes(context._menu);\n      if (composedPath.includes(context._element) || context._config.autoClose === 'inside' && !isMenuTarget || context._config.autoClose === 'outside' && isMenuTarget) {\n        continue;\n      }\n\n      // Tab navigation through the dropdown menu or events from contained inputs shouldn't close the menu\n      if (context._menu.contains(event.target) && (event.type === 'keyup' && event.key === TAB_KEY$1 || /input|select|option|textarea|form/i.test(event.target.tagName))) {\n        continue;\n      }\n      const relatedTarget = {\n        relatedTarget: context._element\n      };\n      if (event.type === 'click') {\n        relatedTarget.clickEvent = event;\n      }\n      context._completeHide(relatedTarget);\n    }\n  }\n  static dataApiKeydownHandler(event) {\n    // If not an UP | DOWN | ESCAPE key => not a dropdown command\n    // If input/textarea && if key is other than ESCAPE => not a dropdown command\n\n    const isInput = /input|textarea/i.test(event.target.tagName);\n    const isEscapeEvent = event.key === ESCAPE_KEY$2;\n    const isUpOrDownEvent = [ARROW_UP_KEY$1, ARROW_DOWN_KEY$1].includes(event.key);\n    if (!isUpOrDownEvent && !isEscapeEvent) {\n      return;\n    }\n    if (isInput && !isEscapeEvent) {\n      return;\n    }\n    event.preventDefault();\n\n    // TODO: v6 revert #37011 & change markup https://getbootstrap.com/docs/5.3/forms/input-group/\n    const getToggleButton = this.matches(SELECTOR_DATA_TOGGLE$3) ? this : SelectorEngine.prev(this, SELECTOR_DATA_TOGGLE$3)[0] || SelectorEngine.next(this, SELECTOR_DATA_TOGGLE$3)[0] || SelectorEngine.findOne(SELECTOR_DATA_TOGGLE$3, event.delegateTarget.parentNode);\n    const instance = Dropdown.getOrCreateInstance(getToggleButton);\n    if (isUpOrDownEvent) {\n      event.stopPropagation();\n      instance.show();\n      instance._selectMenuItem(event);\n      return;\n    }\n    if (instance._isShown()) {\n      // else is escape and we check if it is shown\n      event.stopPropagation();\n      instance.hide();\n      getToggleButton.focus();\n    }\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_KEYDOWN_DATA_API, SELECTOR_DATA_TOGGLE$3, Dropdown.dataApiKeydownHandler);\nEventHandler.on(document, EVENT_KEYDOWN_DATA_API, SELECTOR_MENU, Dropdown.dataApiKeydownHandler);\nEventHandler.on(document, EVENT_CLICK_DATA_API$3, Dropdown.clearMenus);\nEventHandler.on(document, EVENT_KEYUP_DATA_API, Dropdown.clearMenus);\nEventHandler.on(document, EVENT_CLICK_DATA_API$3, SELECTOR_DATA_TOGGLE$3, function (event) {\n  event.preventDefault();\n  Dropdown.getOrCreateInstance(this).toggle();\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Dropdown);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/backdrop.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$9 = 'backdrop';\nconst CLASS_NAME_FADE$4 = 'fade';\nconst CLASS_NAME_SHOW$5 = 'show';\nconst EVENT_MOUSEDOWN = `mousedown.bs.${NAME$9}`;\nconst Default$8 = {\n  className: 'modal-backdrop',\n  clickCallback: null,\n  isAnimated: false,\n  isVisible: true,\n  // if false, we use the backdrop helper without adding any element to the dom\n  rootElement: 'body' // give the choice to place backdrop under different elements\n};\n\nconst DefaultType$8 = {\n  className: 'string',\n  clickCallback: '(function|null)',\n  isAnimated: 'boolean',\n  isVisible: 'boolean',\n  rootElement: '(element|string)'\n};\n\n/**\n * Class definition\n */\n\nclass Backdrop extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n    this._isAppended = false;\n    this._element = null;\n  }\n\n  // Getters\n  static get Default() {\n    return Default$8;\n  }\n  static get DefaultType() {\n    return DefaultType$8;\n  }\n  static get NAME() {\n    return NAME$9;\n  }\n\n  // Public\n  show(callback) {\n    if (!this._config.isVisible) {\n      execute(callback);\n      return;\n    }\n    this._append();\n    const element = this._getElement();\n    if (this._config.isAnimated) {\n      reflow(element);\n    }\n    element.classList.add(CLASS_NAME_SHOW$5);\n    this._emulateAnimation(() => {\n      execute(callback);\n    });\n  }\n  hide(callback) {\n    if (!this._config.isVisible) {\n      execute(callback);\n      return;\n    }\n    this._getElement().classList.remove(CLASS_NAME_SHOW$5);\n    this._emulateAnimation(() => {\n      this.dispose();\n      execute(callback);\n    });\n  }\n  dispose() {\n    if (!this._isAppended) {\n      return;\n    }\n    EventHandler.off(this._element, EVENT_MOUSEDOWN);\n    this._element.remove();\n    this._isAppended = false;\n  }\n\n  // Private\n  _getElement() {\n    if (!this._element) {\n      const backdrop = document.createElement('div');\n      backdrop.className = this._config.className;\n      if (this._config.isAnimated) {\n        backdrop.classList.add(CLASS_NAME_FADE$4);\n      }\n      this._element = backdrop;\n    }\n    return this._element;\n  }\n  _configAfterMerge(config) {\n    // use getElement() with the default \"body\" to get a fresh Element on each instantiation\n    config.rootElement = getElement(config.rootElement);\n    return config;\n  }\n  _append() {\n    if (this._isAppended) {\n      return;\n    }\n    const element = this._getElement();\n    this._config.rootElement.append(element);\n    EventHandler.on(element, EVENT_MOUSEDOWN, () => {\n      execute(this._config.clickCallback);\n    });\n    this._isAppended = true;\n  }\n  _emulateAnimation(callback) {\n    executeAfterTransition(callback, this._getElement(), this._config.isAnimated);\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/focustrap.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$8 = 'focustrap';\nconst DATA_KEY$5 = 'bs.focustrap';\nconst EVENT_KEY$5 = `.${DATA_KEY$5}`;\nconst EVENT_FOCUSIN$2 = `focusin${EVENT_KEY$5}`;\nconst EVENT_KEYDOWN_TAB = `keydown.tab${EVENT_KEY$5}`;\nconst TAB_KEY = 'Tab';\nconst TAB_NAV_FORWARD = 'forward';\nconst TAB_NAV_BACKWARD = 'backward';\nconst Default$7 = {\n  autofocus: true,\n  trapElement: null // The element to trap focus inside of\n};\n\nconst DefaultType$7 = {\n  autofocus: 'boolean',\n  trapElement: 'element'\n};\n\n/**\n * Class definition\n */\n\nclass FocusTrap extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n    this._isActive = false;\n    this._lastTabNavDirection = null;\n  }\n\n  // Getters\n  static get Default() {\n    return Default$7;\n  }\n  static get DefaultType() {\n    return DefaultType$7;\n  }\n  static get NAME() {\n    return NAME$8;\n  }\n\n  // Public\n  activate() {\n    if (this._isActive) {\n      return;\n    }\n    if (this._config.autofocus) {\n      this._config.trapElement.focus();\n    }\n    EventHandler.off(document, EVENT_KEY$5); // guard against infinite focus loop\n    EventHandler.on(document, EVENT_FOCUSIN$2, event => this._handleFocusin(event));\n    EventHandler.on(document, EVENT_KEYDOWN_TAB, event => this._handleKeydown(event));\n    this._isActive = true;\n  }\n  deactivate() {\n    if (!this._isActive) {\n      return;\n    }\n    this._isActive = false;\n    EventHandler.off(document, EVENT_KEY$5);\n  }\n\n  // Private\n  _handleFocusin(event) {\n    const {\n      trapElement\n    } = this._config;\n    if (event.target === document || event.target === trapElement || trapElement.contains(event.target)) {\n      return;\n    }\n    const elements = SelectorEngine.focusableChildren(trapElement);\n    if (elements.length === 0) {\n      trapElement.focus();\n    } else if (this._lastTabNavDirection === TAB_NAV_BACKWARD) {\n      elements[elements.length - 1].focus();\n    } else {\n      elements[0].focus();\n    }\n  }\n  _handleKeydown(event) {\n    if (event.key !== TAB_KEY) {\n      return;\n    }\n    this._lastTabNavDirection = event.shiftKey ? TAB_NAV_BACKWARD : TAB_NAV_FORWARD;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/scrollBar.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst SELECTOR_FIXED_CONTENT = '.fixed-top, .fixed-bottom, .is-fixed, .sticky-top';\nconst SELECTOR_STICKY_CONTENT = '.sticky-top';\nconst PROPERTY_PADDING = 'padding-right';\nconst PROPERTY_MARGIN = 'margin-right';\n\n/**\n * Class definition\n */\n\nclass ScrollBarHelper {\n  constructor() {\n    this._element = document.body;\n  }\n\n  // Public\n  getWidth() {\n    // https://developer.mozilla.org/en-US/docs/Web/API/Window/innerWidth#usage_notes\n    const documentWidth = document.documentElement.clientWidth;\n    return Math.abs(window.innerWidth - documentWidth);\n  }\n  hide() {\n    const width = this.getWidth();\n    this._disableOverFlow();\n    // give padding to element to balance the hidden scrollbar width\n    this._setElementAttributes(this._element, PROPERTY_PADDING, calculatedValue => calculatedValue + width);\n    // trick: We adjust positive paddingRight and negative marginRight to sticky-top elements to keep showing fullwidth\n    this._setElementAttributes(SELECTOR_FIXED_CONTENT, PROPERTY_PADDING, calculatedValue => calculatedValue + width);\n    this._setElementAttributes(SELECTOR_STICKY_CONTENT, PROPERTY_MARGIN, calculatedValue => calculatedValue - width);\n  }\n  reset() {\n    this._resetElementAttributes(this._element, 'overflow');\n    this._resetElementAttributes(this._element, PROPERTY_PADDING);\n    this._resetElementAttributes(SELECTOR_FIXED_CONTENT, PROPERTY_PADDING);\n    this._resetElementAttributes(SELECTOR_STICKY_CONTENT, PROPERTY_MARGIN);\n  }\n  isOverflowing() {\n    return this.getWidth() > 0;\n  }\n\n  // Private\n  _disableOverFlow() {\n    this._saveInitialAttribute(this._element, 'overflow');\n    this._element.style.overflow = 'hidden';\n  }\n  _setElementAttributes(selector, styleProperty, callback) {\n    const scrollbarWidth = this.getWidth();\n    const manipulationCallBack = element => {\n      if (element !== this._element && window.innerWidth > element.clientWidth + scrollbarWidth) {\n        return;\n      }\n      this._saveInitialAttribute(element, styleProperty);\n      const calculatedValue = window.getComputedStyle(element).getPropertyValue(styleProperty);\n      element.style.setProperty(styleProperty, `${callback(Number.parseFloat(calculatedValue))}px`);\n    };\n    this._applyManipulationCallback(selector, manipulationCallBack);\n  }\n  _saveInitialAttribute(element, styleProperty) {\n    const actualValue = element.style.getPropertyValue(styleProperty);\n    if (actualValue) {\n      Manipulator.setDataAttribute(element, styleProperty, actualValue);\n    }\n  }\n  _resetElementAttributes(selector, styleProperty) {\n    const manipulationCallBack = element => {\n      const value = Manipulator.getDataAttribute(element, styleProperty);\n      // We only want to remove the property if the value is `null`; the value can also be zero\n      if (value === null) {\n        element.style.removeProperty(styleProperty);\n        return;\n      }\n      Manipulator.removeDataAttribute(element, styleProperty);\n      element.style.setProperty(styleProperty, value);\n    };\n    this._applyManipulationCallback(selector, manipulationCallBack);\n  }\n  _applyManipulationCallback(selector, callBack) {\n    if (isElement(selector)) {\n      callBack(selector);\n      return;\n    }\n    for (const sel of SelectorEngine.find(selector, this._element)) {\n      callBack(sel);\n    }\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap modal.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$7 = 'modal';\nconst DATA_KEY$4 = 'bs.modal';\nconst EVENT_KEY$4 = `.${DATA_KEY$4}`;\nconst DATA_API_KEY$2 = '.data-api';\nconst ESCAPE_KEY$1 = 'Escape';\nconst EVENT_HIDE$4 = `hide${EVENT_KEY$4}`;\nconst EVENT_HIDE_PREVENTED$1 = `hidePrevented${EVENT_KEY$4}`;\nconst EVENT_HIDDEN$4 = `hidden${EVENT_KEY$4}`;\nconst EVENT_SHOW$4 = `show${EVENT_KEY$4}`;\nconst EVENT_SHOWN$4 = `shown${EVENT_KEY$4}`;\nconst EVENT_RESIZE$1 = `resize${EVENT_KEY$4}`;\nconst EVENT_CLICK_DISMISS = `click.dismiss${EVENT_KEY$4}`;\nconst EVENT_MOUSEDOWN_DISMISS = `mousedown.dismiss${EVENT_KEY$4}`;\nconst EVENT_KEYDOWN_DISMISS$1 = `keydown.dismiss${EVENT_KEY$4}`;\nconst EVENT_CLICK_DATA_API$2 = `click${EVENT_KEY$4}${DATA_API_KEY$2}`;\nconst CLASS_NAME_OPEN = 'modal-open';\nconst CLASS_NAME_FADE$3 = 'fade';\nconst CLASS_NAME_SHOW$4 = 'show';\nconst CLASS_NAME_STATIC = 'modal-static';\nconst OPEN_SELECTOR$1 = '.modal.show';\nconst SELECTOR_DIALOG = '.modal-dialog';\nconst SELECTOR_MODAL_BODY = '.modal-body';\nconst SELECTOR_DATA_TOGGLE$2 = '[data-bs-toggle=\"modal\"]';\nconst Default$6 = {\n  backdrop: true,\n  focus: true,\n  keyboard: true\n};\nconst DefaultType$6 = {\n  backdrop: '(boolean|string)',\n  focus: 'boolean',\n  keyboard: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Modal extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._dialog = SelectorEngine.findOne(SELECTOR_DIALOG, this._element);\n    this._backdrop = this._initializeBackDrop();\n    this._focustrap = this._initializeFocusTrap();\n    this._isShown = false;\n    this._isTransitioning = false;\n    this._scrollBar = new ScrollBarHelper();\n    this._addEventListeners();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$6;\n  }\n  static get DefaultType() {\n    return DefaultType$6;\n  }\n  static get NAME() {\n    return NAME$7;\n  }\n\n  // Public\n  toggle(relatedTarget) {\n    return this._isShown ? this.hide() : this.show(relatedTarget);\n  }\n  show(relatedTarget) {\n    if (this._isShown || this._isTransitioning) {\n      return;\n    }\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$4, {\n      relatedTarget\n    });\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._isShown = true;\n    this._isTransitioning = true;\n    this._scrollBar.hide();\n    document.body.classList.add(CLASS_NAME_OPEN);\n    this._adjustDialog();\n    this._backdrop.show(() => this._showElement(relatedTarget));\n  }\n  hide() {\n    if (!this._isShown || this._isTransitioning) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$4);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    this._isShown = false;\n    this._isTransitioning = true;\n    this._focustrap.deactivate();\n    this._element.classList.remove(CLASS_NAME_SHOW$4);\n    this._queueCallback(() => this._hideModal(), this._element, this._isAnimated());\n  }\n  dispose() {\n    EventHandler.off(window, EVENT_KEY$4);\n    EventHandler.off(this._dialog, EVENT_KEY$4);\n    this._backdrop.dispose();\n    this._focustrap.deactivate();\n    super.dispose();\n  }\n  handleUpdate() {\n    this._adjustDialog();\n  }\n\n  // Private\n  _initializeBackDrop() {\n    return new Backdrop({\n      isVisible: Boolean(this._config.backdrop),\n      // 'static' option will be translated to true, and booleans will keep their value,\n      isAnimated: this._isAnimated()\n    });\n  }\n  _initializeFocusTrap() {\n    return new FocusTrap({\n      trapElement: this._element\n    });\n  }\n  _showElement(relatedTarget) {\n    // try to append dynamic modal\n    if (!document.body.contains(this._element)) {\n      document.body.append(this._element);\n    }\n    this._element.style.display = 'block';\n    this._element.removeAttribute('aria-hidden');\n    this._element.setAttribute('aria-modal', true);\n    this._element.setAttribute('role', 'dialog');\n    this._element.scrollTop = 0;\n    const modalBody = SelectorEngine.findOne(SELECTOR_MODAL_BODY, this._dialog);\n    if (modalBody) {\n      modalBody.scrollTop = 0;\n    }\n    reflow(this._element);\n    this._element.classList.add(CLASS_NAME_SHOW$4);\n    const transitionComplete = () => {\n      if (this._config.focus) {\n        this._focustrap.activate();\n      }\n      this._isTransitioning = false;\n      EventHandler.trigger(this._element, EVENT_SHOWN$4, {\n        relatedTarget\n      });\n    };\n    this._queueCallback(transitionComplete, this._dialog, this._isAnimated());\n  }\n  _addEventListeners() {\n    EventHandler.on(this._element, EVENT_KEYDOWN_DISMISS$1, event => {\n      if (event.key !== ESCAPE_KEY$1) {\n        return;\n      }\n      if (this._config.keyboard) {\n        this.hide();\n        return;\n      }\n      this._triggerBackdropTransition();\n    });\n    EventHandler.on(window, EVENT_RESIZE$1, () => {\n      if (this._isShown && !this._isTransitioning) {\n        this._adjustDialog();\n      }\n    });\n    EventHandler.on(this._element, EVENT_MOUSEDOWN_DISMISS, event => {\n      // a bad trick to segregate clicks that may start inside dialog but end outside, and avoid listen to scrollbar clicks\n      EventHandler.one(this._element, EVENT_CLICK_DISMISS, event2 => {\n        if (this._element !== event.target || this._element !== event2.target) {\n          return;\n        }\n        if (this._config.backdrop === 'static') {\n          this._triggerBackdropTransition();\n          return;\n        }\n        if (this._config.backdrop) {\n          this.hide();\n        }\n      });\n    });\n  }\n  _hideModal() {\n    this._element.style.display = 'none';\n    this._element.setAttribute('aria-hidden', true);\n    this._element.removeAttribute('aria-modal');\n    this._element.removeAttribute('role');\n    this._isTransitioning = false;\n    this._backdrop.hide(() => {\n      document.body.classList.remove(CLASS_NAME_OPEN);\n      this._resetAdjustments();\n      this._scrollBar.reset();\n      EventHandler.trigger(this._element, EVENT_HIDDEN$4);\n    });\n  }\n  _isAnimated() {\n    return this._element.classList.contains(CLASS_NAME_FADE$3);\n  }\n  _triggerBackdropTransition() {\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED$1);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    const isModalOverflowing = this._element.scrollHeight > document.documentElement.clientHeight;\n    const initialOverflowY = this._element.style.overflowY;\n    // return if the following background transition hasn't yet completed\n    if (initialOverflowY === 'hidden' || this._element.classList.contains(CLASS_NAME_STATIC)) {\n      return;\n    }\n    if (!isModalOverflowing) {\n      this._element.style.overflowY = 'hidden';\n    }\n    this._element.classList.add(CLASS_NAME_STATIC);\n    this._queueCallback(() => {\n      this._element.classList.remove(CLASS_NAME_STATIC);\n      this._queueCallback(() => {\n        this._element.style.overflowY = initialOverflowY;\n      }, this._dialog);\n    }, this._dialog);\n    this._element.focus();\n  }\n\n  /**\n   * The following methods are used to handle overflowing modals\n   */\n\n  _adjustDialog() {\n    const isModalOverflowing = this._element.scrollHeight > document.documentElement.clientHeight;\n    const scrollbarWidth = this._scrollBar.getWidth();\n    const isBodyOverflowing = scrollbarWidth > 0;\n    if (isBodyOverflowing && !isModalOverflowing) {\n      const property = isRTL() ? 'paddingLeft' : 'paddingRight';\n      this._element.style[property] = `${scrollbarWidth}px`;\n    }\n    if (!isBodyOverflowing && isModalOverflowing) {\n      const property = isRTL() ? 'paddingRight' : 'paddingLeft';\n      this._element.style[property] = `${scrollbarWidth}px`;\n    }\n  }\n  _resetAdjustments() {\n    this._element.style.paddingLeft = '';\n    this._element.style.paddingRight = '';\n  }\n\n  // Static\n  static jQueryInterface(config, relatedTarget) {\n    return this.each(function () {\n      const data = Modal.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config](relatedTarget);\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$2, SELECTOR_DATA_TOGGLE$2, function (event) {\n  const target = SelectorEngine.getElementFromSelector(this);\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n  EventHandler.one(target, EVENT_SHOW$4, showEvent => {\n    if (showEvent.defaultPrevented) {\n      // only register focus restorer if modal will actually get shown\n      return;\n    }\n    EventHandler.one(target, EVENT_HIDDEN$4, () => {\n      if (isVisible(this)) {\n        this.focus();\n      }\n    });\n  });\n\n  // avoid conflict when clicking modal toggler while another one is open\n  const alreadyOpen = SelectorEngine.findOne(OPEN_SELECTOR$1);\n  if (alreadyOpen) {\n    Modal.getInstance(alreadyOpen).hide();\n  }\n  const data = Modal.getOrCreateInstance(target);\n  data.toggle(this);\n});\nenableDismissTrigger(Modal);\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Modal);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap offcanvas.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$6 = 'offcanvas';\nconst DATA_KEY$3 = 'bs.offcanvas';\nconst EVENT_KEY$3 = `.${DATA_KEY$3}`;\nconst DATA_API_KEY$1 = '.data-api';\nconst EVENT_LOAD_DATA_API$2 = `load${EVENT_KEY$3}${DATA_API_KEY$1}`;\nconst ESCAPE_KEY = 'Escape';\nconst CLASS_NAME_SHOW$3 = 'show';\nconst CLASS_NAME_SHOWING$1 = 'showing';\nconst CLASS_NAME_HIDING = 'hiding';\nconst CLASS_NAME_BACKDROP = 'offcanvas-backdrop';\nconst OPEN_SELECTOR = '.offcanvas.show';\nconst EVENT_SHOW$3 = `show${EVENT_KEY$3}`;\nconst EVENT_SHOWN$3 = `shown${EVENT_KEY$3}`;\nconst EVENT_HIDE$3 = `hide${EVENT_KEY$3}`;\nconst EVENT_HIDE_PREVENTED = `hidePrevented${EVENT_KEY$3}`;\nconst EVENT_HIDDEN$3 = `hidden${EVENT_KEY$3}`;\nconst EVENT_RESIZE = `resize${EVENT_KEY$3}`;\nconst EVENT_CLICK_DATA_API$1 = `click${EVENT_KEY$3}${DATA_API_KEY$1}`;\nconst EVENT_KEYDOWN_DISMISS = `keydown.dismiss${EVENT_KEY$3}`;\nconst SELECTOR_DATA_TOGGLE$1 = '[data-bs-toggle=\"offcanvas\"]';\nconst Default$5 = {\n  backdrop: true,\n  keyboard: true,\n  scroll: false\n};\nconst DefaultType$5 = {\n  backdrop: '(boolean|string)',\n  keyboard: 'boolean',\n  scroll: 'boolean'\n};\n\n/**\n * Class definition\n */\n\nclass Offcanvas extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._isShown = false;\n    this._backdrop = this._initializeBackDrop();\n    this._focustrap = this._initializeFocusTrap();\n    this._addEventListeners();\n  }\n\n  // Getters\n  static get Default() {\n    return Default$5;\n  }\n  static get DefaultType() {\n    return DefaultType$5;\n  }\n  static get NAME() {\n    return NAME$6;\n  }\n\n  // Public\n  toggle(relatedTarget) {\n    return this._isShown ? this.hide() : this.show(relatedTarget);\n  }\n  show(relatedTarget) {\n    if (this._isShown) {\n      return;\n    }\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW$3, {\n      relatedTarget\n    });\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._isShown = true;\n    this._backdrop.show();\n    if (!this._config.scroll) {\n      new ScrollBarHelper().hide();\n    }\n    this._element.setAttribute('aria-modal', true);\n    this._element.setAttribute('role', 'dialog');\n    this._element.classList.add(CLASS_NAME_SHOWING$1);\n    const completeCallBack = () => {\n      if (!this._config.scroll || this._config.backdrop) {\n        this._focustrap.activate();\n      }\n      this._element.classList.add(CLASS_NAME_SHOW$3);\n      this._element.classList.remove(CLASS_NAME_SHOWING$1);\n      EventHandler.trigger(this._element, EVENT_SHOWN$3, {\n        relatedTarget\n      });\n    };\n    this._queueCallback(completeCallBack, this._element, true);\n  }\n  hide() {\n    if (!this._isShown) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE$3);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    this._focustrap.deactivate();\n    this._element.blur();\n    this._isShown = false;\n    this._element.classList.add(CLASS_NAME_HIDING);\n    this._backdrop.hide();\n    const completeCallback = () => {\n      this._element.classList.remove(CLASS_NAME_SHOW$3, CLASS_NAME_HIDING);\n      this._element.removeAttribute('aria-modal');\n      this._element.removeAttribute('role');\n      if (!this._config.scroll) {\n        new ScrollBarHelper().reset();\n      }\n      EventHandler.trigger(this._element, EVENT_HIDDEN$3);\n    };\n    this._queueCallback(completeCallback, this._element, true);\n  }\n  dispose() {\n    this._backdrop.dispose();\n    this._focustrap.deactivate();\n    super.dispose();\n  }\n\n  // Private\n  _initializeBackDrop() {\n    const clickCallback = () => {\n      if (this._config.backdrop === 'static') {\n        EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED);\n        return;\n      }\n      this.hide();\n    };\n\n    // 'static' option will be translated to true, and booleans will keep their value\n    const isVisible = Boolean(this._config.backdrop);\n    return new Backdrop({\n      className: CLASS_NAME_BACKDROP,\n      isVisible,\n      isAnimated: true,\n      rootElement: this._element.parentNode,\n      clickCallback: isVisible ? clickCallback : null\n    });\n  }\n  _initializeFocusTrap() {\n    return new FocusTrap({\n      trapElement: this._element\n    });\n  }\n  _addEventListeners() {\n    EventHandler.on(this._element, EVENT_KEYDOWN_DISMISS, event => {\n      if (event.key !== ESCAPE_KEY) {\n        return;\n      }\n      if (this._config.keyboard) {\n        this.hide();\n        return;\n      }\n      EventHandler.trigger(this._element, EVENT_HIDE_PREVENTED);\n    });\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Offcanvas.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config](this);\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API$1, SELECTOR_DATA_TOGGLE$1, function (event) {\n  const target = SelectorEngine.getElementFromSelector(this);\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n  if (isDisabled(this)) {\n    return;\n  }\n  EventHandler.one(target, EVENT_HIDDEN$3, () => {\n    // focus on trigger when it is closed\n    if (isVisible(this)) {\n      this.focus();\n    }\n  });\n\n  // avoid conflict when clicking a toggler of an offcanvas, while another is open\n  const alreadyOpen = SelectorEngine.findOne(OPEN_SELECTOR);\n  if (alreadyOpen && alreadyOpen !== target) {\n    Offcanvas.getInstance(alreadyOpen).hide();\n  }\n  const data = Offcanvas.getOrCreateInstance(target);\n  data.toggle(this);\n});\nEventHandler.on(window, EVENT_LOAD_DATA_API$2, () => {\n  for (const selector of SelectorEngine.find(OPEN_SELECTOR)) {\n    Offcanvas.getOrCreateInstance(selector).show();\n  }\n});\nEventHandler.on(window, EVENT_RESIZE, () => {\n  for (const element of SelectorEngine.find('[aria-modal][class*=show][class*=offcanvas-]')) {\n    if (getComputedStyle(element).position !== 'fixed') {\n      Offcanvas.getOrCreateInstance(element).hide();\n    }\n  }\n});\nenableDismissTrigger(Offcanvas);\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Offcanvas);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/sanitizer.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n// js-docs-start allow-list\nconst ARIA_ATTRIBUTE_PATTERN = /^aria-[\\w-]*$/i;\nconst DefaultAllowlist = {\n  // Global attributes allowed on any supplied element below.\n  '*': ['class', 'dir', 'id', 'lang', 'role', ARIA_ATTRIBUTE_PATTERN],\n  a: ['target', 'href', 'title', 'rel'],\n  area: [],\n  b: [],\n  br: [],\n  col: [],\n  code: [],\n  div: [],\n  em: [],\n  hr: [],\n  h1: [],\n  h2: [],\n  h3: [],\n  h4: [],\n  h5: [],\n  h6: [],\n  i: [],\n  img: ['src', 'srcset', 'alt', 'title', 'width', 'height'],\n  li: [],\n  ol: [],\n  p: [],\n  pre: [],\n  s: [],\n  small: [],\n  span: [],\n  sub: [],\n  sup: [],\n  strong: [],\n  u: [],\n  ul: []\n};\n// js-docs-end allow-list\n\nconst uriAttributes = new Set(['background', 'cite', 'href', 'itemtype', 'longdesc', 'poster', 'src', 'xlink:href']);\n\n/**\n * A pattern that recognizes URLs that are safe wrt. XSS in URL navigation\n * contexts.\n *\n * Shout-out to Angular https://github.com/angular/angular/blob/15.2.8/packages/core/src/sanitization/url_sanitizer.ts#L38\n */\n// eslint-disable-next-line unicorn/better-regex\nconst SAFE_URL_PATTERN = /^(?!javascript:)(?:[a-z0-9+.-]+:|[^&:/?#]*(?:[/?#]|$))/i;\nconst allowedAttribute = (attribute, allowedAttributeList) => {\n  const attributeName = attribute.nodeName.toLowerCase();\n  if (allowedAttributeList.includes(attributeName)) {\n    if (uriAttributes.has(attributeName)) {\n      return Boolean(SAFE_URL_PATTERN.test(attribute.nodeValue));\n    }\n    return true;\n  }\n\n  // Check if a regular expression validates the attribute.\n  return allowedAttributeList.filter(attributeRegex => attributeRegex instanceof RegExp).some(regex => regex.test(attributeName));\n};\nfunction sanitizeHtml(unsafeHtml, allowList, sanitizeFunction) {\n  if (!unsafeHtml.length) {\n    return unsafeHtml;\n  }\n  if (sanitizeFunction && typeof sanitizeFunction === 'function') {\n    return sanitizeFunction(unsafeHtml);\n  }\n  const domParser = new window.DOMParser();\n  const createdDocument = domParser.parseFromString(unsafeHtml, 'text/html');\n  const elements = [].concat(...createdDocument.body.querySelectorAll('*'));\n  for (const element of elements) {\n    const elementName = element.nodeName.toLowerCase();\n    if (!Object.keys(allowList).includes(elementName)) {\n      element.remove();\n      continue;\n    }\n    const attributeList = [].concat(...element.attributes);\n    const allowedAttributes = [].concat(allowList['*'] || [], allowList[elementName] || []);\n    for (const attribute of attributeList) {\n      if (!allowedAttribute(attribute, allowedAttributes)) {\n        element.removeAttribute(attribute.nodeName);\n      }\n    }\n  }\n  return createdDocument.body.innerHTML;\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap util/template-factory.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$5 = 'TemplateFactory';\nconst Default$4 = {\n  allowList: DefaultAllowlist,\n  content: {},\n  // { selector : text ,  selector2 : text2 , }\n  extraClass: '',\n  html: false,\n  sanitize: true,\n  sanitizeFn: null,\n  template: '<div></div>'\n};\nconst DefaultType$4 = {\n  allowList: 'object',\n  content: 'object',\n  extraClass: '(string|function)',\n  html: 'boolean',\n  sanitize: 'boolean',\n  sanitizeFn: '(null|function)',\n  template: 'string'\n};\nconst DefaultContentType = {\n  entry: '(string|element|function|null)',\n  selector: '(string|element)'\n};\n\n/**\n * Class definition\n */\n\nclass TemplateFactory extends Config {\n  constructor(config) {\n    super();\n    this._config = this._getConfig(config);\n  }\n\n  // Getters\n  static get Default() {\n    return Default$4;\n  }\n  static get DefaultType() {\n    return DefaultType$4;\n  }\n  static get NAME() {\n    return NAME$5;\n  }\n\n  // Public\n  getContent() {\n    return Object.values(this._config.content).map(config => this._resolvePossibleFunction(config)).filter(Boolean);\n  }\n  hasContent() {\n    return this.getContent().length > 0;\n  }\n  changeContent(content) {\n    this._checkContent(content);\n    this._config.content = {\n      ...this._config.content,\n      ...content\n    };\n    return this;\n  }\n  toHtml() {\n    const templateWrapper = document.createElement('div');\n    templateWrapper.innerHTML = this._maybeSanitize(this._config.template);\n    for (const [selector, text] of Object.entries(this._config.content)) {\n      this._setContent(templateWrapper, text, selector);\n    }\n    const template = templateWrapper.children[0];\n    const extraClass = this._resolvePossibleFunction(this._config.extraClass);\n    if (extraClass) {\n      template.classList.add(...extraClass.split(' '));\n    }\n    return template;\n  }\n\n  // Private\n  _typeCheckConfig(config) {\n    super._typeCheckConfig(config);\n    this._checkContent(config.content);\n  }\n  _checkContent(arg) {\n    for (const [selector, content] of Object.entries(arg)) {\n      super._typeCheckConfig({\n        selector,\n        entry: content\n      }, DefaultContentType);\n    }\n  }\n  _setContent(template, content, selector) {\n    const templateElement = SelectorEngine.findOne(selector, template);\n    if (!templateElement) {\n      return;\n    }\n    content = this._resolvePossibleFunction(content);\n    if (!content) {\n      templateElement.remove();\n      return;\n    }\n    if (isElement(content)) {\n      this._putElementInTemplate(getElement(content), templateElement);\n      return;\n    }\n    if (this._config.html) {\n      templateElement.innerHTML = this._maybeSanitize(content);\n      return;\n    }\n    templateElement.textContent = content;\n  }\n  _maybeSanitize(arg) {\n    return this._config.sanitize ? sanitizeHtml(arg, this._config.allowList, this._config.sanitizeFn) : arg;\n  }\n  _resolvePossibleFunction(arg) {\n    return execute(arg, [this]);\n  }\n  _putElementInTemplate(element, templateElement) {\n    if (this._config.html) {\n      templateElement.innerHTML = '';\n      templateElement.append(element);\n      return;\n    }\n    templateElement.textContent = element.textContent;\n  }\n}\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap tooltip.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$4 = 'tooltip';\nconst DISALLOWED_ATTRIBUTES = new Set(['sanitize', 'allowList', 'sanitizeFn']);\nconst CLASS_NAME_FADE$2 = 'fade';\nconst CLASS_NAME_MODAL = 'modal';\nconst CLASS_NAME_SHOW$2 = 'show';\nconst SELECTOR_TOOLTIP_INNER = '.tooltip-inner';\nconst SELECTOR_MODAL = `.${CLASS_NAME_MODAL}`;\nconst EVENT_MODAL_HIDE = 'hide.bs.modal';\nconst TRIGGER_HOVER = 'hover';\nconst TRIGGER_FOCUS = 'focus';\nconst TRIGGER_CLICK = 'click';\nconst TRIGGER_MANUAL = 'manual';\nconst EVENT_HIDE$2 = 'hide';\nconst EVENT_HIDDEN$2 = 'hidden';\nconst EVENT_SHOW$2 = 'show';\nconst EVENT_SHOWN$2 = 'shown';\nconst EVENT_INSERTED = 'inserted';\nconst EVENT_CLICK$1 = 'click';\nconst EVENT_FOCUSIN$1 = 'focusin';\nconst EVENT_FOCUSOUT$1 = 'focusout';\nconst EVENT_MOUSEENTER = 'mouseenter';\nconst EVENT_MOUSELEAVE = 'mouseleave';\nconst AttachmentMap = {\n  AUTO: 'auto',\n  TOP: 'top',\n  RIGHT: isRTL() ? 'left' : 'right',\n  BOTTOM: 'bottom',\n  LEFT: isRTL() ? 'right' : 'left'\n};\nconst Default$3 = {\n  allowList: DefaultAllowlist,\n  animation: true,\n  boundary: 'clippingParents',\n  container: false,\n  customClass: '',\n  delay: 0,\n  fallbackPlacements: ['top', 'right', 'bottom', 'left'],\n  html: false,\n  offset: [0, 6],\n  placement: 'top',\n  popperConfig: null,\n  sanitize: true,\n  sanitizeFn: null,\n  selector: false,\n  template: '<div class=\"tooltip\" role=\"tooltip\">' + '<div class=\"tooltip-arrow\"></div>' + '<div class=\"tooltip-inner\"></div>' + '</div>',\n  title: '',\n  trigger: 'hover focus'\n};\nconst DefaultType$3 = {\n  allowList: 'object',\n  animation: 'boolean',\n  boundary: '(string|element)',\n  container: '(string|element|boolean)',\n  customClass: '(string|function)',\n  delay: '(number|object)',\n  fallbackPlacements: 'array',\n  html: 'boolean',\n  offset: '(array|string|function)',\n  placement: '(string|function)',\n  popperConfig: '(null|object|function)',\n  sanitize: 'boolean',\n  sanitizeFn: '(null|function)',\n  selector: '(string|boolean)',\n  template: 'string',\n  title: '(string|element|function)',\n  trigger: 'string'\n};\n\n/**\n * Class definition\n */\n\nclass Tooltip extends BaseComponent {\n  constructor(element, config) {\n    if (typeof Popper === 'undefined') {\n      throw new TypeError('Bootstrap\\'s tooltips require Popper (https://popper.js.org)');\n    }\n    super(element, config);\n\n    // Private\n    this._isEnabled = true;\n    this._timeout = 0;\n    this._isHovered = null;\n    this._activeTrigger = {};\n    this._popper = null;\n    this._templateFactory = null;\n    this._newContent = null;\n\n    // Protected\n    this.tip = null;\n    this._setListeners();\n    if (!this._config.selector) {\n      this._fixTitle();\n    }\n  }\n\n  // Getters\n  static get Default() {\n    return Default$3;\n  }\n  static get DefaultType() {\n    return DefaultType$3;\n  }\n  static get NAME() {\n    return NAME$4;\n  }\n\n  // Public\n  enable() {\n    this._isEnabled = true;\n  }\n  disable() {\n    this._isEnabled = false;\n  }\n  toggleEnabled() {\n    this._isEnabled = !this._isEnabled;\n  }\n  toggle() {\n    if (!this._isEnabled) {\n      return;\n    }\n    this._activeTrigger.click = !this._activeTrigger.click;\n    if (this._isShown()) {\n      this._leave();\n      return;\n    }\n    this._enter();\n  }\n  dispose() {\n    clearTimeout(this._timeout);\n    EventHandler.off(this._element.closest(SELECTOR_MODAL), EVENT_MODAL_HIDE, this._hideModalHandler);\n    if (this._element.getAttribute('data-bs-original-title')) {\n      this._element.setAttribute('title', this._element.getAttribute('data-bs-original-title'));\n    }\n    this._disposePopper();\n    super.dispose();\n  }\n  show() {\n    if (this._element.style.display === 'none') {\n      throw new Error('Please use show on visible elements');\n    }\n    if (!(this._isWithContent() && this._isEnabled)) {\n      return;\n    }\n    const showEvent = EventHandler.trigger(this._element, this.constructor.eventName(EVENT_SHOW$2));\n    const shadowRoot = findShadowRoot(this._element);\n    const isInTheDom = (shadowRoot || this._element.ownerDocument.documentElement).contains(this._element);\n    if (showEvent.defaultPrevented || !isInTheDom) {\n      return;\n    }\n\n    // TODO: v6 remove this or make it optional\n    this._disposePopper();\n    const tip = this._getTipElement();\n    this._element.setAttribute('aria-describedby', tip.getAttribute('id'));\n    const {\n      container\n    } = this._config;\n    if (!this._element.ownerDocument.documentElement.contains(this.tip)) {\n      container.append(tip);\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_INSERTED));\n    }\n    this._popper = this._createPopper(tip);\n    tip.classList.add(CLASS_NAME_SHOW$2);\n\n    // If this is a touch-enabled device we add extra\n    // empty mouseover listeners to the body's immediate children;\n    // only needed because of broken event delegation on iOS\n    // https://www.quirksmode.org/blog/archives/2014/02/mouse_event_bub.html\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.on(element, 'mouseover', noop);\n      }\n    }\n    const complete = () => {\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_SHOWN$2));\n      if (this._isHovered === false) {\n        this._leave();\n      }\n      this._isHovered = false;\n    };\n    this._queueCallback(complete, this.tip, this._isAnimated());\n  }\n  hide() {\n    if (!this._isShown()) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, this.constructor.eventName(EVENT_HIDE$2));\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    const tip = this._getTipElement();\n    tip.classList.remove(CLASS_NAME_SHOW$2);\n\n    // If this is a touch-enabled device we remove the extra\n    // empty mouseover listeners we added for iOS support\n    if ('ontouchstart' in document.documentElement) {\n      for (const element of [].concat(...document.body.children)) {\n        EventHandler.off(element, 'mouseover', noop);\n      }\n    }\n    this._activeTrigger[TRIGGER_CLICK] = false;\n    this._activeTrigger[TRIGGER_FOCUS] = false;\n    this._activeTrigger[TRIGGER_HOVER] = false;\n    this._isHovered = null; // it is a trick to support manual triggering\n\n    const complete = () => {\n      if (this._isWithActiveTrigger()) {\n        return;\n      }\n      if (!this._isHovered) {\n        this._disposePopper();\n      }\n      this._element.removeAttribute('aria-describedby');\n      EventHandler.trigger(this._element, this.constructor.eventName(EVENT_HIDDEN$2));\n    };\n    this._queueCallback(complete, this.tip, this._isAnimated());\n  }\n  update() {\n    if (this._popper) {\n      this._popper.update();\n    }\n  }\n\n  // Protected\n  _isWithContent() {\n    return Boolean(this._getTitle());\n  }\n  _getTipElement() {\n    if (!this.tip) {\n      this.tip = this._createTipElement(this._newContent || this._getContentForTemplate());\n    }\n    return this.tip;\n  }\n  _createTipElement(content) {\n    const tip = this._getTemplateFactory(content).toHtml();\n\n    // TODO: remove this check in v6\n    if (!tip) {\n      return null;\n    }\n    tip.classList.remove(CLASS_NAME_FADE$2, CLASS_NAME_SHOW$2);\n    // TODO: v6 the following can be achieved with CSS only\n    tip.classList.add(`bs-${this.constructor.NAME}-auto`);\n    const tipId = getUID(this.constructor.NAME).toString();\n    tip.setAttribute('id', tipId);\n    if (this._isAnimated()) {\n      tip.classList.add(CLASS_NAME_FADE$2);\n    }\n    return tip;\n  }\n  setContent(content) {\n    this._newContent = content;\n    if (this._isShown()) {\n      this._disposePopper();\n      this.show();\n    }\n  }\n  _getTemplateFactory(content) {\n    if (this._templateFactory) {\n      this._templateFactory.changeContent(content);\n    } else {\n      this._templateFactory = new TemplateFactory({\n        ...this._config,\n        // the `content` var has to be after `this._config`\n        // to override config.content in case of popover\n        content,\n        extraClass: this._resolvePossibleFunction(this._config.customClass)\n      });\n    }\n    return this._templateFactory;\n  }\n  _getContentForTemplate() {\n    return {\n      [SELECTOR_TOOLTIP_INNER]: this._getTitle()\n    };\n  }\n  _getTitle() {\n    return this._resolvePossibleFunction(this._config.title) || this._element.getAttribute('data-bs-original-title');\n  }\n\n  // Private\n  _initializeOnDelegatedTarget(event) {\n    return this.constructor.getOrCreateInstance(event.delegateTarget, this._getDelegateConfig());\n  }\n  _isAnimated() {\n    return this._config.animation || this.tip && this.tip.classList.contains(CLASS_NAME_FADE$2);\n  }\n  _isShown() {\n    return this.tip && this.tip.classList.contains(CLASS_NAME_SHOW$2);\n  }\n  _createPopper(tip) {\n    const placement = execute(this._config.placement, [this, tip, this._element]);\n    const attachment = AttachmentMap[placement.toUpperCase()];\n    return Popper.createPopper(this._element, tip, this._getPopperConfig(attachment));\n  }\n  _getOffset() {\n    const {\n      offset\n    } = this._config;\n    if (typeof offset === 'string') {\n      return offset.split(',').map(value => Number.parseInt(value, 10));\n    }\n    if (typeof offset === 'function') {\n      return popperData => offset(popperData, this._element);\n    }\n    return offset;\n  }\n  _resolvePossibleFunction(arg) {\n    return execute(arg, [this._element]);\n  }\n  _getPopperConfig(attachment) {\n    const defaultBsPopperConfig = {\n      placement: attachment,\n      modifiers: [{\n        name: 'flip',\n        options: {\n          fallbackPlacements: this._config.fallbackPlacements\n        }\n      }, {\n        name: 'offset',\n        options: {\n          offset: this._getOffset()\n        }\n      }, {\n        name: 'preventOverflow',\n        options: {\n          boundary: this._config.boundary\n        }\n      }, {\n        name: 'arrow',\n        options: {\n          element: `.${this.constructor.NAME}-arrow`\n        }\n      }, {\n        name: 'preSetPlacement',\n        enabled: true,\n        phase: 'beforeMain',\n        fn: data => {\n          // Pre-set Popper's placement attribute in order to read the arrow sizes properly.\n          // Otherwise, Popper mixes up the width and height dimensions since the initial arrow style is for top placement\n          this._getTipElement().setAttribute('data-popper-placement', data.state.placement);\n        }\n      }]\n    };\n    return {\n      ...defaultBsPopperConfig,\n      ...execute(this._config.popperConfig, [defaultBsPopperConfig])\n    };\n  }\n  _setListeners() {\n    const triggers = this._config.trigger.split(' ');\n    for (const trigger of triggers) {\n      if (trigger === 'click') {\n        EventHandler.on(this._element, this.constructor.eventName(EVENT_CLICK$1), this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n          context.toggle();\n        });\n      } else if (trigger !== TRIGGER_MANUAL) {\n        const eventIn = trigger === TRIGGER_HOVER ? this.constructor.eventName(EVENT_MOUSEENTER) : this.constructor.eventName(EVENT_FOCUSIN$1);\n        const eventOut = trigger === TRIGGER_HOVER ? this.constructor.eventName(EVENT_MOUSELEAVE) : this.constructor.eventName(EVENT_FOCUSOUT$1);\n        EventHandler.on(this._element, eventIn, this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n          context._activeTrigger[event.type === 'focusin' ? TRIGGER_FOCUS : TRIGGER_HOVER] = true;\n          context._enter();\n        });\n        EventHandler.on(this._element, eventOut, this._config.selector, event => {\n          const context = this._initializeOnDelegatedTarget(event);\n          context._activeTrigger[event.type === 'focusout' ? TRIGGER_FOCUS : TRIGGER_HOVER] = context._element.contains(event.relatedTarget);\n          context._leave();\n        });\n      }\n    }\n    this._hideModalHandler = () => {\n      if (this._element) {\n        this.hide();\n      }\n    };\n    EventHandler.on(this._element.closest(SELECTOR_MODAL), EVENT_MODAL_HIDE, this._hideModalHandler);\n  }\n  _fixTitle() {\n    const title = this._element.getAttribute('title');\n    if (!title) {\n      return;\n    }\n    if (!this._element.getAttribute('aria-label') && !this._element.textContent.trim()) {\n      this._element.setAttribute('aria-label', title);\n    }\n    this._element.setAttribute('data-bs-original-title', title); // DO NOT USE IT. Is only for backwards compatibility\n    this._element.removeAttribute('title');\n  }\n  _enter() {\n    if (this._isShown() || this._isHovered) {\n      this._isHovered = true;\n      return;\n    }\n    this._isHovered = true;\n    this._setTimeout(() => {\n      if (this._isHovered) {\n        this.show();\n      }\n    }, this._config.delay.show);\n  }\n  _leave() {\n    if (this._isWithActiveTrigger()) {\n      return;\n    }\n    this._isHovered = false;\n    this._setTimeout(() => {\n      if (!this._isHovered) {\n        this.hide();\n      }\n    }, this._config.delay.hide);\n  }\n  _setTimeout(handler, timeout) {\n    clearTimeout(this._timeout);\n    this._timeout = setTimeout(handler, timeout);\n  }\n  _isWithActiveTrigger() {\n    return Object.values(this._activeTrigger).includes(true);\n  }\n  _getConfig(config) {\n    const dataAttributes = Manipulator.getDataAttributes(this._element);\n    for (const dataAttribute of Object.keys(dataAttributes)) {\n      if (DISALLOWED_ATTRIBUTES.has(dataAttribute)) {\n        delete dataAttributes[dataAttribute];\n      }\n    }\n    config = {\n      ...dataAttributes,\n      ...(typeof config === 'object' && config ? config : {})\n    };\n    config = this._mergeConfigObj(config);\n    config = this._configAfterMerge(config);\n    this._typeCheckConfig(config);\n    return config;\n  }\n  _configAfterMerge(config) {\n    config.container = config.container === false ? document.body : getElement(config.container);\n    if (typeof config.delay === 'number') {\n      config.delay = {\n        show: config.delay,\n        hide: config.delay\n      };\n    }\n    if (typeof config.title === 'number') {\n      config.title = config.title.toString();\n    }\n    if (typeof config.content === 'number') {\n      config.content = config.content.toString();\n    }\n    return config;\n  }\n  _getDelegateConfig() {\n    const config = {};\n    for (const [key, value] of Object.entries(this._config)) {\n      if (this.constructor.Default[key] !== value) {\n        config[key] = value;\n      }\n    }\n    config.selector = false;\n    config.trigger = 'manual';\n\n    // In the future can be replaced with:\n    // const keysWithDifferentValues = Object.entries(this._config).filter(entry => this.constructor.Default[entry[0]] !== this._config[entry[0]])\n    // `Object.fromEntries(keysWithDifferentValues)`\n    return config;\n  }\n  _disposePopper() {\n    if (this._popper) {\n      this._popper.destroy();\n      this._popper = null;\n    }\n    if (this.tip) {\n      this.tip.remove();\n      this.tip = null;\n    }\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Tooltip.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Tooltip);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap popover.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$3 = 'popover';\nconst SELECTOR_TITLE = '.popover-header';\nconst SELECTOR_CONTENT = '.popover-body';\nconst Default$2 = {\n  ...Tooltip.Default,\n  content: '',\n  offset: [0, 8],\n  placement: 'right',\n  template: '<div class=\"popover\" role=\"tooltip\">' + '<div class=\"popover-arrow\"></div>' + '<h3 class=\"popover-header\"></h3>' + '<div class=\"popover-body\"></div>' + '</div>',\n  trigger: 'click'\n};\nconst DefaultType$2 = {\n  ...Tooltip.DefaultType,\n  content: '(null|string|element|function)'\n};\n\n/**\n * Class definition\n */\n\nclass Popover extends Tooltip {\n  // Getters\n  static get Default() {\n    return Default$2;\n  }\n  static get DefaultType() {\n    return DefaultType$2;\n  }\n  static get NAME() {\n    return NAME$3;\n  }\n\n  // Overrides\n  _isWithContent() {\n    return this._getTitle() || this._getContent();\n  }\n\n  // Private\n  _getContentForTemplate() {\n    return {\n      [SELECTOR_TITLE]: this._getTitle(),\n      [SELECTOR_CONTENT]: this._getContent()\n    };\n  }\n  _getContent() {\n    return this._resolvePossibleFunction(this._config.content);\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Popover.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (typeof data[config] === 'undefined') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Popover);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap scrollspy.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$2 = 'scrollspy';\nconst DATA_KEY$2 = 'bs.scrollspy';\nconst EVENT_KEY$2 = `.${DATA_KEY$2}`;\nconst DATA_API_KEY = '.data-api';\nconst EVENT_ACTIVATE = `activate${EVENT_KEY$2}`;\nconst EVENT_CLICK = `click${EVENT_KEY$2}`;\nconst EVENT_LOAD_DATA_API$1 = `load${EVENT_KEY$2}${DATA_API_KEY}`;\nconst CLASS_NAME_DROPDOWN_ITEM = 'dropdown-item';\nconst CLASS_NAME_ACTIVE$1 = 'active';\nconst SELECTOR_DATA_SPY = '[data-bs-spy=\"scroll\"]';\nconst SELECTOR_TARGET_LINKS = '[href]';\nconst SELECTOR_NAV_LIST_GROUP = '.nav, .list-group';\nconst SELECTOR_NAV_LINKS = '.nav-link';\nconst SELECTOR_NAV_ITEMS = '.nav-item';\nconst SELECTOR_LIST_ITEMS = '.list-group-item';\nconst SELECTOR_LINK_ITEMS = `${SELECTOR_NAV_LINKS}, ${SELECTOR_NAV_ITEMS} > ${SELECTOR_NAV_LINKS}, ${SELECTOR_LIST_ITEMS}`;\nconst SELECTOR_DROPDOWN = '.dropdown';\nconst SELECTOR_DROPDOWN_TOGGLE$1 = '.dropdown-toggle';\nconst Default$1 = {\n  offset: null,\n  // TODO: v6 @deprecated, keep it for backwards compatibility reasons\n  rootMargin: '0px 0px -25%',\n  smoothScroll: false,\n  target: null,\n  threshold: [0.1, 0.5, 1]\n};\nconst DefaultType$1 = {\n  offset: '(number|null)',\n  // TODO v6 @deprecated, keep it for backwards compatibility reasons\n  rootMargin: 'string',\n  smoothScroll: 'boolean',\n  target: 'element',\n  threshold: 'array'\n};\n\n/**\n * Class definition\n */\n\nclass ScrollSpy extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n\n    // this._element is the observablesContainer and config.target the menu links wrapper\n    this._targetLinks = new Map();\n    this._observableSections = new Map();\n    this._rootElement = getComputedStyle(this._element).overflowY === 'visible' ? null : this._element;\n    this._activeTarget = null;\n    this._observer = null;\n    this._previousScrollData = {\n      visibleEntryTop: 0,\n      parentScrollTop: 0\n    };\n    this.refresh(); // initialize\n  }\n\n  // Getters\n  static get Default() {\n    return Default$1;\n  }\n  static get DefaultType() {\n    return DefaultType$1;\n  }\n  static get NAME() {\n    return NAME$2;\n  }\n\n  // Public\n  refresh() {\n    this._initializeTargetsAndObservables();\n    this._maybeEnableSmoothScroll();\n    if (this._observer) {\n      this._observer.disconnect();\n    } else {\n      this._observer = this._getNewObserver();\n    }\n    for (const section of this._observableSections.values()) {\n      this._observer.observe(section);\n    }\n  }\n  dispose() {\n    this._observer.disconnect();\n    super.dispose();\n  }\n\n  // Private\n  _configAfterMerge(config) {\n    // TODO: on v6 target should be given explicitly & remove the {target: 'ss-target'} case\n    config.target = getElement(config.target) || document.body;\n\n    // TODO: v6 Only for backwards compatibility reasons. Use rootMargin only\n    config.rootMargin = config.offset ? `${config.offset}px 0px -30%` : config.rootMargin;\n    if (typeof config.threshold === 'string') {\n      config.threshold = config.threshold.split(',').map(value => Number.parseFloat(value));\n    }\n    return config;\n  }\n  _maybeEnableSmoothScroll() {\n    if (!this._config.smoothScroll) {\n      return;\n    }\n\n    // unregister any previous listeners\n    EventHandler.off(this._config.target, EVENT_CLICK);\n    EventHandler.on(this._config.target, EVENT_CLICK, SELECTOR_TARGET_LINKS, event => {\n      const observableSection = this._observableSections.get(event.target.hash);\n      if (observableSection) {\n        event.preventDefault();\n        const root = this._rootElement || window;\n        const height = observableSection.offsetTop - this._element.offsetTop;\n        if (root.scrollTo) {\n          root.scrollTo({\n            top: height,\n            behavior: 'smooth'\n          });\n          return;\n        }\n\n        // Chrome 60 doesn't support `scrollTo`\n        root.scrollTop = height;\n      }\n    });\n  }\n  _getNewObserver() {\n    const options = {\n      root: this._rootElement,\n      threshold: this._config.threshold,\n      rootMargin: this._config.rootMargin\n    };\n    return new IntersectionObserver(entries => this._observerCallback(entries), options);\n  }\n\n  // The logic of selection\n  _observerCallback(entries) {\n    const targetElement = entry => this._targetLinks.get(`#${entry.target.id}`);\n    const activate = entry => {\n      this._previousScrollData.visibleEntryTop = entry.target.offsetTop;\n      this._process(targetElement(entry));\n    };\n    const parentScrollTop = (this._rootElement || document.documentElement).scrollTop;\n    const userScrollsDown = parentScrollTop >= this._previousScrollData.parentScrollTop;\n    this._previousScrollData.parentScrollTop = parentScrollTop;\n    for (const entry of entries) {\n      if (!entry.isIntersecting) {\n        this._activeTarget = null;\n        this._clearActiveClass(targetElement(entry));\n        continue;\n      }\n      const entryIsLowerThanPrevious = entry.target.offsetTop >= this._previousScrollData.visibleEntryTop;\n      // if we are scrolling down, pick the bigger offsetTop\n      if (userScrollsDown && entryIsLowerThanPrevious) {\n        activate(entry);\n        // if parent isn't scrolled, let's keep the first visible item, breaking the iteration\n        if (!parentScrollTop) {\n          return;\n        }\n        continue;\n      }\n\n      // if we are scrolling up, pick the smallest offsetTop\n      if (!userScrollsDown && !entryIsLowerThanPrevious) {\n        activate(entry);\n      }\n    }\n  }\n  _initializeTargetsAndObservables() {\n    this._targetLinks = new Map();\n    this._observableSections = new Map();\n    const targetLinks = SelectorEngine.find(SELECTOR_TARGET_LINKS, this._config.target);\n    for (const anchor of targetLinks) {\n      // ensure that the anchor has an id and is not disabled\n      if (!anchor.hash || isDisabled(anchor)) {\n        continue;\n      }\n      const observableSection = SelectorEngine.findOne(decodeURI(anchor.hash), this._element);\n\n      // ensure that the observableSection exists & is visible\n      if (isVisible(observableSection)) {\n        this._targetLinks.set(decodeURI(anchor.hash), anchor);\n        this._observableSections.set(anchor.hash, observableSection);\n      }\n    }\n  }\n  _process(target) {\n    if (this._activeTarget === target) {\n      return;\n    }\n    this._clearActiveClass(this._config.target);\n    this._activeTarget = target;\n    target.classList.add(CLASS_NAME_ACTIVE$1);\n    this._activateParents(target);\n    EventHandler.trigger(this._element, EVENT_ACTIVATE, {\n      relatedTarget: target\n    });\n  }\n  _activateParents(target) {\n    // Activate dropdown parents\n    if (target.classList.contains(CLASS_NAME_DROPDOWN_ITEM)) {\n      SelectorEngine.findOne(SELECTOR_DROPDOWN_TOGGLE$1, target.closest(SELECTOR_DROPDOWN)).classList.add(CLASS_NAME_ACTIVE$1);\n      return;\n    }\n    for (const listGroup of SelectorEngine.parents(target, SELECTOR_NAV_LIST_GROUP)) {\n      // Set triggered links parents as active\n      // With both <ul> and <nav> markup a parent is the previous sibling of any nav ancestor\n      for (const item of SelectorEngine.prev(listGroup, SELECTOR_LINK_ITEMS)) {\n        item.classList.add(CLASS_NAME_ACTIVE$1);\n      }\n    }\n  }\n  _clearActiveClass(parent) {\n    parent.classList.remove(CLASS_NAME_ACTIVE$1);\n    const activeNodes = SelectorEngine.find(`${SELECTOR_TARGET_LINKS}.${CLASS_NAME_ACTIVE$1}`, parent);\n    for (const node of activeNodes) {\n      node.classList.remove(CLASS_NAME_ACTIVE$1);\n    }\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = ScrollSpy.getOrCreateInstance(this, config);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(window, EVENT_LOAD_DATA_API$1, () => {\n  for (const spy of SelectorEngine.find(SELECTOR_DATA_SPY)) {\n    ScrollSpy.getOrCreateInstance(spy);\n  }\n});\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(ScrollSpy);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap tab.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME$1 = 'tab';\nconst DATA_KEY$1 = 'bs.tab';\nconst EVENT_KEY$1 = `.${DATA_KEY$1}`;\nconst EVENT_HIDE$1 = `hide${EVENT_KEY$1}`;\nconst EVENT_HIDDEN$1 = `hidden${EVENT_KEY$1}`;\nconst EVENT_SHOW$1 = `show${EVENT_KEY$1}`;\nconst EVENT_SHOWN$1 = `shown${EVENT_KEY$1}`;\nconst EVENT_CLICK_DATA_API = `click${EVENT_KEY$1}`;\nconst EVENT_KEYDOWN = `keydown${EVENT_KEY$1}`;\nconst EVENT_LOAD_DATA_API = `load${EVENT_KEY$1}`;\nconst ARROW_LEFT_KEY = 'ArrowLeft';\nconst ARROW_RIGHT_KEY = 'ArrowRight';\nconst ARROW_UP_KEY = 'ArrowUp';\nconst ARROW_DOWN_KEY = 'ArrowDown';\nconst HOME_KEY = 'Home';\nconst END_KEY = 'End';\nconst CLASS_NAME_ACTIVE = 'active';\nconst CLASS_NAME_FADE$1 = 'fade';\nconst CLASS_NAME_SHOW$1 = 'show';\nconst CLASS_DROPDOWN = 'dropdown';\nconst SELECTOR_DROPDOWN_TOGGLE = '.dropdown-toggle';\nconst SELECTOR_DROPDOWN_MENU = '.dropdown-menu';\nconst NOT_SELECTOR_DROPDOWN_TOGGLE = `:not(${SELECTOR_DROPDOWN_TOGGLE})`;\nconst SELECTOR_TAB_PANEL = '.list-group, .nav, [role=\"tablist\"]';\nconst SELECTOR_OUTER = '.nav-item, .list-group-item';\nconst SELECTOR_INNER = `.nav-link${NOT_SELECTOR_DROPDOWN_TOGGLE}, .list-group-item${NOT_SELECTOR_DROPDOWN_TOGGLE}, [role=\"tab\"]${NOT_SELECTOR_DROPDOWN_TOGGLE}`;\nconst SELECTOR_DATA_TOGGLE = '[data-bs-toggle=\"tab\"], [data-bs-toggle=\"pill\"], [data-bs-toggle=\"list\"]'; // TODO: could only be `tab` in v6\nconst SELECTOR_INNER_ELEM = `${SELECTOR_INNER}, ${SELECTOR_DATA_TOGGLE}`;\nconst SELECTOR_DATA_TOGGLE_ACTIVE = `.${CLASS_NAME_ACTIVE}[data-bs-toggle=\"tab\"], .${CLASS_NAME_ACTIVE}[data-bs-toggle=\"pill\"], .${CLASS_NAME_ACTIVE}[data-bs-toggle=\"list\"]`;\n\n/**\n * Class definition\n */\n\nclass Tab extends BaseComponent {\n  constructor(element) {\n    super(element);\n    this._parent = this._element.closest(SELECTOR_TAB_PANEL);\n    if (!this._parent) {\n      return;\n      // TODO: should throw exception in v6\n      // throw new TypeError(`${element.outerHTML} has not a valid parent ${SELECTOR_INNER_ELEM}`)\n    }\n\n    // Set up initial aria attributes\n    this._setInitialAttributes(this._parent, this._getChildren());\n    EventHandler.on(this._element, EVENT_KEYDOWN, event => this._keydown(event));\n  }\n\n  // Getters\n  static get NAME() {\n    return NAME$1;\n  }\n\n  // Public\n  show() {\n    // Shows this elem and deactivate the active sibling if exists\n    const innerElem = this._element;\n    if (this._elemIsActive(innerElem)) {\n      return;\n    }\n\n    // Search for active tab on same parent to deactivate it\n    const active = this._getActiveElem();\n    const hideEvent = active ? EventHandler.trigger(active, EVENT_HIDE$1, {\n      relatedTarget: innerElem\n    }) : null;\n    const showEvent = EventHandler.trigger(innerElem, EVENT_SHOW$1, {\n      relatedTarget: active\n    });\n    if (showEvent.defaultPrevented || hideEvent && hideEvent.defaultPrevented) {\n      return;\n    }\n    this._deactivate(active, innerElem);\n    this._activate(innerElem, active);\n  }\n\n  // Private\n  _activate(element, relatedElem) {\n    if (!element) {\n      return;\n    }\n    element.classList.add(CLASS_NAME_ACTIVE);\n    this._activate(SelectorEngine.getElementFromSelector(element)); // Search and activate/show the proper section\n\n    const complete = () => {\n      if (element.getAttribute('role') !== 'tab') {\n        element.classList.add(CLASS_NAME_SHOW$1);\n        return;\n      }\n      element.removeAttribute('tabindex');\n      element.setAttribute('aria-selected', true);\n      this._toggleDropDown(element, true);\n      EventHandler.trigger(element, EVENT_SHOWN$1, {\n        relatedTarget: relatedElem\n      });\n    };\n    this._queueCallback(complete, element, element.classList.contains(CLASS_NAME_FADE$1));\n  }\n  _deactivate(element, relatedElem) {\n    if (!element) {\n      return;\n    }\n    element.classList.remove(CLASS_NAME_ACTIVE);\n    element.blur();\n    this._deactivate(SelectorEngine.getElementFromSelector(element)); // Search and deactivate the shown section too\n\n    const complete = () => {\n      if (element.getAttribute('role') !== 'tab') {\n        element.classList.remove(CLASS_NAME_SHOW$1);\n        return;\n      }\n      element.setAttribute('aria-selected', false);\n      element.setAttribute('tabindex', '-1');\n      this._toggleDropDown(element, false);\n      EventHandler.trigger(element, EVENT_HIDDEN$1, {\n        relatedTarget: relatedElem\n      });\n    };\n    this._queueCallback(complete, element, element.classList.contains(CLASS_NAME_FADE$1));\n  }\n  _keydown(event) {\n    if (![ARROW_LEFT_KEY, ARROW_RIGHT_KEY, ARROW_UP_KEY, ARROW_DOWN_KEY, HOME_KEY, END_KEY].includes(event.key)) {\n      return;\n    }\n    event.stopPropagation(); // stopPropagation/preventDefault both added to support up/down keys without scrolling the page\n    event.preventDefault();\n    const children = this._getChildren().filter(element => !isDisabled(element));\n    let nextActiveElement;\n    if ([HOME_KEY, END_KEY].includes(event.key)) {\n      nextActiveElement = children[event.key === HOME_KEY ? 0 : children.length - 1];\n    } else {\n      const isNext = [ARROW_RIGHT_KEY, ARROW_DOWN_KEY].includes(event.key);\n      nextActiveElement = getNextActiveElement(children, event.target, isNext, true);\n    }\n    if (nextActiveElement) {\n      nextActiveElement.focus({\n        preventScroll: true\n      });\n      Tab.getOrCreateInstance(nextActiveElement).show();\n    }\n  }\n  _getChildren() {\n    // collection of inner elements\n    return SelectorEngine.find(SELECTOR_INNER_ELEM, this._parent);\n  }\n  _getActiveElem() {\n    return this._getChildren().find(child => this._elemIsActive(child)) || null;\n  }\n  _setInitialAttributes(parent, children) {\n    this._setAttributeIfNotExists(parent, 'role', 'tablist');\n    for (const child of children) {\n      this._setInitialAttributesOnChild(child);\n    }\n  }\n  _setInitialAttributesOnChild(child) {\n    child = this._getInnerElement(child);\n    const isActive = this._elemIsActive(child);\n    const outerElem = this._getOuterElement(child);\n    child.setAttribute('aria-selected', isActive);\n    if (outerElem !== child) {\n      this._setAttributeIfNotExists(outerElem, 'role', 'presentation');\n    }\n    if (!isActive) {\n      child.setAttribute('tabindex', '-1');\n    }\n    this._setAttributeIfNotExists(child, 'role', 'tab');\n\n    // set attributes to the related panel too\n    this._setInitialAttributesOnTargetPanel(child);\n  }\n  _setInitialAttributesOnTargetPanel(child) {\n    const target = SelectorEngine.getElementFromSelector(child);\n    if (!target) {\n      return;\n    }\n    this._setAttributeIfNotExists(target, 'role', 'tabpanel');\n    if (child.id) {\n      this._setAttributeIfNotExists(target, 'aria-labelledby', `${child.id}`);\n    }\n  }\n  _toggleDropDown(element, open) {\n    const outerElem = this._getOuterElement(element);\n    if (!outerElem.classList.contains(CLASS_DROPDOWN)) {\n      return;\n    }\n    const toggle = (selector, className) => {\n      const element = SelectorEngine.findOne(selector, outerElem);\n      if (element) {\n        element.classList.toggle(className, open);\n      }\n    };\n    toggle(SELECTOR_DROPDOWN_TOGGLE, CLASS_NAME_ACTIVE);\n    toggle(SELECTOR_DROPDOWN_MENU, CLASS_NAME_SHOW$1);\n    outerElem.setAttribute('aria-expanded', open);\n  }\n  _setAttributeIfNotExists(element, attribute, value) {\n    if (!element.hasAttribute(attribute)) {\n      element.setAttribute(attribute, value);\n    }\n  }\n  _elemIsActive(elem) {\n    return elem.classList.contains(CLASS_NAME_ACTIVE);\n  }\n\n  // Try to get the inner element (usually the .nav-link)\n  _getInnerElement(elem) {\n    return elem.matches(SELECTOR_INNER_ELEM) ? elem : SelectorEngine.findOne(SELECTOR_INNER_ELEM, elem);\n  }\n\n  // Try to get the outer element (usually the .nav-item)\n  _getOuterElement(elem) {\n    return elem.closest(SELECTOR_OUTER) || elem;\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Tab.getOrCreateInstance(this);\n      if (typeof config !== 'string') {\n        return;\n      }\n      if (data[config] === undefined || config.startsWith('_') || config === 'constructor') {\n        throw new TypeError(`No method named \"${config}\"`);\n      }\n      data[config]();\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nEventHandler.on(document, EVENT_CLICK_DATA_API, SELECTOR_DATA_TOGGLE, function (event) {\n  if (['A', 'AREA'].includes(this.tagName)) {\n    event.preventDefault();\n  }\n  if (isDisabled(this)) {\n    return;\n  }\n  Tab.getOrCreateInstance(this).show();\n});\n\n/**\n * Initialize on focus\n */\nEventHandler.on(window, EVENT_LOAD_DATA_API, () => {\n  for (const element of SelectorEngine.find(SELECTOR_DATA_TOGGLE_ACTIVE)) {\n    Tab.getOrCreateInstance(element);\n  }\n});\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Tab);\n\n/**\n * --------------------------------------------------------------------------\n * Bootstrap toast.js\n * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n * --------------------------------------------------------------------------\n */\n\n\n/**\n * Constants\n */\n\nconst NAME = 'toast';\nconst DATA_KEY = 'bs.toast';\nconst EVENT_KEY = `.${DATA_KEY}`;\nconst EVENT_MOUSEOVER = `mouseover${EVENT_KEY}`;\nconst EVENT_MOUSEOUT = `mouseout${EVENT_KEY}`;\nconst EVENT_FOCUSIN = `focusin${EVENT_KEY}`;\nconst EVENT_FOCUSOUT = `focusout${EVENT_KEY}`;\nconst EVENT_HIDE = `hide${EVENT_KEY}`;\nconst EVENT_HIDDEN = `hidden${EVENT_KEY}`;\nconst EVENT_SHOW = `show${EVENT_KEY}`;\nconst EVENT_SHOWN = `shown${EVENT_KEY}`;\nconst CLASS_NAME_FADE = 'fade';\nconst CLASS_NAME_HIDE = 'hide'; // @deprecated - kept here only for backwards compatibility\nconst CLASS_NAME_SHOW = 'show';\nconst CLASS_NAME_SHOWING = 'showing';\nconst DefaultType = {\n  animation: 'boolean',\n  autohide: 'boolean',\n  delay: 'number'\n};\nconst Default = {\n  animation: true,\n  autohide: true,\n  delay: 5000\n};\n\n/**\n * Class definition\n */\n\nclass Toast extends BaseComponent {\n  constructor(element, config) {\n    super(element, config);\n    this._timeout = null;\n    this._hasMouseInteraction = false;\n    this._hasKeyboardInteraction = false;\n    this._setListeners();\n  }\n\n  // Getters\n  static get Default() {\n    return Default;\n  }\n  static get DefaultType() {\n    return DefaultType;\n  }\n  static get NAME() {\n    return NAME;\n  }\n\n  // Public\n  show() {\n    const showEvent = EventHandler.trigger(this._element, EVENT_SHOW);\n    if (showEvent.defaultPrevented) {\n      return;\n    }\n    this._clearTimeout();\n    if (this._config.animation) {\n      this._element.classList.add(CLASS_NAME_FADE);\n    }\n    const complete = () => {\n      this._element.classList.remove(CLASS_NAME_SHOWING);\n      EventHandler.trigger(this._element, EVENT_SHOWN);\n      this._maybeScheduleHide();\n    };\n    this._element.classList.remove(CLASS_NAME_HIDE); // @deprecated\n    reflow(this._element);\n    this._element.classList.add(CLASS_NAME_SHOW, CLASS_NAME_SHOWING);\n    this._queueCallback(complete, this._element, this._config.animation);\n  }\n  hide() {\n    if (!this.isShown()) {\n      return;\n    }\n    const hideEvent = EventHandler.trigger(this._element, EVENT_HIDE);\n    if (hideEvent.defaultPrevented) {\n      return;\n    }\n    const complete = () => {\n      this._element.classList.add(CLASS_NAME_HIDE); // @deprecated\n      this._element.classList.remove(CLASS_NAME_SHOWING, CLASS_NAME_SHOW);\n      EventHandler.trigger(this._element, EVENT_HIDDEN);\n    };\n    this._element.classList.add(CLASS_NAME_SHOWING);\n    this._queueCallback(complete, this._element, this._config.animation);\n  }\n  dispose() {\n    this._clearTimeout();\n    if (this.isShown()) {\n      this._element.classList.remove(CLASS_NAME_SHOW);\n    }\n    super.dispose();\n  }\n  isShown() {\n    return this._element.classList.contains(CLASS_NAME_SHOW);\n  }\n\n  // Private\n\n  _maybeScheduleHide() {\n    if (!this._config.autohide) {\n      return;\n    }\n    if (this._hasMouseInteraction || this._hasKeyboardInteraction) {\n      return;\n    }\n    this._timeout = setTimeout(() => {\n      this.hide();\n    }, this._config.delay);\n  }\n  _onInteraction(event, isInteracting) {\n    switch (event.type) {\n      case 'mouseover':\n      case 'mouseout':\n        {\n          this._hasMouseInteraction = isInteracting;\n          break;\n        }\n      case 'focusin':\n      case 'focusout':\n        {\n          this._hasKeyboardInteraction = isInteracting;\n          break;\n        }\n    }\n    if (isInteracting) {\n      this._clearTimeout();\n      return;\n    }\n    const nextElement = event.relatedTarget;\n    if (this._element === nextElement || this._element.contains(nextElement)) {\n      return;\n    }\n    this._maybeScheduleHide();\n  }\n  _setListeners() {\n    EventHandler.on(this._element, EVENT_MOUSEOVER, event => this._onInteraction(event, true));\n    EventHandler.on(this._element, EVENT_MOUSEOUT, event => this._onInteraction(event, false));\n    EventHandler.on(this._element, EVENT_FOCUSIN, event => this._onInteraction(event, true));\n    EventHandler.on(this._element, EVENT_FOCUSOUT, event => this._onInteraction(event, false));\n  }\n  _clearTimeout() {\n    clearTimeout(this._timeout);\n    this._timeout = null;\n  }\n\n  // Static\n  static jQueryInterface(config) {\n    return this.each(function () {\n      const data = Toast.getOrCreateInstance(this, config);\n      if (typeof config === 'string') {\n        if (typeof data[config] === 'undefined') {\n          throw new TypeError(`No method named \"${config}\"`);\n        }\n        data[config](this);\n      }\n    });\n  }\n}\n\n/**\n * Data API implementation\n */\n\nenableDismissTrigger(Toast);\n\n/**\n * jQuery\n */\n\ndefineJQueryPlugin(Toast);\n\nexport { Alert, Button, Carousel, Collapse, Dropdown, Modal, Offcanvas, Popover, ScrollSpy, Tab, Toast, Tooltip };\n//# sourceMappingURL=bootstrap.esm.js.map\n","/* define several functions to replace jQuery methods\n * inspired by https://tobiasahlin.com/blog/move-from-jquery-to-vanilla-javascript/\n */\n\n/**\n * Execute a method if DOM has finished loading\n *\n * @param {function} callback the method to execute\n */\nexport function documentReady(callback) {\n  if 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Symbol(\"webpack error\") : \"__webpack_error__\";\nvar resolveQueue = (queue) => {\n\tif(queue && queue.d < 1) {\n\t\tqueue.d = 1;\n\t\tqueue.forEach((fn) => (fn.r--));\n\t\tqueue.forEach((fn) => (fn.r-- ? fn.r++ : fn()));\n\t}\n}\nvar wrapDeps = (deps) => (deps.map((dep) => {\n\tif(dep !== null && typeof dep === \"object\") {\n\t\tif(dep[webpackQueues]) return dep;\n\t\tif(dep.then) {\n\t\t\tvar queue = [];\n\t\t\tqueue.d = 0;\n\t\t\tdep.then((r) => {\n\t\t\t\tobj[webpackExports] = r;\n\t\t\t\tresolveQueue(queue);\n\t\t\t}, (e) => {\n\t\t\t\tobj[webpackError] = e;\n\t\t\t\tresolveQueue(queue);\n\t\t\t});\n\t\t\tvar obj = {};\n\t\t\tobj[webpackQueues] = (fn) => (fn(queue));\n\t\t\treturn obj;\n\t\t}\n\t}\n\tvar ret = {};\n\tret[webpackQueues] = x => {};\n\tret[webpackExports] = dep;\n\treturn ret;\n}));\n__webpack_require__.a = (module, body, hasAwait) => {\n\tvar queue;\n\thasAwait && ((queue = []).d = -1);\n\tvar depQueues = new Set();\n\tvar exports = module.exports;\n\tvar currentDeps;\n\tvar outerResolve;\n\tvar reject;\n\tvar promise = new Promise((resolve, rej) => {\n\t\treject = rej;\n\t\touterResolve = resolve;\n\t});\n\tpromise[webpackExports] = exports;\n\tpromise[webpackQueues] = (fn) => (queue && fn(queue), depQueues.forEach(fn), promise[\"catch\"](x => {}));\n\tmodule.exports = promise;\n\tbody((deps) => {\n\t\tcurrentDeps = wrapDeps(deps);\n\t\tvar fn;\n\t\tvar getResult = () => (currentDeps.map((d) => {\n\t\t\tif(d[webpackError]) throw d[webpackError];\n\t\t\treturn d[webpackExports];\n\t\t}))\n\t\tvar promise = new Promise((resolve) => {\n\t\t\tfn = () => (resolve(getResult));\n\t\t\tfn.r = 0;\n\t\t\tvar fnQueue = (q) => (q !== queue && !depQueues.has(q) && (depQueues.add(q), q && !q.d && (fn.r++, q.push(fn))));\n\t\t\tcurrentDeps.map((dep) => (dep[webpackQueues](fnQueue)));\n\t\t});\n\t\treturn fn.r ? promise : getResult();\n\t}, (err) => ((err ? reject(promise[webpackError] = err) : outerResolve(exports)), resolveQueue(queue)));\n\tqueue && queue.d < 0 && (queue.d = 0);\n};","/**\n * Compare [semver](https://semver.org/) version strings to find greater, equal or lesser.\n * This library supports the full semver specification, including comparing versions with different number of digits like `1.0.0`, `1.0`, `1`, and pre-release versions like `1.0.0-alpha`.\n * @param v1 - First version to compare\n * @param v2 - Second version to compare\n * @returns Numeric value compatible with the [Array.sort(fn) interface](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/sort#Parameters).\n */\nexport const compareVersions = (v1, v2) => {\n    // validate input and split into segments\n    const n1 = validateAndParse(v1);\n    const n2 = validateAndParse(v2);\n    // pop off the patch\n    const p1 = n1.pop();\n    const p2 = n2.pop();\n    // validate numbers\n    const r = compareSegments(n1, n2);\n    if (r !== 0)\n        return r;\n    // validate pre-release\n    if (p1 && p2) {\n        return compareSegments(p1.split('.'), p2.split('.'));\n    }\n    else if (p1 || p2) {\n        return p1 ? -1 : 1;\n    }\n    return 0;\n};\n/**\n * Validate [semver](https://semver.org/) version strings.\n *\n * @param version Version number to validate\n * @returns `true` if the version number is a valid semver version number, `false` otherwise.\n *\n * @example\n * ```\n * validate('1.0.0-rc.1'); // return true\n * validate('1.0-rc.1'); // return false\n * validate('foo'); // return false\n * ```\n */\nexport const validate = (version) => typeof version === 'string' && /^[v\\d]/.test(version) && semver.test(version);\n/**\n * Compare [semver](https://semver.org/) version strings using the specified operator.\n *\n * @param v1 First version to compare\n * @param v2 Second version to compare\n * @param operator Allowed arithmetic operator to use\n * @returns `true` if the comparison between the firstVersion and the secondVersion satisfies the operator, `false` otherwise.\n *\n * @example\n * ```\n * compare('10.1.8', '10.0.4', '>'); // return true\n * compare('10.0.1', '10.0.1', '='); // return true\n * compare('10.1.1', '10.2.2', '<'); // return true\n * compare('10.1.1', '10.2.2', '<='); // return true\n * compare('10.1.1', '10.2.2', '>='); // return false\n * ```\n */\nexport const compare = (v1, v2, operator) => {\n    // validate input operator\n    assertValidOperator(operator);\n    // since result of compareVersions can only be -1 or 0 or 1\n    // a simple map can be used to replace switch\n    const res = compareVersions(v1, v2);\n    return operatorResMap[operator].includes(res);\n};\n/**\n * Match [npm semver](https://docs.npmjs.com/cli/v6/using-npm/semver) version range.\n *\n * @param version Version number to match\n * @param range Range pattern for version\n * @returns `true` if the version number is within the range, `false` otherwise.\n *\n * @example\n * ```\n * satisfies('1.1.0', '^1.0.0'); // return true\n * satisfies('1.1.0', '~1.0.0'); // return false\n * ```\n */\nexport const satisfies = (version, range) => {\n    // if no range operator then \"=\"\n    const m = range.match(/^([<>=~^]+)/);\n    const op = m ? m[1] : '=';\n    // if gt/lt/eq then operator compare\n    if (op !== '^' && op !== '~')\n        return compare(version, range, op);\n    // else range of either \"~\" or \"^\" is assumed\n    const [v1, v2, v3, , vp] = validateAndParse(version);\n    const [r1, r2, r3, , rp] = validateAndParse(range);\n    const v = [v1, v2, v3];\n    const r = [r1, r2 !== null && r2 !== void 0 ? r2 : 'x', r3 !== null && r3 !== void 0 ? r3 : 'x'];\n    // validate pre-release\n    if (rp) {\n        if (!vp)\n            return false;\n        if (compareSegments(v, r) !== 0)\n            return false;\n        if (compareSegments(vp.split('.'), rp.split('.')) === -1)\n            return false;\n    }\n    // first non-zero number\n    const nonZero = r.findIndex((v) => v !== '0') + 1;\n    // pointer to where segments can be >=\n    const i = op === '~' ? 2 : nonZero > 1 ? nonZero : 1;\n    // before pointer must be equal\n    if (compareSegments(v.slice(0, i), r.slice(0, i)) !== 0)\n        return false;\n    // after pointer must be >=\n    if (compareSegments(v.slice(i), r.slice(i)) === -1)\n        return false;\n    return true;\n};\nconst semver = /^[v^~<>=]*?(\\d+)(?:\\.([x*]|\\d+)(?:\\.([x*]|\\d+)(?:\\.([x*]|\\d+))?(?:-([\\da-z\\-]+(?:\\.[\\da-z\\-]+)*))?(?:\\+[\\da-z\\-]+(?:\\.[\\da-z\\-]+)*)?)?)?$/i;\nconst validateAndParse = (version) => {\n    if (typeof version !== 'string') {\n        throw new TypeError('Invalid argument expected string');\n    }\n    const match = version.match(semver);\n    if (!match) {\n        throw new Error(`Invalid argument not valid semver ('${version}' received)`);\n    }\n    match.shift();\n    return match;\n};\nconst isWildcard = (s) => s === '*' || s === 'x' || s === 'X';\nconst tryParse = (v) => {\n    const n = parseInt(v, 10);\n    return isNaN(n) ? v : n;\n};\nconst forceType = (a, b) => typeof a !== typeof b ? [String(a), String(b)] : [a, b];\nconst compareStrings = (a, b) => {\n    if (isWildcard(a) || isWildcard(b))\n        return 0;\n    const [ap, bp] = forceType(tryParse(a), tryParse(b));\n    if (ap > bp)\n        return 1;\n    if (ap < bp)\n        return -1;\n    return 0;\n};\nconst compareSegments = (a, b) => {\n    for (let i = 0; i < Math.max(a.length, b.length); i++) {\n        const r = compareStrings(a[i] || '0', b[i] || '0');\n        if (r !== 0)\n            return r;\n    }\n    return 0;\n};\nconst operatorResMap = {\n    '>': [1],\n    '>=': [0, 1],\n    '=': [0],\n    '<=': [-1, 0],\n    '<': [-1],\n};\nconst allowedOperators = Object.keys(operatorResMap);\nconst assertValidOperator = (op) => {\n    if (typeof op !== 'string') {\n        throw new TypeError(`Invalid operator type, expected string but got ${typeof op}`);\n    }\n    if (allowedOperators.indexOf(op) === -1) {\n        throw new Error(`Invalid operator, expected one of ${allowedOperators.join('|')}`);\n    }\n};\n//# sourceMappingURL=index.js.map","/* define several functions to replace jQuery methods\n * inspired by https://tobiasahlin.com/blog/move-from-jquery-to-vanilla-javascript/\n */\n\n/**\n * Execute a method if DOM has finished loading\n *\n * @param {function} callback the method to execute\n */\nexport function documentReady(callback) {\n  if (document.readyState != \"loading\") callback();\n  else document.addEventListener(\"DOMContentLoaded\", callback);\n}\n","// Define the custom behavior of the page\nimport { documentReady } from \"./mixin\";\nimport { compare, validate } from \"compare-versions\";\n\nimport \"../styles/pydata-sphinx-theme.scss\";\n\n/*******************************************************************************\n * Theme interaction\n */\n\nvar prefersDark = window.matchMedia(\"(prefers-color-scheme: dark)\");\n\n/**\n * set the the body theme to the one specified by the user browser\n *\n * @param {event} e\n */\nfunction autoTheme(e) {\n  document.documentElement.dataset.theme = prefersDark.matches\n    ? \"dark\"\n    : \"light\";\n}\n\n/**\n * Set the theme using the specified mode.\n * It can be one of [\"auto\", \"dark\", \"light\"]\n *\n * @param {str} mode\n */\nfunction setTheme(mode) {\n  if (mode !== \"light\" && mode !== \"dark\" && mode !== \"auto\") {\n    console.error(`Got invalid theme mode: ${mode}. Resetting to auto.`);\n    mode = \"auto\";\n  }\n\n  // get the theme\n  var colorScheme = prefersDark.matches ? \"dark\" : \"light\";\n  document.documentElement.dataset.mode = mode;\n  var theme = mode == \"auto\" ? colorScheme : mode;\n  document.documentElement.dataset.theme = theme;\n  // TODO: remove this line after Bootstrap upgrade\n  // v5.3 has a colors mode: https://getbootstrap.com/docs/5.3/customize/color-modes/\n  document.querySelectorAll(\".dropdown-menu\").forEach((el) => {\n    if (theme === \"dark\") {\n      el.classList.add(\"dropdown-menu-dark\");\n    } else {\n      el.classList.remove(\"dropdown-menu-dark\");\n    }\n  });\n\n  // save mode and theme\n  localStorage.setItem(\"mode\", mode);\n  localStorage.setItem(\"theme\", theme);\n  console.log(`[PST]: Changed to ${mode} mode using the ${theme} theme.`);\n\n  // add a listener if set on auto\n  prefersDark.onchange = mode == \"auto\" ? autoTheme : \"\";\n}\n\n/**\n * Change the theme option order so that clicking on the btn is always a change\n * from \"auto\"\n */\nfunction cycleMode() {\n  const defaultMode = document.documentElement.dataset.defaultMode || \"auto\";\n  const currentMode = localStorage.getItem(\"mode\") || defaultMode;\n\n  var loopArray = (arr, current) => {\n    var nextPosition = arr.indexOf(current) + 1;\n    if (nextPosition === arr.length) {\n      nextPosition = 0;\n    }\n    return arr[nextPosition];\n  };\n\n  // make sure the next theme after auto is always a change\n  var modeList = prefersDark.matches\n    ? [\"auto\", \"light\", \"dark\"]\n    : [\"auto\", \"dark\", \"light\"];\n  var newMode = loopArray(modeList, currentMode);\n  setTheme(newMode);\n}\n\n/**\n * add the theme listener on the btns of the navbar\n */\nfunction addModeListener() {\n  // the theme was set a first time using the initial mini-script\n  // running setMode will ensure the use of the dark mode if auto is selected\n  setTheme(document.documentElement.dataset.mode);\n\n  // Attach event handlers for toggling themes colors\n  document.querySelectorAll(\".theme-switch-button\").forEach((el) => {\n    el.addEventListener(\"click\", cycleMode);\n  });\n}\n\n/*******************************************************************************\n * TOC interactivity\n */\n\n/**\n * TOC sidebar - add \"active\" class to parent list\n *\n * Bootstrap's scrollspy adds the active class to the <a> link,\n * but for the automatic collapsing we need this on the parent list item.\n *\n * The event is triggered on \"window\" (and not the nav item as documented),\n * see https://github.com/twbs/bootstrap/issues/20086\n */\nfunction addTOCInteractivity() {\n  window.addEventListener(\"activate.bs.scrollspy\", function () {\n    const navLinks = document.querySelectorAll(\".bd-toc-nav a\");\n\n    navLinks.forEach((navLink) => {\n      navLink.parentElement.classList.remove(\"active\");\n    });\n\n    const activeNavLinks = document.querySelectorAll(\".bd-toc-nav a.active\");\n    activeNavLinks.forEach((navLink) => {\n      navLink.parentElement.classList.add(\"active\");\n    });\n  });\n}\n\n/*******************************************************************************\n * Scroll\n */\n\n/**\n * Navigation sidebar scrolling to active page\n */\nfunction scrollToActive() {\n  // If the docs nav doesn't exist, do nothing (e.g., on search page)\n  if (!document.querySelector(\".bd-docs-nav\")) {\n    return;\n  }\n\n  var sidebar = document.querySelector(\"div.bd-sidebar\");\n\n  // Remember the sidebar scroll position between page loads\n  // Inspired on source of revealjs.com\n  let storedScrollTop = parseInt(\n    sessionStorage.getItem(\"sidebar-scroll-top\"),\n    10\n  );\n\n  if (!isNaN(storedScrollTop)) {\n    // If we've got a saved scroll position, just use that\n    sidebar.scrollTop = storedScrollTop;\n    console.log(\"[PST]: Scrolled sidebar using stored browser position...\");\n  } else {\n    // Otherwise, calculate a position to scroll to based on the lowest `active` link\n    var sidebarNav = document.querySelector(\".bd-docs-nav\");\n    var active_pages = sidebarNav.querySelectorAll(\".active\");\n    if (active_pages.length > 0) {\n      // Use the last active page as the offset since it's the page we're on\n      var latest_active = active_pages[active_pages.length - 1];\n      var offset =\n        latest_active.getBoundingClientRect().y -\n        sidebar.getBoundingClientRect().y;\n      // Only scroll the navbar if the active link is lower than 50% of the page\n      if (latest_active.getBoundingClientRect().y > window.innerHeight * 0.5) {\n        let buffer = 0.25; // Buffer so we have some space above the scrolled item\n        sidebar.scrollTop = offset - sidebar.clientHeight * buffer;\n        console.log(\"[PST]: Scrolled sidebar using last active link...\");\n      }\n    }\n  }\n\n  // Store the sidebar scroll position\n  window.addEventListener(\"beforeunload\", () => {\n    sessionStorage.setItem(\"sidebar-scroll-top\", sidebar.scrollTop);\n  });\n}\n\n/*******************************************************************************\n * Search\n */\n\n/**\n * Find any search forms on the page and return their input element\n */\nvar findSearchInput = () => {\n  let forms = document.querySelectorAll(\"form.bd-search\");\n  if (!forms.length) {\n    // no search form found\n    return;\n  } else {\n    var form;\n    if (forms.length == 1) {\n      // there is exactly one search form (persistent or hidden)\n      form = forms[0];\n    } else {\n      // must be at least one persistent form, use the first persistent one\n      form = document.querySelector(\n        \"div:not(.search-button__search-container) > form.bd-search\"\n      );\n    }\n    return form.querySelector(\"input\");\n  }\n};\n\n/**\n * Activate the search field on the page.\n * - If there is a search field already visible it will be activated.\n * - If not, then a search field will pop up.\n */\nvar toggleSearchField = () => {\n  // Find the search input to highlight\n  let input = findSearchInput();\n\n  // if the input field is the hidden one (the one associated with the\n  // search button) then toggle the button state (to show/hide the field)\n  let searchPopupWrapper = document.querySelector(\".search-button__wrapper\");\n  let hiddenInput = searchPopupWrapper.querySelector(\"input\");\n  if (input === hiddenInput) {\n    searchPopupWrapper.classList.toggle(\"show\");\n  }\n  // when toggling off the search field, remove its focus\n  if (document.activeElement === input) {\n    input.blur();\n  } else {\n    input.focus();\n    input.select();\n    input.scrollIntoView({ block: \"center\" });\n  }\n};\n\n/**\n * Add an event listener for toggleSearchField() for Ctrl/Cmd + K\n */\nvar addEventListenerForSearchKeyboard = () => {\n  window.addEventListener(\n    \"keydown\",\n    (event) => {\n      let input = findSearchInput();\n      // toggle on Ctrl+k or ⌘+k\n      if (\n        // Ignore if shift or alt are pressed\n        !event.shiftKey &&\n        !event.altKey &&\n        // On Mac use ⌘, all other OS use Ctrl\n        (useCommandKey\n          ? event.metaKey && !event.ctrlKey\n          : !event.metaKey && event.ctrlKey) &&\n        // Case-insensitive so the shortcut still works with caps lock\n        /^k$/i.test(event.key)\n      ) {\n        event.preventDefault();\n        toggleSearchField();\n      }\n      // also allow Escape key to hide (but not show) the dynamic search field\n      else if (document.activeElement === input && /Escape/i.test(event.key)) {\n        toggleSearchField();\n      }\n    },\n    true\n  );\n};\n\n/**\n * If the user is on a Mac, use command (⌘) instead of control (ctrl) key\n *\n * Note: `navigator.platform` is deprecated; however MDN still recommends using\n * it for the one specific use case of detecting whether a keyboard shortcut\n * should use control or command:\n * https://developer.mozilla.org/en-US/docs/Web/API/Navigator/platform#examples\n */\nvar useCommandKey =\n  navigator.platform.indexOf(\"Mac\") === 0 || navigator.platform === \"iPhone\";\n\n/**\n * Change the search hint to `meta key` if we are a Mac\n */\n\nvar changeSearchShortcutKey = () => {\n  let shortcuts = document.querySelectorAll(\".search-button__kbd-shortcut\");\n  if (useCommandKey) {\n    shortcuts.forEach(\n      (f) => (f.querySelector(\"kbd.kbd-shortcut__modifier\").innerText = \"⌘\")\n    );\n  }\n};\n\n/**\n * Activate callbacks for search button popup\n */\nvar setupSearchButtons = () => {\n  changeSearchShortcutKey();\n  addEventListenerForSearchKeyboard();\n\n  // Add the search button trigger event callback\n  document.querySelectorAll(\".search-button__button\").forEach((btn) => {\n    btn.onclick = toggleSearchField;\n  });\n\n  // Add the search button overlay event callback\n  let overlay = document.querySelector(\".search-button__overlay\");\n  if (overlay) {\n    overlay.onclick = toggleSearchField;\n  }\n};\n\n/*******************************************************************************\n * Version Switcher\n * Note that this depends on two variables existing that are defined in\n * and `html-page-context` hook:\n *\n * - DOCUMENTATION_OPTIONS.pagename\n * - DOCUMENTATION_OPTIONS.theme_switcher_url\n */\n\n/**\n * Check if corresponding page path exists in other version of docs\n * and, if so, go there instead of the homepage of the other docs version\n *\n * @param {event} event the event that trigger the check\n */\nasync function checkPageExistsAndRedirect(event) {\n  // ensure we don't follow the initial link\n  event.preventDefault();\n  let currentFilePath = `${DOCUMENTATION_OPTIONS.pagename}.html`;\n  let tryUrl = event.currentTarget.getAttribute(\"href\");\n  let otherDocsHomepage = tryUrl.replace(currentFilePath, \"\");\n  try {\n    let head = await fetch(tryUrl, { method: \"HEAD\" });\n    if (head.ok) {\n      location.href = tryUrl; // the page exists, go there\n    } else {\n      location.href = otherDocsHomepage;\n    }\n  } catch (err) {\n    // something went wrong, probably CORS restriction, fallback to other docs homepage\n    location.href = otherDocsHomepage;\n  }\n}\n\n/**\n * Load and parse the version switcher JSON file from an absolute or relative URL.\n *\n * @param {string} url The URL to load version switcher entries from.\n */\nasync function fetchVersionSwitcherJSON(url) {\n  // first check if it's a valid URL\n  try {\n    var result = new URL(url);\n  } catch (err) {\n    if (err instanceof TypeError) {\n      // assume we got a relative path, and fix accordingly. But first, we need to\n      // use `fetch()` to follow redirects so we get the correct final base URL\n      const origin = await fetch(window.location.origin, { method: \"HEAD\" });\n      result = new URL(url, origin.url);\n    } else {\n      // something unexpected happened\n      throw err;\n    }\n  }\n  // load and return the JSON\n  const response = await fetch(result);\n  const data = await response.json();\n  return data;\n}\n\n// Populate the version switcher from the JSON data\nfunction populateVersionSwitcher(data, versionSwitcherBtns) {\n  const currentFilePath = `${DOCUMENTATION_OPTIONS.pagename}.html`;\n  versionSwitcherBtns.forEach((btn) => {\n    // Set empty strings by default so that these attributes exist and can be used in CSS selectors\n    btn.dataset[\"activeVersionName\"] = \"\";\n    btn.dataset[\"activeVersion\"] = \"\";\n  });\n  // in case there are multiple entries with the same version string, this helps us\n  // decide which entry's `name` to put on the button itself. Without this, it would\n  // always be the *last* version-matching entry; now it will be either the\n  // version-matching entry that is also marked as `\"preferred\": true`, or if that\n  // doesn't exist: the *first* version-matching entry.\n  data = data.map((entry) => {\n    // does this entry match the version that we're currently building/viewing?\n    entry.match =\n      entry.version == DOCUMENTATION_OPTIONS.theme_switcher_version_match;\n    entry.preferred = entry.preferred || false;\n    // if no custom name specified (e.g., \"latest\"), use version string\n    if (!(\"name\" in entry)) {\n      entry.name = entry.version;\n    }\n    return entry;\n  });\n  const hasMatchingPreferredEntry = data\n    .map((entry) => entry.preferred && entry.match)\n    .some(Boolean);\n  var foundMatch = false;\n  // create links to the corresponding page in the other docs versions\n  data.forEach((entry) => {\n    // create the node\n    const anchor = document.createElement(\"a\");\n    anchor.setAttribute(\n      \"class\",\n      \"dropdown-item list-group-item list-group-item-action py-1\"\n    );\n    anchor.setAttribute(\"href\", `${entry.url}${currentFilePath}`);\n    anchor.setAttribute(\"role\", \"option\");\n    const span = document.createElement(\"span\");\n    span.textContent = `${entry.name}`;\n    anchor.appendChild(span);\n    // Add dataset values for the version and name in case people want\n    // to apply CSS styling based on this information.\n    anchor.dataset[\"versionName\"] = entry.name;\n    anchor.dataset[\"version\"] = entry.version;\n    // replace dropdown button text with the preferred display name of the\n    // currently-viewed version, rather than using sphinx's {{ version }} variable.\n    // also highlight the dropdown entry for the currently-viewed version's entry\n    let matchesAndIsPreferred = hasMatchingPreferredEntry && entry.preferred;\n    let matchesAndIsFirst =\n      !hasMatchingPreferredEntry && !foundMatch && entry.match;\n    if (matchesAndIsPreferred || matchesAndIsFirst) {\n      anchor.classList.add(\"active\");\n      versionSwitcherBtns.forEach((btn) => {\n        btn.innerText = entry.name;\n        btn.dataset[\"activeVersionName\"] = entry.name;\n        btn.dataset[\"activeVersion\"] = entry.version;\n      });\n      foundMatch = true;\n    }\n    // There may be multiple version-switcher elements, e.g. one\n    // in a slide-over panel displayed on smaller screens.\n    document.querySelectorAll(\".version-switcher__menu\").forEach((menu) => {\n      // we need to clone the node for each menu, but onclick attributes are not\n      // preserved by `.cloneNode()` so we add onclick here after cloning.\n      let node = anchor.cloneNode(true);\n      node.onclick = checkPageExistsAndRedirect;\n      // on click, AJAX calls will check if the linked page exists before\n      // trying to redirect, and if not, will redirect to the homepage\n      // for that version of the docs.\n      menu.append(node);\n    });\n  });\n}\n\n/*******************************************************************************\n * Warning banner when viewing non-stable version of the docs.\n */\n\n/**\n * Show a warning banner when viewing a non-stable version of the docs.\n *\n * adapted 2023-06 from https://mne.tools/versionwarning.js, which was\n * originally adapted 2020-05 from https://scikit-learn.org/versionwarning.js\n *\n * @param {Array} data The version data used to populate the switcher menu.\n */\nfunction showVersionWarningBanner(data) {\n  var version = DOCUMENTATION_OPTIONS.VERSION;\n  // figure out what latest stable version is\n  var preferredEntries = data.filter((entry) => entry.preferred);\n  if (preferredEntries.length !== 1) {\n    const howMany = preferredEntries.length == 0 ? \"No\" : \"Multiple\";\n    console.log(\n      `[PST] ${howMany} versions marked \"preferred\" found in versions JSON, ignoring.`\n    );\n    return;\n  }\n  const preferredVersion = 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https://developer.mozilla.org/en-US/docs/Web/API/Fullscreen_API\n */\nvar toggleFullScreen = () => {\n  var isInFullScreen =\n    (document.fullscreenElement && document.fullscreenElement !== null) ||\n    (document.webkitFullscreenElement &&\n      document.webkitFullscreenElement !== null);\n  let docElm = document.documentElement;\n  if (!isInFullScreen) {\n    console.log(\"[SBT]: Entering full screen\");\n    if (docElm.requestFullscreen) {\n      docElm.requestFullscreen();\n    } else if (docElm.webkitRequestFullscreen) {\n      docElm.webkitRequestFullscreen();\n    }\n  } else {\n    console.log(\"[SBT]: Exiting full screen\");\n    if (document.exitFullscreen) {\n      document.exitFullscreen();\n    } else if (document.webkitExitFullscreen) {\n      document.webkitExitFullscreen();\n    }\n  }\n};\n\n/**\n * Manage scrolling behavior. This is primarily two things:\n *\n * 1. Hide the Table of Contents any time sidebar content is on the screen.\n *\n * This will be triggered any time a sidebar item enters or exits the screen.\n * It adds/removes items from an array if they have entered the screen, and\n * removes them when they exit the screen. It hides the TOC if anything is\n * on-screen.\n *\n * ref: https://developer.mozilla.org/en-US/docs/Web/API/Intersection_Observer_API\n *\n * 2. Add a `scrolled` class to <body> to trigger CSS changes.\n */\nvar initTocHide = () => {\n  var onScreenItems = [];\n  let hideTocCallback = (entries, observer) => {\n    // Check whether any sidebar item is displayed\n    entries.forEach((entry) => {\n      if (entry.isIntersecting) {\n        // If an element just came on screen, add it our list\n        onScreenItems.push(entry.target);\n      } else {\n        // Otherwise, if it's in our list then remove it\n        for (let ii = 0; ii < onScreenItems.length; ii++) {\n          if (onScreenItems[ii] === entry.target) {\n            onScreenItems.splice(ii, 1);\n            break;\n          }\n        }\n      }\n    });\n\n    // Hide the TOC if any margin content is displayed on the screen\n    if (onScreenItems.length > 0) {\n      document.querySelector(\"div.bd-sidebar-secondary\").classList.add(\"hide\");\n    } else {\n      document\n        .querySelector(\"div.bd-sidebar-secondary\")\n        .classList.remove(\"hide\");\n    }\n  };\n  let manageScrolledClassOnBody = (entries, observer) => {\n    // The pixel is at the top, so if we're < 0 that it means we've scrolled\n    if (entries[0].boundingClientRect.y < 0) {\n      document.body.classList.add(\"scrolled\");\n    } else {\n      document.body.classList.remove(\"scrolled\");\n    }\n  };\n\n  // Set up the intersection observer to watch all margin content\n  let options = {\n    // Trigger callback when the top of a margin item is 1/3 up the screen\n    rootMargin: \"0px 0px -33% 0px\",\n  };\n  let tocObserver = new IntersectionObserver(hideTocCallback, options);\n  // TODO: deprecate popout after v0.5.0\n  const selectorClasses = [\n    \"marginnote\",\n    \"sidenote\",\n    \"margin\",\n    \"margin-caption\",\n    \"full-width\",\n    \"sidebar\",\n    \"popout\",\n  ];\n  let marginSelector = [];\n  selectorClasses.forEach((ii) => {\n    // Use three permutations of each class name because `tag_` and `_` used to be supported\n    marginSelector.push(\n      ...[\n        `.${ii}`,\n        `.tag_${ii}`,\n        `.${ii.replace(\"-\", \"_\")}`,\n        `.tag_${ii.replace(\"-\", \"_\")}`,\n      ],\n    );\n  });\n  document.querySelectorAll(marginSelector.join(\", \")).forEach((ii) => {\n    tocObserver.observe(ii);\n  });\n\n  // Set up the observer to check if we've scrolled from top of page\n  let scrollObserver = new IntersectionObserver(manageScrolledClassOnBody);\n  scrollObserver.observe(document.querySelector(\".sbt-scroll-pixel-helper\"));\n};\n\n/**\n * Activate Thebe with a custom button click.\n */\nvar initThebeSBT = () => {\n  var title = document.querySelector(\"section h1\");\n  var sibling = title.nextElementSibling;\n  // If the next element after the title isn't a thebe button, add one now.\n  // That way it is initiatlized when thebe is first-clicked and isn't re-added after.\n  if (!sibling.classList.contains(\"thebe-launch-button\")) {\n    title.insertAdjacentHTML(\n      \"afterend\",\n      \"<button class='thebe-launch-button'></button>\",\n    );\n  }\n  // This function is provided by sphinx-thebe\n  initThebe();\n};\n\n/**\n * Add no print class to certain DOM elements\n */\n\nfunction addNoPrint() {\n  var noPrintSelector = [\n    \".bd-header-announcement\",\n    \".bd-header\",\n    \".bd-header-article\",\n    \".bd-sidebar-primary\",\n    \".bd-sidebar-secondary\",\n    \".bd-footer-article\",\n    \".bd-footer-content\",\n    \".bd-footer\",\n  ].join(\",\");\n  document.querySelectorAll(noPrintSelector).forEach((ii) => {\n    ii.classList.add(\"noprint\");\n  });\n}\n\n/**\n * Set up callback functions for UI click actions\n */\nwindow.initThebeSBT = initThebeSBT;\nwindow.toggleFullScreen = toggleFullScreen;\n\n/**\n * Set up functions to load when the DOM is ready\n */\nsbRunWhenDOMLoaded(initTocHide);\nsbRunWhenDOMLoaded(addNoPrint);\n"],"sourceRoot":""}
\ No newline at end of file
diff --git a/_static/searchtools.js b/_static/searchtools.js
new file mode 100644
index 0000000..7918c3f
--- /dev/null
+++ b/_static/searchtools.js
@@ -0,0 +1,574 @@
+/*
+ * searchtools.js
+ * ~~~~~~~~~~~~~~~~
+ *
+ * Sphinx JavaScript utilities for the full-text search.
+ *
+ * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS.
+ * :license: BSD, see LICENSE for details.
+ *
+ */
+"use strict";
+
+/**
+ * Simple result scoring code.
+ */
+if (typeof Scorer === "undefined") {
+  var Scorer = {
+    // Implement the following function to further tweak the score for each result
+    // The function takes a result array [docname, title, anchor, descr, score, filename]
+    // and returns the new score.
+    /*
+    score: result => {
+      const [docname, title, anchor, descr, score, filename] = result
+      return score
+    },
+    */
+
+    // query matches the full name of an object
+    objNameMatch: 11,
+    // or matches in the last dotted part of the object name
+    objPartialMatch: 6,
+    // Additive scores depending on the priority of the object
+    objPrio: {
+      0: 15, // used to be importantResults
+      1: 5, // used to be objectResults
+      2: -5, // used to be unimportantResults
+    },
+    //  Used when the priority is not in the mapping.
+    objPrioDefault: 0,
+
+    // query found in title
+    title: 15,
+    partialTitle: 7,
+    // query found in terms
+    term: 5,
+    partialTerm: 2,
+  };
+}
+
+const _removeChildren = (element) => {
+  while (element && element.lastChild) element.removeChild(element.lastChild);
+};
+
+/**
+ * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping
+ */
+const _escapeRegExp = (string) =>
+  string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string
+
+const _displayItem = (item, searchTerms, highlightTerms) => {
+  const docBuilder = DOCUMENTATION_OPTIONS.BUILDER;
+  const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX;
+  const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX;
+  const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY;
+  const contentRoot = document.documentElement.dataset.content_root;
+
+  const [docName, title, anchor, descr, score, _filename] = item;
+
+  let listItem = document.createElement("li");
+  let requestUrl;
+  let linkUrl;
+  if (docBuilder === "dirhtml") {
+    // dirhtml builder
+    let dirname = docName + "/";
+    if (dirname.match(/\/index\/$/))
+      dirname = dirname.substring(0, dirname.length - 6);
+    else if (dirname === "index/") dirname = "";
+    requestUrl = contentRoot + dirname;
+    linkUrl = requestUrl;
+  } else {
+    // normal html builders
+    requestUrl = contentRoot + docName + docFileSuffix;
+    linkUrl = docName + docLinkSuffix;
+  }
+  let linkEl = listItem.appendChild(document.createElement("a"));
+  linkEl.href = linkUrl + anchor;
+  linkEl.dataset.score = score;
+  linkEl.innerHTML = title;
+  if (descr) {
+    listItem.appendChild(document.createElement("span")).innerHTML =
+      " (" + descr + ")";
+    // highlight search terms in the description
+    if (SPHINX_HIGHLIGHT_ENABLED)  // set in sphinx_highlight.js
+      highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted"));
+  }
+  else if (showSearchSummary)
+    fetch(requestUrl)
+      .then((responseData) => responseData.text())
+      .then((data) => {
+        if (data)
+          listItem.appendChild(
+            Search.makeSearchSummary(data, searchTerms)
+          );
+        // highlight search terms in the summary
+        if (SPHINX_HIGHLIGHT_ENABLED)  // set in sphinx_highlight.js
+          highlightTerms.forEach((term) => _highlightText(listItem, term, "highlighted"));
+      });
+  Search.output.appendChild(listItem);
+};
+const _finishSearch = (resultCount) => {
+  Search.stopPulse();
+  Search.title.innerText = _("Search Results");
+  if (!resultCount)
+    Search.status.innerText = Documentation.gettext(
+      "Your search did not match any documents. Please make sure that all words are spelled correctly and that you've selected enough categories."
+    );
+  else
+    Search.status.innerText = _(
+      `Search finished, found ${resultCount} page(s) matching the search query.`
+    );
+};
+const _displayNextItem = (
+  results,
+  resultCount,
+  searchTerms,
+  highlightTerms,
+) => {
+  // results left, load the summary and display it
+  // this is intended to be dynamic (don't sub resultsCount)
+  if (results.length) {
+    _displayItem(results.pop(), searchTerms, highlightTerms);
+    setTimeout(
+      () => _displayNextItem(results, resultCount, searchTerms, highlightTerms),
+      5
+    );
+  }
+  // search finished, update title and status message
+  else _finishSearch(resultCount);
+};
+
+/**
+ * Default splitQuery function. Can be overridden in ``sphinx.search`` with a
+ * custom function per language.
+ *
+ * The regular expression works by splitting the string on consecutive characters
+ * that are not Unicode letters, numbers, underscores, or emoji characters.
+ * This is the same as ``\W+`` in Python, preserving the surrogate pair area.
+ */
+if (typeof splitQuery === "undefined") {
+  var splitQuery = (query) => query
+      .split(/[^\p{Letter}\p{Number}_\p{Emoji_Presentation}]+/gu)
+      .filter(term => term)  // remove remaining empty strings
+}
+
+/**
+ * Search Module
+ */
+const Search = {
+  _index: null,
+  _queued_query: null,
+  _pulse_status: -1,
+
+  htmlToText: (htmlString) => {
+    const htmlElement = new DOMParser().parseFromString(htmlString, 'text/html');
+    htmlElement.querySelectorAll(".headerlink").forEach((el) => { el.remove() });
+    const docContent = htmlElement.querySelector('[role="main"]');
+    if (docContent !== undefined) return docContent.textContent;
+    console.warn(
+      "Content block not found. Sphinx search tries to obtain it via '[role=main]'. Could you check your theme or template."
+    );
+    return "";
+  },
+
+  init: () => {
+    const query = new URLSearchParams(window.location.search).get("q");
+    document
+      .querySelectorAll('input[name="q"]')
+      .forEach((el) => (el.value = query));
+    if (query) Search.performSearch(query);
+  },
+
+  loadIndex: (url) =>
+    (document.body.appendChild(document.createElement("script")).src = url),
+
+  setIndex: (index) => {
+    Search._index = index;
+    if (Search._queued_query !== null) {
+      const query = Search._queued_query;
+      Search._queued_query = null;
+      Search.query(query);
+    }
+  },
+
+  hasIndex: () => Search._index !== null,
+
+  deferQuery: (query) => (Search._queued_query = query),
+
+  stopPulse: () => (Search._pulse_status = -1),
+
+  startPulse: () => {
+    if (Search._pulse_status >= 0) return;
+
+    const pulse = () => {
+      Search._pulse_status = (Search._pulse_status + 1) % 4;
+      Search.dots.innerText = ".".repeat(Search._pulse_status);
+      if (Search._pulse_status >= 0) window.setTimeout(pulse, 500);
+    };
+    pulse();
+  },
+
+  /**
+   * perform a search for something (or wait until index is loaded)
+   */
+  performSearch: (query) => {
+    // create the required interface elements
+    const searchText = document.createElement("h2");
+    searchText.textContent = _("Searching");
+    const searchSummary = document.createElement("p");
+    searchSummary.classList.add("search-summary");
+    searchSummary.innerText = "";
+    const searchList = document.createElement("ul");
+    searchList.classList.add("search");
+
+    const out = document.getElementById("search-results");
+    Search.title = out.appendChild(searchText);
+    Search.dots = Search.title.appendChild(document.createElement("span"));
+    Search.status = out.appendChild(searchSummary);
+    Search.output = out.appendChild(searchList);
+
+    const searchProgress = document.getElementById("search-progress");
+    // Some themes don't use the search progress node
+    if (searchProgress) {
+      searchProgress.innerText = _("Preparing search...");
+    }
+    Search.startPulse();
+
+    // index already loaded, the browser was quick!
+    if (Search.hasIndex()) Search.query(query);
+    else Search.deferQuery(query);
+  },
+
+  /**
+   * execute search (requires search index to be loaded)
+   */
+  query: (query) => {
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const titles = Search._index.titles;
+    const allTitles = Search._index.alltitles;
+    const indexEntries = Search._index.indexentries;
+
+    // stem the search terms and add them to the correct list
+    const stemmer = new Stemmer();
+    const searchTerms = new Set();
+    const excludedTerms = new Set();
+    const highlightTerms = new Set();
+    const objectTerms = new Set(splitQuery(query.toLowerCase().trim()));
+    splitQuery(query.trim()).forEach((queryTerm) => {
+      const queryTermLower = queryTerm.toLowerCase();
+
+      // maybe skip this "word"
+      // stopwords array is from language_data.js
+      if (
+        stopwords.indexOf(queryTermLower) !== -1 ||
+        queryTerm.match(/^\d+$/)
+      )
+        return;
+
+      // stem the word
+      let word = stemmer.stemWord(queryTermLower);
+      // select the correct list
+      if (word[0] === "-") excludedTerms.add(word.substr(1));
+      else {
+        searchTerms.add(word);
+        highlightTerms.add(queryTermLower);
+      }
+    });
+
+    if (SPHINX_HIGHLIGHT_ENABLED) {  // set in sphinx_highlight.js
+      localStorage.setItem("sphinx_highlight_terms", [...highlightTerms].join(" "))
+    }
+
+    // console.debug("SEARCH: searching for:");
+    // console.info("required: ", [...searchTerms]);
+    // console.info("excluded: ", [...excludedTerms]);
+
+    // array of [docname, title, anchor, descr, score, filename]
+    let results = [];
+    _removeChildren(document.getElementById("search-progress"));
+
+    const queryLower = query.toLowerCase();
+    for (const [title, foundTitles] of Object.entries(allTitles)) {
+      if (title.toLowerCase().includes(queryLower) && (queryLower.length >= title.length/2)) {
+        for (const [file, id] of foundTitles) {
+          let score = Math.round(100 * queryLower.length / title.length)
+          results.push([
+            docNames[file],
+            titles[file] !== title ? `${titles[file]} > ${title}` : title,
+            id !== null ? "#" + id : "",
+            null,
+            score,
+            filenames[file],
+          ]);
+        }
+      }
+    }
+
+    // search for explicit entries in index directives
+    for (const [entry, foundEntries] of Object.entries(indexEntries)) {
+      if (entry.includes(queryLower) && (queryLower.length >= entry.length/2)) {
+        for (const [file, id] of foundEntries) {
+          let score = Math.round(100 * queryLower.length / entry.length)
+          results.push([
+            docNames[file],
+            titles[file],
+            id ? "#" + id : "",
+            null,
+            score,
+            filenames[file],
+          ]);
+        }
+      }
+    }
+
+    // lookup as object
+    objectTerms.forEach((term) =>
+      results.push(...Search.performObjectSearch(term, objectTerms))
+    );
+
+    // lookup as search terms in fulltext
+    results.push(...Search.performTermsSearch(searchTerms, excludedTerms));
+
+    // let the scorer override scores with a custom scoring function
+    if (Scorer.score) results.forEach((item) => (item[4] = Scorer.score(item)));
+
+    // now sort the results by score (in opposite order of appearance, since the
+    // display function below uses pop() to retrieve items) and then
+    // alphabetically
+    results.sort((a, b) => {
+      const leftScore = a[4];
+      const rightScore = b[4];
+      if (leftScore === rightScore) {
+        // same score: sort alphabetically
+        const leftTitle = a[1].toLowerCase();
+        const rightTitle = b[1].toLowerCase();
+        if (leftTitle === rightTitle) return 0;
+        return leftTitle > rightTitle ? -1 : 1; // inverted is intentional
+      }
+      return leftScore > rightScore ? 1 : -1;
+    });
+
+    // remove duplicate search results
+    // note the reversing of results, so that in the case of duplicates, the highest-scoring entry is kept
+    let seen = new Set();
+    results = results.reverse().reduce((acc, result) => {
+      let resultStr = result.slice(0, 4).concat([result[5]]).map(v => String(v)).join(',');
+      if (!seen.has(resultStr)) {
+        acc.push(result);
+        seen.add(resultStr);
+      }
+      return acc;
+    }, []);
+
+    results = results.reverse();
+
+    // for debugging
+    //Search.lastresults = results.slice();  // a copy
+    // console.info("search results:", Search.lastresults);
+
+    // print the results
+    _displayNextItem(results, results.length, searchTerms, highlightTerms);
+  },
+
+  /**
+   * search for object names
+   */
+  performObjectSearch: (object, objectTerms) => {
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const objects = Search._index.objects;
+    const objNames = Search._index.objnames;
+    const titles = Search._index.titles;
+
+    const results = [];
+
+    const objectSearchCallback = (prefix, match) => {
+      const name = match[4]
+      const fullname = (prefix ? prefix + "." : "") + name;
+      const fullnameLower = fullname.toLowerCase();
+      if (fullnameLower.indexOf(object) < 0) return;
+
+      let score = 0;
+      const parts = fullnameLower.split(".");
+
+      // check for different match types: exact matches of full name or
+      // "last name" (i.e. last dotted part)
+      if (fullnameLower === object || parts.slice(-1)[0] === object)
+        score += Scorer.objNameMatch;
+      else if (parts.slice(-1)[0].indexOf(object) > -1)
+        score += Scorer.objPartialMatch; // matches in last name
+
+      const objName = objNames[match[1]][2];
+      const title = titles[match[0]];
+
+      // If more than one term searched for, we require other words to be
+      // found in the name/title/description
+      const otherTerms = new Set(objectTerms);
+      otherTerms.delete(object);
+      if (otherTerms.size > 0) {
+        const haystack = `${prefix} ${name} ${objName} ${title}`.toLowerCase();
+        if (
+          [...otherTerms].some((otherTerm) => haystack.indexOf(otherTerm) < 0)
+        )
+          return;
+      }
+
+      let anchor = match[3];
+      if (anchor === "") anchor = fullname;
+      else if (anchor === "-") anchor = objNames[match[1]][1] + "-" + fullname;
+
+      const descr = objName + _(", in ") + title;
+
+      // add custom score for some objects according to scorer
+      if (Scorer.objPrio.hasOwnProperty(match[2]))
+        score += Scorer.objPrio[match[2]];
+      else score += Scorer.objPrioDefault;
+
+      results.push([
+        docNames[match[0]],
+        fullname,
+        "#" + anchor,
+        descr,
+        score,
+        filenames[match[0]],
+      ]);
+    };
+    Object.keys(objects).forEach((prefix) =>
+      objects[prefix].forEach((array) =>
+        objectSearchCallback(prefix, array)
+      )
+    );
+    return results;
+  },
+
+  /**
+   * search for full-text terms in the index
+   */
+  performTermsSearch: (searchTerms, excludedTerms) => {
+    // prepare search
+    const terms = Search._index.terms;
+    const titleTerms = Search._index.titleterms;
+    const filenames = Search._index.filenames;
+    const docNames = Search._index.docnames;
+    const titles = Search._index.titles;
+
+    const scoreMap = new Map();
+    const fileMap = new Map();
+
+    // perform the search on the required terms
+    searchTerms.forEach((word) => {
+      const files = [];
+      const arr = [
+        { files: terms[word], score: Scorer.term },
+        { files: titleTerms[word], score: Scorer.title },
+      ];
+      // add support for partial matches
+      if (word.length > 2) {
+        const escapedWord = _escapeRegExp(word);
+        Object.keys(terms).forEach((term) => {
+          if (term.match(escapedWord) && !terms[word])
+            arr.push({ files: terms[term], score: Scorer.partialTerm });
+        });
+        Object.keys(titleTerms).forEach((term) => {
+          if (term.match(escapedWord) && !titleTerms[word])
+            arr.push({ files: titleTerms[word], score: Scorer.partialTitle });
+        });
+      }
+
+      // no match but word was a required one
+      if (arr.every((record) => record.files === undefined)) return;
+
+      // found search word in contents
+      arr.forEach((record) => {
+        if (record.files === undefined) return;
+
+        let recordFiles = record.files;
+        if (recordFiles.length === undefined) recordFiles = [recordFiles];
+        files.push(...recordFiles);
+
+        // set score for the word in each file
+        recordFiles.forEach((file) => {
+          if (!scoreMap.has(file)) scoreMap.set(file, {});
+          scoreMap.get(file)[word] = record.score;
+        });
+      });
+
+      // create the mapping
+      files.forEach((file) => {
+        if (fileMap.has(file) && fileMap.get(file).indexOf(word) === -1)
+          fileMap.get(file).push(word);
+        else fileMap.set(file, [word]);
+      });
+    });
+
+    // now check if the files don't contain excluded terms
+    const results = [];
+    for (const [file, wordList] of fileMap) {
+      // check if all requirements are matched
+
+      // as search terms with length < 3 are discarded
+      const filteredTermCount = [...searchTerms].filter(
+        (term) => term.length > 2
+      ).length;
+      if (
+        wordList.length !== searchTerms.size &&
+        wordList.length !== filteredTermCount
+      )
+        continue;
+
+      // ensure that none of the excluded terms is in the search result
+      if (
+        [...excludedTerms].some(
+          (term) =>
+            terms[term] === file ||
+            titleTerms[term] === file ||
+            (terms[term] || []).includes(file) ||
+            (titleTerms[term] || []).includes(file)
+        )
+      )
+        break;
+
+      // select one (max) score for the file.
+      const score = Math.max(...wordList.map((w) => scoreMap.get(file)[w]));
+      // add result to the result list
+      results.push([
+        docNames[file],
+        titles[file],
+        "",
+        null,
+        score,
+        filenames[file],
+      ]);
+    }
+    return results;
+  },
+
+  /**
+   * helper function to return a node containing the
+   * search summary for a given text. keywords is a list
+   * of stemmed words.
+   */
+  makeSearchSummary: (htmlText, keywords) => {
+    const text = Search.htmlToText(htmlText);
+    if (text === "") return null;
+
+    const textLower = text.toLowerCase();
+    const actualStartPosition = [...keywords]
+      .map((k) => textLower.indexOf(k.toLowerCase()))
+      .filter((i) => i > -1)
+      .slice(-1)[0];
+    const startWithContext = Math.max(actualStartPosition - 120, 0);
+
+    const top = startWithContext === 0 ? "" : "...";
+    const tail = startWithContext + 240 < text.length ? "..." : "";
+
+    let summary = document.createElement("p");
+    summary.classList.add("context");
+    summary.textContent = top + text.substr(startWithContext, 240).trim() + tail;
+
+    return summary;
+  },
+};
+
+_ready(Search.init);
diff --git a/_static/sphinx-thebe.css b/_static/sphinx-thebe.css
new file mode 100644
index 0000000..1da2793
--- /dev/null
+++ b/_static/sphinx-thebe.css
@@ -0,0 +1,129 @@
+/* Thebelab Buttons */
+.thebelab-button {
+  z-index: 999;
+  display: inline-block;
+  padding: 0.35em 1.2em;
+  margin: 0px 1px;
+  border-radius: 0.12em;
+  box-sizing: border-box;
+  text-decoration: none;
+  font-family: "Roboto", sans-serif;
+  font-weight: 300;
+  text-align: center;
+  transition: all 0.2s;
+  background-color: #dddddd;
+  border: 0.05em solid white;
+  color: #000000;
+}
+
+.thebelab-button:hover {
+  border: 0.05em solid black;
+  background-color: #fcfcfc;
+}
+
+.thebe-launch-button {
+  height: 2.2em;
+  font-size: 0.8em;
+  border: 1px black solid;
+}
+
+/* Thebelab Cell */
+.thebelab-cell pre {
+  background: none;
+}
+
+.thebelab-cell .thebelab-input {
+  padding-left: 1em;
+  margin-bottom: 0.5em;
+  margin-top: 0.5em;
+}
+
+.thebelab-cell .jp-OutputArea {
+  margin-top: 0.5em;
+  margin-left: 1em;
+}
+
+button.thebelab-button.thebelab-run-button {
+  margin-left: 1.5em;
+  margin-bottom: 0.5em;
+}
+
+/* Loading button */
+button.thebe-launch-button div.spinner {
+  float: left;
+  margin-right: 1em;
+}
+
+/* Remove the spinner when thebelab is ready */
+.thebe-launch-button.thebe-status-ready .spinner {
+  display: none;
+}
+
+.thebe-launch-button span.status {
+  font-family: monospace;
+  font-weight: bold;
+}
+
+.thebe-launch-button.thebe-status-ready span.status {
+  color: green;
+}
+
+.spinner {
+  height: 2em;
+  text-align: center;
+  font-size: 0.7em;
+}
+
+.spinner > div {
+  background-color: #f37726;
+  height: 100%;
+  width: 6px;
+  display: inline-block;
+
+  -webkit-animation: sk-stretchdelay 1.2s infinite ease-in-out;
+  animation: sk-stretchdelay 1.2s infinite ease-in-out;
+}
+
+.spinner .rect2 {
+  -webkit-animation-delay: -1.1s;
+  animation-delay: -1.1s;
+}
+
+.spinner .rect3 {
+  -webkit-animation-delay: -1s;
+  animation-delay: -1s;
+}
+
+.spinner .rect4 {
+  -webkit-animation-delay: -0.9s;
+  animation-delay: -0.9s;
+}
+
+.spinner .rect5 {
+  -webkit-animation-delay: -0.8s;
+  animation-delay: -0.8s;
+}
+
+@-webkit-keyframes sk-stretchdelay {
+  0%,
+  40%,
+  100% {
+    -webkit-transform: scaleY(0.4);
+  }
+  20% {
+    -webkit-transform: scaleY(1);
+  }
+}
+
+@keyframes sk-stretchdelay {
+  0%,
+  40%,
+  100% {
+    transform: scaleY(0.4);
+    -webkit-transform: scaleY(0.4);
+  }
+  20% {
+    transform: scaleY(1);
+    -webkit-transform: scaleY(1);
+  }
+}
diff --git a/_static/sphinx-thebe.js b/_static/sphinx-thebe.js
new file mode 100644
index 0000000..d92b1d3
--- /dev/null
+++ b/_static/sphinx-thebe.js
@@ -0,0 +1,127 @@
+/**
+ * Add attributes to Thebe blocks to initialize thebe properly
+ */
+var configureThebe = () => {
+  // Load thebe config in case we want to update it as some point
+  console.log("[sphinx-thebe]: Loading thebe config...");
+  thebe_config = document.querySelector("script[type=\"text/x-thebe-config\"]");
+
+  // If we already detect a Thebe cell, don't re-run
+  if (document.querySelectorAll("div.thebe-cell").length > 0) {
+    return;
+  }
+
+  // Update thebe buttons with loading message
+  document.querySelectorAll(".thebe-launch-button").forEach((button) => {
+    button.innerHTML = `
+        <div class="spinner">
+            <div class="rect1"></div>
+            <div class="rect2"></div>
+            <div class="rect3"></div>
+            <div class="rect4"></div>
+        </div>
+        <span class="loading-text"></span>`;
+  });
+
+  // Set thebe event hooks
+  var thebeStatus;
+  thebelab.on("status", function (evt, data) {
+    console.log("Status changed:", data.status, data.message);
+
+    const button = document.querySelector(".thebe-launch-button ");
+    button.classList.replace(
+	    `thebe-status-${thebeStatus}`, 
+	    `thebe-status-${data.status}`
+    )
+    button.querySelector(".loading-text")
+      .innerHTML = (
+        `<span class='launch_msg'>Launching from mybinder.org: </span>
+	 <span class='status'>${data.status}</span>`
+      );
+
+    // Now update our thebe status
+    thebeStatus = data.status;
+
+    // Find any cells with an initialization tag and ask thebe to run them when ready
+    if (data.status === "ready") {
+      var thebeInitCells = document.querySelectorAll(
+        ".thebe-init, .tag_thebe-init"
+      );
+      thebeInitCells.forEach((cell) => {
+        console.log("Initializing Thebe with cell: " + cell.id);
+        cell.querySelector(".thebelab-run-button").click();
+      });
+    }
+  });
+};
+
+/**
+ * Update the page DOM to use Thebe elements
+ */
+var modifyDOMForThebe = () => {
+  // Find all code cells, replace with Thebe interactive code cells
+  const codeCells = document.querySelectorAll(thebe_selector);
+  codeCells.forEach((codeCell, index) => {
+    const codeCellId = (index) => `codecell${index}`;
+    codeCell.id = codeCellId(index);
+    codeCellText = codeCell.querySelector(thebe_selector_input);
+    codeCellOutput = codeCell.querySelector(thebe_selector_output);
+
+    // Clean up the language to make it work w/ CodeMirror and add it to the cell
+    dataLanguage = detectLanguage(kernelName);
+
+    // Re-arrange the cell and add metadata
+    if (codeCellText) {
+      codeCellText.setAttribute("data-language", dataLanguage);
+      codeCellText.setAttribute("data-executable", "true");
+
+      // If we had an output, insert it just after the `pre` cell
+      if (codeCellOutput) {
+        codeCellOutput.setAttribute("data-output", "");
+        codeCellText.insertAdjacentElement('afterend', codeCellOutput);
+      }
+    }
+
+    // Remove sphinx-copybutton blocks, which are common in Sphinx
+    codeCell.querySelectorAll("button.copybtn").forEach((el) => {
+      el.remove();
+    });
+  });
+};
+
+var initThebe = () => {
+  // Load thebe dynamically if it's not already loaded
+  if (typeof thebelab === "undefined") {
+    console.log("[sphinx-thebe]: Loading thebe from CDN...");
+    document.querySelector(".thebe-launch-button ").innerText = "Loading thebe from CDN...";
+
+    const script = document.createElement("script");
+    script.src = `${THEBE_JS_URL}`;
+    document.head.appendChild(script);
+
+    // Runs once the script has finished loading
+    script.addEventListener("load", () => {
+      console.log("[sphinx-thebe]: Finished loading thebe from CDN...");
+      configureThebe();
+      modifyDOMForThebe();
+      thebelab.bootstrap();
+    });
+  } else {
+    console.log(
+      "[sphinx-thebe]: thebe already loaded, not loading from CDN..."
+    );
+    configureThebe();
+    modifyDOMForThebe();
+    thebelab.bootstrap();
+  }
+};
+
+// Helper function to munge the language name
+var detectLanguage = (language) => {
+  if (language.indexOf("python") > -1) {
+    language = "python";
+  } else if (language === "ir") {
+    language = "r";
+  }
+  return language;
+};
diff --git a/_static/sphinx_highlight.js b/_static/sphinx_highlight.js
new file mode 100644
index 0000000..8a96c69
--- /dev/null
+++ b/_static/sphinx_highlight.js
@@ -0,0 +1,154 @@
+/* Highlighting utilities for Sphinx HTML documentation. */
+"use strict";
+
+const SPHINX_HIGHLIGHT_ENABLED = true
+
+/**
+ * highlight a given string on a node by wrapping it in
+ * span elements with the given class name.
+ */
+const _highlight = (node, addItems, text, className) => {
+  if (node.nodeType === Node.TEXT_NODE) {
+    const val = node.nodeValue;
+    const parent = node.parentNode;
+    const pos = val.toLowerCase().indexOf(text);
+    if (
+      pos >= 0 &&
+      !parent.classList.contains(className) &&
+      !parent.classList.contains("nohighlight")
+    ) {
+      let span;
+
+      const closestNode = parent.closest("body, svg, foreignObject");
+      const isInSVG = closestNode && closestNode.matches("svg");
+      if (isInSVG) {
+        span = document.createElementNS("http://www.w3.org/2000/svg", "tspan");
+      } else {
+        span = document.createElement("span");
+        span.classList.add(className);
+      }
+
+      span.appendChild(document.createTextNode(val.substr(pos, text.length)));
+      const rest = document.createTextNode(val.substr(pos + text.length));
+      parent.insertBefore(
+        span,
+        parent.insertBefore(
+          rest,
+          node.nextSibling
+        )
+      );
+      node.nodeValue = val.substr(0, pos);
+      /* There may be more occurrences of search term in this node. So call this
+       * function recursively on the remaining fragment.
+       */
+      _highlight(rest, addItems, text, className);
+
+      if (isInSVG) {
+        const rect = document.createElementNS(
+          "http://www.w3.org/2000/svg",
+          "rect"
+        );
+        const bbox = parent.getBBox();
+        rect.x.baseVal.value = bbox.x;
+        rect.y.baseVal.value = bbox.y;
+        rect.width.baseVal.value = bbox.width;
+        rect.height.baseVal.value = bbox.height;
+        rect.setAttribute("class", className);
+        addItems.push({ parent: parent, target: rect });
+      }
+    }
+  } else if (node.matches && !node.matches("button, select, textarea")) {
+    node.childNodes.forEach((el) => _highlight(el, addItems, text, className));
+  }
+};
+const _highlightText = (thisNode, text, className) => {
+  let addItems = [];
+  _highlight(thisNode, addItems, text, className);
+  addItems.forEach((obj) =>
+    obj.parent.insertAdjacentElement("beforebegin", obj.target)
+  );
+};
+
+/**
+ * Small JavaScript module for the documentation.
+ */
+const SphinxHighlight = {
+
+  /**
+   * highlight the search words provided in localstorage in the text
+   */
+  highlightSearchWords: () => {
+    if (!SPHINX_HIGHLIGHT_ENABLED) return;  // bail if no highlight
+
+    // get and clear terms from localstorage
+    const url = new URL(window.location);
+    const highlight =
+        localStorage.getItem("sphinx_highlight_terms")
+        || url.searchParams.get("highlight")
+        || "";
+    localStorage.removeItem("sphinx_highlight_terms")
+    url.searchParams.delete("highlight");
+    window.history.replaceState({}, "", url);
+
+    // get individual terms from highlight string
+    const terms = highlight.toLowerCase().split(/\s+/).filter(x => x);
+    if (terms.length === 0) return; // nothing to do
+
+    // There should never be more than one element matching "div.body"
+    const divBody = document.querySelectorAll("div.body");
+    const body = divBody.length ? divBody[0] : document.querySelector("body");
+    window.setTimeout(() => {
+      terms.forEach((term) => _highlightText(body, term, "highlighted"));
+    }, 10);
+
+    const searchBox = document.getElementById("searchbox");
+    if (searchBox === null) return;
+    searchBox.appendChild(
+      document
+        .createRange()
+        .createContextualFragment(
+          '<p class="highlight-link">' +
+            '<a href="javascript:SphinxHighlight.hideSearchWords()">' +
+            _("Hide Search Matches") +
+            "</a></p>"
+        )
+    );
+  },
+
+  /**
+   * helper function to hide the search marks again
+   */
+  hideSearchWords: () => {
+    document
+      .querySelectorAll("#searchbox .highlight-link")
+      .forEach((el) => el.remove());
+    document
+      .querySelectorAll("span.highlighted")
+      .forEach((el) => el.classList.remove("highlighted"));
+    localStorage.removeItem("sphinx_highlight_terms")
+  },
+
+  initEscapeListener: () => {
+    // only install a listener if it is really needed
+    if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) return;
+
+    document.addEventListener("keydown", (event) => {
+      // bail for input elements
+      if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return;
+      // bail with special keys
+      if (event.shiftKey || event.altKey || event.ctrlKey || event.metaKey) return;
+      if (DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS && (event.key === "Escape")) {
+        SphinxHighlight.hideSearchWords();
+        event.preventDefault();
+      }
+    });
+  },
+};
+
+_ready(() => {
+  /* Do not call highlightSearchWords() when we are on the search page.
+   * It will highlight words from the *previous* search query.
+   */
+  if (typeof Search === "undefined") SphinxHighlight.highlightSearchWords();
+  SphinxHighlight.initEscapeListener();
+});
diff --git a/_static/styles/bootstrap.css b/_static/styles/bootstrap.css
new file mode 100644
index 0000000..82b03c0
--- /dev/null
+++ b/_static/styles/bootstrap.css
@@ -0,0 +1,6 @@
+/*!
+ * Bootstrap  v5.3.2 (https://getbootstrap.com/)
+ * Copyright 2011-2023 The Bootstrap Authors
+ * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)
+ */:root,[data-bs-theme=light]{--bs-blue:#0d6efd;--bs-indigo:#6610f2;--bs-purple:#6f42c1;--bs-pink:#d63384;--bs-red:#dc3545;--bs-orange:#fd7e14;--bs-yellow:#ffc107;--bs-green:#198754;--bs-teal:#20c997;--bs-cyan:#0dcaf0;--bs-black:#000;--bs-white:#fff;--bs-gray:#6c757d;--bs-gray-dark:#343a40;--bs-gray-100:#f8f9fa;--bs-gray-200:#e9ecef;--bs-gray-300:#dee2e6;--bs-gray-400:#ced4da;--bs-gray-500:#adb5bd;--bs-gray-600:#6c757d;--bs-gray-700:#495057;--bs-gray-800:#343a40;--bs-gray-900:#212529;--bs-primary:#0d6efd;--bs-secondary:#6c757d;--bs-success:#198754;--bs-info:#0dcaf0;--bs-warning:#ffc107;--bs-danger:#dc3545;--bs-light:#f8f9fa;--bs-dark:#212529;--bs-primary-rgb:13,110,253;--bs-secondary-rgb:108,117,125;--bs-success-rgb:25,135,84;--bs-info-rgb:13,202,240;--bs-warning-rgb:255,193,7;--bs-danger-rgb:220,53,69;--bs-light-rgb:248,249,250;--bs-dark-rgb:33,37,41;--bs-primary-text-emphasis:#052c65;--bs-secondary-text-emphasis:#2b2f32;--bs-success-text-emphasis:#0a3622;--bs-info-text-emphasis:#055160;--bs-warning-text-emphasis:#664d03;--bs-danger-text-emphasis:#58151c;--bs-light-text-emphasis:#495057;--bs-dark-text-emphasis:#495057;--bs-primary-bg-subtle:#cfe2ff;--bs-secondary-bg-subtle:#e2e3e5;--bs-success-bg-subtle:#d1e7dd;--bs-info-bg-subtle:#cff4fc;--bs-warning-bg-subtle:#fff3cd;--bs-danger-bg-subtle:#f8d7da;--bs-light-bg-subtle:#fcfcfd;--bs-dark-bg-subtle:#ced4da;--bs-primary-border-subtle:#9ec5fe;--bs-secondary-border-subtle:#c4c8cb;--bs-success-border-subtle:#a3cfbb;--bs-info-border-subtle:#9eeaf9;--bs-warning-border-subtle:#ffe69c;--bs-danger-border-subtle:#f1aeb5;--bs-light-border-subtle:#e9ecef;--bs-dark-border-subtle:#adb5bd;--bs-white-rgb:255,255,255;--bs-black-rgb:0,0,0;--bs-font-sans-serif:system-ui,-apple-system,"Segoe UI",Roboto,"Helvetica Neue","Noto Sans","Liberation Sans",Arial,sans-serif,"Apple Color Emoji","Segoe UI Emoji","Segoe UI Symbol","Noto Color Emoji";--bs-font-monospace:SFMono-Regular,Menlo,Monaco,Consolas,"Liberation Mono","Courier 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.accordion-collapse{border-bottom-left-radius:var(--bs-accordion-border-radius);border-bottom-right-radius:var(--bs-accordion-border-radius)}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-left:0;border-radius:0;border-right:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}.accordion-flush .accordion-item .accordion-button,.accordion-flush .accordion-item .accordion-button.collapsed{border-radius:0}[data-bs-theme=dark] .accordion-button:after{--bs-accordion-btn-icon:url("data:image/svg+xml;charset=utf-8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3E%3Cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 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var(--bs-breadcrumb-padding-x)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item:before{color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider,"/");float:left;padding-right:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x:0.75rem;--bs-pagination-padding-y:0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color:var(--bs-link-color);--bs-pagination-bg:var(--bs-body-bg);--bs-pagination-border-width:var(--bs-border-width);--bs-pagination-border-color:var(--bs-border-color);--bs-pagination-border-radius:var(--bs-border-radius);--bs-pagination-hover-color:var(--bs-link-hover-color);--bs-pagination-hover-bg:var(--bs-tertiary-bg);--bs-pagination-hover-border-color:var(--bs-border-color);--bs-pagination-focus-color:var(--bs-link-hover-color);--bs-pagination-focus-bg:var(--bs-secondary-bg);--bs-pagination-focus-box-shadow:0 0 0 0.25rem rgba(13,110,253,.25);--bs-pagination-active-color:#fff;--bs-pagination-active-bg:#0d6efd;--bs-pagination-active-border-color:#0d6efd;--bs-pagination-disabled-color:var(--bs-secondary-color);--bs-pagination-disabled-bg:var(--bs-secondary-bg);--bs-pagination-disabled-border-color:var(--bs-border-color);display:flex;list-style:none;padding-left:0}.page-link{background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);color:var(--bs-pagination-color);display:block;font-size:var(--bs-pagination-font-size);padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);position:relative;text-decoration:none;transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s 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.page-link{border-bottom-right-radius:var(--bs-pagination-border-radius);border-top-right-radius:var(--bs-pagination-border-radius)}.pagination-lg{--bs-pagination-padding-x:1.5rem;--bs-pagination-padding-y:0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius:var(--bs-border-radius-lg)}.pagination-sm{--bs-pagination-padding-x:0.5rem;--bs-pagination-padding-y:0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius:var(--bs-border-radius-sm)}.badge{--bs-badge-padding-x:0.65em;--bs-badge-padding-y:0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight:700;--bs-badge-color:#fff;--bs-badge-border-radius:var(--bs-border-radius);border-radius:var(--bs-badge-border-radius);color:var(--bs-badge-color);display:inline-block;font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);text-align:center;vertical-align:baseline;white-space:nowrap}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg:transparent;--bs-alert-padding-x:1rem;--bs-alert-padding-y:1rem;--bs-alert-margin-bottom:1rem;--bs-alert-color:inherit;--bs-alert-border-color:transparent;--bs-alert-border:var(--bs-border-width) solid var(--bs-alert-border-color);--bs-alert-border-radius:var(--bs-border-radius);--bs-alert-link-color:inherit;background-color:var(--bs-alert-bg);border:var(--bs-alert-border);border-radius:var(--bs-alert-border-radius);color:var(--bs-alert-color);margin-bottom:var(--bs-alert-margin-bottom);padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);position:relative}.alert-heading{color:inherit}.alert-link{color:var(--bs-alert-link-color);font-weight:700}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{padding:1.25rem 1rem;position:absolute;right:0;top:0;z-index:2}.alert-primary{--bs-alert-color:var(--bs-primary-text-emphasis);--bs-alert-bg:var(--bs-primary-bg-subtle);--bs-alert-border-color:var(--bs-primary-border-subtle);--bs-alert-link-color:var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color:var(--bs-secondary-text-emphasis);--bs-alert-bg:var(--bs-secondary-bg-subtle);--bs-alert-border-color:var(--bs-secondary-border-subtle);--bs-alert-link-color:var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color:var(--bs-success-text-emphasis);--bs-alert-bg:var(--bs-success-bg-subtle);--bs-alert-border-color:var(--bs-success-border-subtle);--bs-alert-link-color:var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color:var(--bs-info-text-emphasis);--bs-alert-bg:var(--bs-info-bg-subtle);--bs-alert-border-color:var(--bs-info-border-subtle);--bs-alert-link-color:var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color:var(--bs-warning-text-emphasis);--bs-alert-bg:var(--bs-warning-bg-subtle);--bs-alert-border-color:var(--bs-warning-border-subtle);--bs-alert-link-color:var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color:var(--bs-danger-text-emphasis);--bs-alert-bg:var(--bs-danger-bg-subtle);--bs-alert-border-color:var(--bs-danger-border-subtle);--bs-alert-link-color:var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color:var(--bs-light-text-emphasis);--bs-alert-bg:var(--bs-light-bg-subtle);--bs-alert-border-color:var(--bs-light-border-subtle);--bs-alert-link-color:var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color:var(--bs-dark-text-emphasis);--bs-alert-bg:var(--bs-dark-bg-subtle);--bs-alert-border-color:var(--bs-dark-border-subtle);--bs-alert-link-color:var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:1rem}}.progress,.progress-stacked{--bs-progress-height:1rem;--bs-progress-font-size:0.75rem;--bs-progress-bg:var(--bs-secondary-bg);--bs-progress-border-radius:var(--bs-border-radius);--bs-progress-box-shadow:var(--bs-box-shadow-inset);--bs-progress-bar-color:#fff;--bs-progress-bar-bg:#0d6efd;--bs-progress-bar-transition:width 0.6s ease;background-color:var(--bs-progress-bg);border-radius:var(--bs-progress-border-radius);display:flex;font-size:var(--bs-progress-font-size);height:var(--bs-progress-height);overflow:hidden}.progress-bar{background-color:var(--bs-progress-bar-bg);color:var(--bs-progress-bar-color);display:flex;flex-direction:column;justify-content:center;overflow:hidden;text-align:center;transition:var(--bs-progress-bar-transition);white-space:nowrap}@media(prefers-reduced-motion:reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg,hsla(0,0%,100%,.15) 25%,transparent 0,transparent 50%,hsla(0,0%,100%,.15) 0,hsla(0,0%,100%,.15) 75%,transparent 0,transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:progress-bar-stripes 1s linear infinite}@media(prefers-reduced-motion:reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color:var(--bs-body-color);--bs-list-group-bg:var(--bs-body-bg);--bs-list-group-border-color:var(--bs-border-color);--bs-list-group-border-width:var(--bs-border-width);--bs-list-group-border-radius:var(--bs-border-radius);--bs-list-group-item-padding-x:1rem;--bs-list-group-item-padding-y:0.5rem;--bs-list-group-action-color:var(--bs-secondary-color);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-tertiary-bg);--bs-list-group-action-active-color:var(--bs-body-color);--bs-list-group-action-active-bg:var(--bs-secondary-bg);--bs-list-group-disabled-color:var(--bs-secondary-color);--bs-list-group-disabled-bg:var(--bs-body-bg);--bs-list-group-active-color:#fff;--bs-list-group-active-bg:#0d6efd;--bs-list-group-active-border-color:#0d6efd;border-radius:var(--bs-list-group-border-radius);display:flex;flex-direction:column;margin-bottom:0;padding-left:0}.list-group-numbered{counter-reset:section;list-style-type:none}.list-group-numbered>.list-group-item:before{content:counters(section,".") ". ";counter-increment:section}.list-group-item-action{color:var(--bs-list-group-action-color);text-align:inherit;width:100%}.list-group-item-action:focus,.list-group-item-action:hover{background-color:var(--bs-list-group-action-hover-bg);color:var(--bs-list-group-action-hover-color);text-decoration:none;z-index:1}.list-group-item-action:active{background-color:var(--bs-list-group-action-active-bg);color:var(--bs-list-group-action-active-color)}.list-group-item{background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color);color:var(--bs-list-group-color);display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);position:relative;text-decoration:none}.list-group-item:first-child{border-top-left-radius:inherit;border-top-right-radius:inherit}.list-group-item:last-child{border-bottom-left-radius:inherit;border-bottom-right-radius:inherit}.list-group-item.disabled,.list-group-item:disabled{background-color:var(--bs-list-group-disabled-bg);color:var(--bs-list-group-disabled-color);pointer-events:none}.list-group-item.active{background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color);color:var(--bs-list-group-active-color);z-index:2}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{border-top-width:var(--bs-list-group-border-width);margin-top:calc(var(--bs-list-group-border-width)*-1)}.list-group-horizontal{flex-direction:row}.list-group-horizontal>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}@media(min-width:540px){.list-group-horizontal-sm{flex-direction:row}.list-group-horizontal-sm>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-sm>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}@media(min-width:720px){.list-group-horizontal-md{flex-direction:row}.list-group-horizontal-md>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-md>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-md>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}@media(min-width:960px){.list-group-horizontal-lg{flex-direction:row}.list-group-horizontal-lg>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-lg>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}@media(min-width:1200px){.list-group-horizontal-xl{flex-direction:row}.list-group-horizontal-xl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xl>.list-group-item:last-child:not(:first-child){border-bottom-left-radius:0;border-top-right-radius:var(--bs-list-group-border-radius)}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-left-width:0;border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{border-left-width:var(--bs-list-group-border-width);margin-left:calc(var(--bs-list-group-border-width)*-1)}}.list-group-flush{border-radius:0}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-primary{--bs-list-group-color:var(--bs-primary-text-emphasis);--bs-list-group-bg:var(--bs-primary-bg-subtle);--bs-list-group-border-color:var(--bs-primary-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-primary-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-primary-border-subtle);--bs-list-group-active-color:var(--bs-primary-bg-subtle);--bs-list-group-active-bg:var(--bs-primary-text-emphasis);--bs-list-group-active-border-color:var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color:var(--bs-secondary-text-emphasis);--bs-list-group-bg:var(--bs-secondary-bg-subtle);--bs-list-group-border-color:var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-secondary-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-secondary-border-subtle);--bs-list-group-active-color:var(--bs-secondary-bg-subtle);--bs-list-group-active-bg:var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color:var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color:var(--bs-success-text-emphasis);--bs-list-group-bg:var(--bs-success-bg-subtle);--bs-list-group-border-color:var(--bs-success-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-success-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-success-border-subtle);--bs-list-group-active-color:var(--bs-success-bg-subtle);--bs-list-group-active-bg:var(--bs-success-text-emphasis);--bs-list-group-active-border-color:var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color:var(--bs-info-text-emphasis);--bs-list-group-bg:var(--bs-info-bg-subtle);--bs-list-group-border-color:var(--bs-info-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-info-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-info-border-subtle);--bs-list-group-active-color:var(--bs-info-bg-subtle);--bs-list-group-active-bg:var(--bs-info-text-emphasis);--bs-list-group-active-border-color:var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color:var(--bs-warning-text-emphasis);--bs-list-group-bg:var(--bs-warning-bg-subtle);--bs-list-group-border-color:var(--bs-warning-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-warning-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-warning-border-subtle);--bs-list-group-active-color:var(--bs-warning-bg-subtle);--bs-list-group-active-bg:var(--bs-warning-text-emphasis);--bs-list-group-active-border-color:var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color:var(--bs-danger-text-emphasis);--bs-list-group-bg:var(--bs-danger-bg-subtle);--bs-list-group-border-color:var(--bs-danger-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-danger-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-danger-border-subtle);--bs-list-group-active-color:var(--bs-danger-bg-subtle);--bs-list-group-active-bg:var(--bs-danger-text-emphasis);--bs-list-group-active-border-color:var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color:var(--bs-light-text-emphasis);--bs-list-group-bg:var(--bs-light-bg-subtle);--bs-list-group-border-color:var(--bs-light-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-light-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-light-border-subtle);--bs-list-group-active-color:var(--bs-light-bg-subtle);--bs-list-group-active-bg:var(--bs-light-text-emphasis);--bs-list-group-active-border-color:var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color:var(--bs-dark-text-emphasis);--bs-list-group-bg:var(--bs-dark-bg-subtle);--bs-list-group-border-color:var(--bs-dark-border-subtle);--bs-list-group-action-hover-color:var(--bs-emphasis-color);--bs-list-group-action-hover-bg:var(--bs-dark-border-subtle);--bs-list-group-action-active-color:var(--bs-emphasis-color);--bs-list-group-action-active-bg:var(--bs-dark-border-subtle);--bs-list-group-active-color:var(--bs-dark-bg-subtle);--bs-list-group-active-bg:var(--bs-dark-text-emphasis);--bs-list-group-active-border-color:var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color:#000;--bs-btn-close-bg:url("data:image/svg+xml;charset=utf-8,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16'%3E%3Cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3E%3C/svg%3E");--bs-btn-close-opacity:0.5;--bs-btn-close-hover-opacity:0.75;--bs-btn-close-focus-shadow:0 0 0 0.25rem rgba(13,110,253,.25);--bs-btn-close-focus-opacity:1;--bs-btn-close-disabled-opacity:0.25;--bs-btn-close-white-filter:invert(1) grayscale(100%) brightness(200%);background:transparent var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;border-radius:.375rem;box-sizing:content-box;height:1em;opacity:var(--bs-btn-close-opacity);padding:.25em;width:1em}.btn-close,.btn-close:hover{color:var(--bs-btn-close-color)}.btn-close:hover{opacity:var(--bs-btn-close-hover-opacity);text-decoration:none}.btn-close:focus{box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity);outline:0}.btn-close.disabled,.btn-close:disabled{opacity:var(--bs-btn-close-disabled-opacity);pointer-events:none;user-select:none}.btn-close-white,[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex:1090;--bs-toast-padding-x:0.75rem;--bs-toast-padding-y:0.5rem;--bs-toast-spacing:1.5rem;--bs-toast-max-width:350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg:rgba(var(--bs-body-bg-rgb),0.85);--bs-toast-border-width:var(--bs-border-width);--bs-toast-border-color:var(--bs-border-color-translucent);--bs-toast-border-radius:var(--bs-border-radius);--bs-toast-box-shadow:var(--bs-box-shadow);--bs-toast-header-color:var(--bs-secondary-color);--bs-toast-header-bg:rgba(var(--bs-body-bg-rgb),0.85);--bs-toast-header-border-color:var(--bs-border-color-translucent);background-clip:padding-box;background-color:var(--bs-toast-bg);border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);border-radius:var(--bs-toast-border-radius);box-shadow:var(--bs-toast-box-shadow);color:var(--bs-toast-color);font-size:var(--bs-toast-font-size);max-width:100%;pointer-events:auto;width:var(--bs-toast-max-width)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex:1090;max-width:100%;pointer-events:none;position:absolute;width:max-content;z-index:var(--bs-toast-zindex)}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{align-items:center;background-clip:padding-box;background-color:var(--bs-toast-header-bg);border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color);border-top-left-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width));border-top-right-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width));color:var(--bs-toast-header-color);display:flex;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x)}.toast-header .btn-close{margin-left:var(--bs-toast-padding-x);margin-right:calc(var(--bs-toast-padding-x)*-.5)}.toast-body{word-wrap:break-word;padding:var(--bs-toast-padding-x)}.modal{--bs-modal-zindex:1055;--bs-modal-width:500px;--bs-modal-padding:1rem;--bs-modal-margin:0.5rem;--bs-modal-color: ;--bs-modal-bg:var(--bs-body-bg);--bs-modal-border-color:var(--bs-border-color-translucent);--bs-modal-border-width:var(--bs-border-width);--bs-modal-border-radius:var(--bs-border-radius-lg);--bs-modal-box-shadow:var(--bs-box-shadow-sm);--bs-modal-inner-border-radius:calc(var(--bs-border-radius-lg) - var(--bs-border-width));--bs-modal-header-padding-x:1rem;--bs-modal-header-padding-y:1rem;--bs-modal-header-padding:1rem 1rem;--bs-modal-header-border-color:var(--bs-border-color);--bs-modal-header-border-width:var(--bs-border-width);--bs-modal-title-line-height:1.5;--bs-modal-footer-gap:0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color:var(--bs-border-color);--bs-modal-footer-border-width:var(--bs-border-width);display:none;height:100%;left:0;outline:0;overflow-x:hidden;overflow-y:auto;position:fixed;top:0;width:100%;z-index:var(--bs-modal-zindex)}.modal-dialog{margin:var(--bs-modal-margin);pointer-events:none;position:relative;width:auto}.modal.fade .modal-dialog{transform:translateY(-50px);transition:transform .3s ease-out}@media(prefers-reduced-motion:reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static .modal-dialog{transform:scale(1.02)}.modal-dialog-scrollable{height:calc(100% - var(--bs-modal-margin)*2)}.modal-dialog-scrollable .modal-content{max-height:100%;overflow:hidden}.modal-dialog-scrollable .modal-body{overflow-y:auto}.modal-dialog-centered{align-items:center;display:flex;min-height:calc(100% - 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0)}.dropdown-item:hover,.dropdown-item:focus{color:var(--bs-dropdown-link-hover-color);background-color:var(--bs-dropdown-link-hover-bg)}.dropdown-item.active,.dropdown-item:active{color:var(--bs-dropdown-link-active-color);text-decoration:none;background-color:var(--bs-dropdown-link-active-bg)}.dropdown-item.disabled,.dropdown-item:disabled{color:var(--bs-dropdown-link-disabled-color);pointer-events:none;background-color:rgba(0,0,0,0)}.dropdown-menu.show{display:block}.dropdown-header{display:block;padding:var(--bs-dropdown-header-padding-y) var(--bs-dropdown-header-padding-x);margin-bottom:0;font-size:0.875rem;color:var(--bs-dropdown-header-color);white-space:nowrap}.dropdown-item-text{display:block;padding:var(--bs-dropdown-item-padding-y) var(--bs-dropdown-item-padding-x);color:var(--bs-dropdown-link-color)}.dropdown-menu-dark{--bs-dropdown-color: #dee2e6;--bs-dropdown-bg: #343a40;--bs-dropdown-border-color: var(--bs-border-color-translucent);--bs-dropdown-box-shadow: ;--bs-dropdown-link-color: #dee2e6;--bs-dropdown-link-hover-color: #fff;--bs-dropdown-divider-bg: var(--bs-border-color-translucent);--bs-dropdown-link-hover-bg: var(--pst-color-surface);--bs-dropdown-link-active-color: #fff;--bs-dropdown-link-active-bg: var(--pst-color-surface);--bs-dropdown-link-disabled-color: #adb5bd;--bs-dropdown-header-color: #adb5bd}.btn-group,.btn-group-vertical{position:relative;display:inline-flex;vertical-align:middle}.btn-group>.btn,.btn-group-vertical>.btn{position:relative;flex:1 1 auto}.btn-group>.btn-check:checked+.btn,.btn-group>.btn-check:focus+.btn,.btn-group>.btn:hover,.btn-group>.btn:focus,.btn-group>.btn:active,.btn-group>.btn.active,.btn-group-vertical>.btn-check:checked+.btn,.btn-group-vertical>.btn-check:focus+.btn,.btn-group-vertical>.btn:hover,.btn-group-vertical>.btn:focus,.btn-group-vertical>.btn:active,.btn-group-vertical>.btn.active{z-index:1}.btn-toolbar{display:flex;flex-wrap:wrap;justify-content:flex-start}.btn-toolbar .input-group{width:auto}.btn-group{border-radius:var(--bs-border-radius)}.btn-group>:not(.btn-check:first-child)+.btn,.btn-group>.btn-group:not(:first-child){margin-left:calc(var(--bs-border-width)*-1)}.btn-group>.btn:not(:last-child):not(.dropdown-toggle),.btn-group>.btn.dropdown-toggle-split:first-child,.btn-group>.btn-group:not(:last-child)>.btn{border-top-right-radius:0;border-bottom-right-radius:0}.btn-group>.btn:nth-child(n+3),.btn-group>:not(.btn-check)+.btn,.btn-group>.btn-group:not(:first-child)>.btn{border-top-left-radius:0;border-bottom-left-radius:0}.dropdown-toggle-split{padding-right:.5625rem;padding-left:.5625rem}.dropdown-toggle-split::after,.dropup .dropdown-toggle-split::after,.dropend .dropdown-toggle-split::after{margin-left:0}.dropstart .dropdown-toggle-split::before{margin-right:0}.btn-sm+.dropdown-toggle-split,.btn-group-sm>.btn+.dropdown-toggle-split{padding-right:.375rem;padding-left:.375rem}.btn-lg+.dropdown-toggle-split,.btn-group-lg>.btn+.dropdown-toggle-split{padding-right:.75rem;padding-left:.75rem}.btn-group-vertical{flex-direction:column;align-items:flex-start;justify-content:center}.btn-group-vertical>.btn,.btn-group-vertical>.btn-group{width:100%}.btn-group-vertical>.btn:not(:first-child),.btn-group-vertical>.btn-group:not(:first-child){margin-top:calc(var(--bs-border-width)*-1)}.btn-group-vertical>.btn:not(:last-child):not(.dropdown-toggle),.btn-group-vertical>.btn-group:not(:last-child)>.btn{border-bottom-right-radius:0;border-bottom-left-radius:0}.btn-group-vertical>.btn~.btn,.btn-group-vertical>.btn-group:not(:first-child)>.btn{border-top-left-radius:0;border-top-right-radius:0}.nav{--bs-nav-link-padding-x: 1rem;--bs-nav-link-padding-y: 0.5rem;--bs-nav-link-font-weight: ;--bs-nav-link-color: 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1.25rem;--bs-navbar-brand-color: rgba(var(--bs-emphasis-color-rgb), 1);--bs-navbar-brand-hover-color: rgba(var(--bs-emphasis-color-rgb), 1);--bs-navbar-nav-link-padding-x: 0.5rem;--bs-navbar-toggler-padding-y: 0.25rem;--bs-navbar-toggler-padding-x: 0.75rem;--bs-navbar-toggler-font-size: 1.25rem;--bs-navbar-toggler-icon-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%2833, 37, 41, 0.75%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e\");--bs-navbar-toggler-border-color: rgba(var(--bs-emphasis-color-rgb), 0.15);--bs-navbar-toggler-border-radius: var(--bs-border-radius);--bs-navbar-toggler-focus-width: 0.25rem;--bs-navbar-toggler-transition: box-shadow 0.15s ease-in-out;position:relative;display:flex;flex-wrap:wrap;align-items:center;justify-content:space-between;padding:var(--bs-navbar-padding-y) 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.offcanvas-body{display:flex;flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: rgba(255, 255, 255, 0.55);--bs-navbar-hover-color: rgba(255, 255, 255, 0.75);--bs-navbar-disabled-color: rgba(255, 255, 255, 0.25);--bs-navbar-active-color: #fff;--bs-navbar-brand-color: #fff;--bs-navbar-brand-hover-color: #fff;--bs-navbar-toggler-border-color: rgba(255, 255, 255, 0.1);--bs-navbar-toggler-icon-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%28255, 255, 255, 0.55%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e\")}[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='rgba%28255, 255, 255, 0.55%29' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e\")}.card{--bs-card-spacer-y: 1rem;--bs-card-spacer-x: 1rem;--bs-card-title-spacer-y: 0.5rem;--bs-card-title-color: ;--bs-card-subtitle-color: ;--bs-card-border-width: var(--bs-border-width);--bs-card-border-color: var(--bs-border-color-translucent);--bs-card-border-radius: var(--bs-border-radius);--bs-card-box-shadow: ;--bs-card-inner-border-radius: calc(var(--bs-border-radius) - (var(--bs-border-width)));--bs-card-cap-padding-y: 0.5rem;--bs-card-cap-padding-x: 1rem;--bs-card-cap-bg: rgba(var(--bs-body-color-rgb), 0.03);--bs-card-cap-color: ;--bs-card-height: ;--bs-card-color: ;--bs-card-bg: var(--bs-body-bg);--bs-card-img-overlay-padding: 1rem;--bs-card-group-margin: 0.75rem;position:relative;display:flex;flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color);border-radius:var(--bs-card-border-radius)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0;border-top-left-radius:var(--bs-card-inner-border-radius);border-top-right-radius:var(--bs-card-inner-border-radius)}.card>.list-group:last-child{border-bottom-width:0;border-bottom-right-radius:var(--bs-card-inner-border-radius);border-bottom-left-radius:var(--bs-card-inner-border-radius)}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header:first-child{border-radius:var(--bs-card-inner-border-radius) var(--bs-card-inner-border-radius) 0 0}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer:last-child{border-radius:0 0 var(--bs-card-inner-border-radius) var(--bs-card-inner-border-radius)}.card-header-tabs{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-bottom:calc(-1*var(--bs-card-cap-padding-y));margin-left:calc(-0.5*var(--bs-card-cap-padding-x));border-bottom:0}.card-header-tabs .nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding);border-radius:var(--bs-card-inner-border-radius)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-img,.card-img-top{border-top-left-radius:var(--bs-card-inner-border-radius);border-top-right-radius:var(--bs-card-inner-border-radius)}.card-img,.card-img-bottom{border-bottom-right-radius:var(--bs-card-inner-border-radius);border-bottom-left-radius:var(--bs-card-inner-border-radius)}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 540px){.card-group{display:flex;flex-flow:row wrap}.card-group>.card{flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}.card-group>.card:not(:last-child){border-top-right-radius:0;border-bottom-right-radius:0}.card-group>.card:not(:last-child) .card-img-top,.card-group>.card:not(:last-child) .card-header{border-top-right-radius:0}.card-group>.card:not(:last-child) .card-img-bottom,.card-group>.card:not(:last-child) .card-footer{border-bottom-right-radius:0}.card-group>.card:not(:first-child){border-top-left-radius:0;border-bottom-left-radius:0}.card-group>.card:not(:first-child) .card-img-top,.card-group>.card:not(:first-child) .card-header{border-top-left-radius:0}.card-group>.card:not(:first-child) .card-img-bottom,.card-group>.card:not(:first-child) .card-footer{border-bottom-left-radius:0}}.accordion{--bs-accordion-color: var(--bs-body-color);--bs-accordion-bg: var(--bs-body-bg);--bs-accordion-transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, border-radius 0.15s ease;--bs-accordion-border-color: var(--bs-border-color);--bs-accordion-border-width: var(--bs-border-width);--bs-accordion-border-radius: var(--bs-border-radius);--bs-accordion-inner-border-radius: calc(var(--bs-border-radius) - (var(--bs-border-width)));--bs-accordion-btn-padding-x: 1.25rem;--bs-accordion-btn-padding-y: 1rem;--bs-accordion-btn-color: var(--bs-body-color);--bs-accordion-btn-bg: var(--bs-accordion-bg);--bs-accordion-btn-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23212529'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e\");--bs-accordion-btn-icon-width: 1.25rem;--bs-accordion-btn-icon-transform: rotate(-180deg);--bs-accordion-btn-icon-transition: transform 0.2s ease-in-out;--bs-accordion-btn-active-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23052c65'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e\");--bs-accordion-btn-focus-border-color: #86b7fe;--bs-accordion-btn-focus-box-shadow: 0 0 0 0.25rem rgba(13, 110, 253, 0.25);--bs-accordion-body-padding-x: 1.25rem;--bs-accordion-body-padding-y: 1rem;--bs-accordion-active-color: var(--bs-primary-text-emphasis);--bs-accordion-active-bg: var(--bs-primary-bg-subtle)}.accordion-button{position:relative;display:flex;align-items:center;width:100%;padding:var(--bs-accordion-btn-padding-y) var(--bs-accordion-btn-padding-x);font-size:1rem;color:var(--bs-accordion-btn-color);text-align:left;background-color:var(--bs-accordion-btn-bg);border:0;border-radius:0;overflow-anchor:none;transition:var(--bs-accordion-transition)}@media(prefers-reduced-motion: reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:\"\";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:first-of-type{border-top-left-radius:var(--bs-accordion-border-radius);border-top-right-radius:var(--bs-accordion-border-radius)}.accordion-item:first-of-type .accordion-button{border-top-left-radius:var(--bs-accordion-inner-border-radius);border-top-right-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:not(:first-of-type){border-top:0}.accordion-item:last-of-type{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-item:last-of-type .accordion-button.collapsed{border-bottom-right-radius:var(--bs-accordion-inner-border-radius);border-bottom-left-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:last-of-type .accordion-collapse{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0;border-radius:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}.accordion-flush .accordion-item .accordion-button,.accordion-flush .accordion-item .accordion-button.collapsed{border-radius:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e\");--bs-accordion-btn-active-icon: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e\")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: var(--bs-secondary-color);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: var(--bs-secondary-color);display:flex;flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg);border-radius:var(--bs-breadcrumb-border-radius)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, \"/\") /* rtl: var(--bs-breadcrumb-divider, \"/\") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: var(--bs-link-color);--bs-pagination-bg: var(--bs-body-bg);--bs-pagination-border-width: var(--bs-border-width);--bs-pagination-border-color: var(--bs-border-color);--bs-pagination-border-radius: var(--bs-border-radius);--bs-pagination-hover-color: var(--bs-link-hover-color);--bs-pagination-hover-bg: var(--bs-tertiary-bg);--bs-pagination-hover-border-color: var(--bs-border-color);--bs-pagination-focus-color: var(--bs-link-hover-color);--bs-pagination-focus-bg: var(--bs-secondary-bg);--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(13, 110, 253, 0.25);--bs-pagination-active-color: #fff;--bs-pagination-active-bg: #0d6efd;--bs-pagination-active-border-color: #0d6efd;--bs-pagination-disabled-color: var(--bs-secondary-color);--bs-pagination-disabled-bg: var(--bs-secondary-bg);--bs-pagination-disabled-border-color: var(--bs-border-color);display:flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(var(--bs-border-width)*-1)}.page-item:first-child .page-link{border-top-left-radius:var(--bs-pagination-border-radius);border-bottom-left-radius:var(--bs-pagination-border-radius)}.page-item:last-child .page-link{border-top-right-radius:var(--bs-pagination-border-radius);border-bottom-right-radius:var(--bs-pagination-border-radius)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: var(--bs-border-radius-lg)}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: var(--bs-border-radius-sm)}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #fff;--bs-badge-border-radius: var(--bs-border-radius);display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline;border-radius:var(--bs-badge-border-radius)}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: var(--bs-border-width) solid var(--bs-alert-border-color);--bs-alert-border-radius: var(--bs-border-radius);--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border);border-radius:var(--bs-alert-border-radius)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:1rem}}.progress,.progress-stacked{--bs-progress-height: 1rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: var(--bs-secondary-bg);--bs-progress-border-radius: var(--bs-border-radius);--bs-progress-box-shadow: var(--bs-box-shadow-inset);--bs-progress-bar-color: #fff;--bs-progress-bar-bg: #0d6efd;--bs-progress-bar-transition: width 0.6s ease;display:flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg);border-radius:var(--bs-progress-border-radius)}.progress-bar{display:flex;flex-direction:column;justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: var(--bs-body-color);--bs-list-group-bg: var(--bs-body-bg);--bs-list-group-border-color: var(--bs-border-color);--bs-list-group-border-width: var(--bs-border-width);--bs-list-group-border-radius: var(--bs-border-radius);--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: var(--bs-secondary-color);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-tertiary-bg);--bs-list-group-action-active-color: var(--bs-body-color);--bs-list-group-action-active-bg: var(--bs-secondary-bg);--bs-list-group-disabled-color: var(--bs-secondary-color);--bs-list-group-disabled-bg: var(--bs-body-bg);--bs-list-group-active-color: #fff;--bs-list-group-active-bg: #0d6efd;--bs-list-group-active-border-color: 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var(--bs-list-group-border-color)}.list-group-item:first-child{border-top-left-radius:inherit;border-top-right-radius:inherit}.list-group-item:last-child{border-bottom-right-radius:inherit;border-bottom-left-radius:inherit}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row}.list-group-horizontal>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 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960px){.list-group-horizontal-lg{flex-direction:row}.list-group-horizontal-lg>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-lg>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row}.list-group-horizontal-xl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xl>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush{border-radius:0}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url(\"data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e\");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(13, 110, 253, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto 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calc(var(--bs-border-radius-lg) - (var(--bs-border-width)));--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: var(--bs-border-color);--bs-modal-header-border-width: var(--bs-border-width);--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: var(--bs-border-color);--bs-modal-footer-border-width: var(--bs-border-width);position:fixed;top:0;left:0;z-index:var(--bs-modal-zindex);display:none;width:100%;height:100%;overflow-x:hidden;overflow-y:auto;outline:0}.modal-dialog{position:relative;width:auto;margin:var(--bs-modal-margin);pointer-events:none}.modal.fade .modal-dialog{transition:transform .3s ease-out;transform:translate(0, -50px)}@media(prefers-reduced-motion: reduce){.modal.fade .modal-dialog{transition:none}}.modal.show .modal-dialog{transform:none}.modal.modal-static 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!important}.pt-lg-0{padding-top:0 !important}.pt-lg-1{padding-top:.25rem !important}.pt-lg-2{padding-top:.5rem !important}.pt-lg-3{padding-top:1rem !important}.pt-lg-4{padding-top:1.5rem !important}.pt-lg-5{padding-top:3rem !important}.pe-lg-0{padding-right:0 !important}.pe-lg-1{padding-right:.25rem !important}.pe-lg-2{padding-right:.5rem !important}.pe-lg-3{padding-right:1rem !important}.pe-lg-4{padding-right:1.5rem !important}.pe-lg-5{padding-right:3rem !important}.pb-lg-0{padding-bottom:0 !important}.pb-lg-1{padding-bottom:.25rem !important}.pb-lg-2{padding-bottom:.5rem !important}.pb-lg-3{padding-bottom:1rem !important}.pb-lg-4{padding-bottom:1.5rem !important}.pb-lg-5{padding-bottom:3rem !important}.ps-lg-0{padding-left:0 !important}.ps-lg-1{padding-left:.25rem !important}.ps-lg-2{padding-left:.5rem !important}.ps-lg-3{padding-left:1rem !important}.ps-lg-4{padding-left:1.5rem !important}.ps-lg-5{padding-left:3rem !important}.gap-lg-0{gap:0 !important}.gap-lg-1{gap:.25rem !important}.gap-lg-2{gap:.5rem !important}.gap-lg-3{gap:1rem !important}.gap-lg-4{gap:1.5rem !important}.gap-lg-5{gap:3rem !important}.row-gap-lg-0{row-gap:0 !important}.row-gap-lg-1{row-gap:.25rem !important}.row-gap-lg-2{row-gap:.5rem !important}.row-gap-lg-3{row-gap:1rem !important}.row-gap-lg-4{row-gap:1.5rem !important}.row-gap-lg-5{row-gap:3rem !important}.column-gap-lg-0{column-gap:0 !important}.column-gap-lg-1{column-gap:.25rem !important}.column-gap-lg-2{column-gap:.5rem !important}.column-gap-lg-3{column-gap:1rem !important}.column-gap-lg-4{column-gap:1.5rem !important}.column-gap-lg-5{column-gap:3rem !important}.text-lg-start{text-align:left !important}.text-lg-end{text-align:right !important}.text-lg-center{text-align:center !important}}@media(min-width: 1200px){.float-xl-start{float:left !important}.float-xl-end{float:right !important}.float-xl-none{float:none !important}.object-fit-xl-contain{object-fit:contain !important}.object-fit-xl-cover{object-fit:cover !important}.object-fit-xl-fill{object-fit:fill !important}.object-fit-xl-scale{object-fit:scale-down !important}.object-fit-xl-none{object-fit:none !important}.d-xl-inline{display:inline !important}.d-xl-inline-block{display:inline-block !important}.d-xl-block{display:block !important}.d-xl-grid{display:grid !important}.d-xl-inline-grid{display:inline-grid !important}.d-xl-table{display:table !important}.d-xl-table-row{display:table-row !important}.d-xl-table-cell{display:table-cell !important}.d-xl-flex{display:flex !important}.d-xl-inline-flex{display:inline-flex !important}.d-xl-none{display:none !important}.flex-xl-fill{flex:1 1 auto !important}.flex-xl-row{flex-direction:row !important}.flex-xl-column{flex-direction:column !important}.flex-xl-row-reverse{flex-direction:row-reverse !important}.flex-xl-column-reverse{flex-direction:column-reverse !important}.flex-xl-grow-0{flex-grow:0 !important}.flex-xl-grow-1{flex-grow:1 !important}.flex-xl-shrink-0{flex-shrink:0 !important}.flex-xl-shrink-1{flex-shrink:1 !important}.flex-xl-wrap{flex-wrap:wrap !important}.flex-xl-nowrap{flex-wrap:nowrap !important}.flex-xl-wrap-reverse{flex-wrap:wrap-reverse !important}.justify-content-xl-start{justify-content:flex-start !important}.justify-content-xl-end{justify-content:flex-end !important}.justify-content-xl-center{justify-content:center !important}.justify-content-xl-between{justify-content:space-between !important}.justify-content-xl-around{justify-content:space-around !important}.justify-content-xl-evenly{justify-content:space-evenly !important}.align-items-xl-start{align-items:flex-start !important}.align-items-xl-end{align-items:flex-end !important}.align-items-xl-center{align-items:center !important}.align-items-xl-baseline{align-items:baseline !important}.align-items-xl-stretch{align-items:stretch !important}.align-content-xl-start{align-content:flex-start !important}.align-content-xl-end{align-content:flex-end !important}.align-content-xl-center{align-content:center !important}.align-content-xl-between{align-content:space-between !important}.align-content-xl-around{align-content:space-around !important}.align-content-xl-stretch{align-content:stretch !important}.align-self-xl-auto{align-self:auto !important}.align-self-xl-start{align-self:flex-start !important}.align-self-xl-end{align-self:flex-end !important}.align-self-xl-center{align-self:center !important}.align-self-xl-baseline{align-self:baseline !important}.align-self-xl-stretch{align-self:stretch !important}.order-xl-first{order:-1 !important}.order-xl-0{order:0 !important}.order-xl-1{order:1 !important}.order-xl-2{order:2 !important}.order-xl-3{order:3 !important}.order-xl-4{order:4 !important}.order-xl-5{order:5 !important}.order-xl-last{order:6 !important}.m-xl-0{margin:0 !important}.m-xl-1{margin:.25rem !important}.m-xl-2{margin:.5rem !important}.m-xl-3{margin:1rem !important}.m-xl-4{margin:1.5rem !important}.m-xl-5{margin:3rem !important}.m-xl-auto{margin:auto !important}.mx-xl-0{margin-right:0 !important;margin-left:0 !important}.mx-xl-1{margin-right:.25rem !important;margin-left:.25rem !important}.mx-xl-2{margin-right:.5rem !important;margin-left:.5rem !important}.mx-xl-3{margin-right:1rem !important;margin-left:1rem !important}.mx-xl-4{margin-right:1.5rem !important;margin-left:1.5rem !important}.mx-xl-5{margin-right:3rem !important;margin-left:3rem !important}.mx-xl-auto{margin-right:auto !important;margin-left:auto !important}.my-xl-0{margin-top:0 !important;margin-bottom:0 !important}.my-xl-1{margin-top:.25rem !important;margin-bottom:.25rem !important}.my-xl-2{margin-top:.5rem !important;margin-bottom:.5rem !important}.my-xl-3{margin-top:1rem !important;margin-bottom:1rem !important}.my-xl-4{margin-top:1.5rem !important;margin-bottom:1.5rem !important}.my-xl-5{margin-top:3rem !important;margin-bottom:3rem !important}.my-xl-auto{margin-top:auto !important;margin-bottom:auto !important}.mt-xl-0{margin-top:0 !important}.mt-xl-1{margin-top:.25rem !important}.mt-xl-2{margin-top:.5rem !important}.mt-xl-3{margin-top:1rem !important}.mt-xl-4{margin-top:1.5rem !important}.mt-xl-5{margin-top:3rem !important}.mt-xl-auto{margin-top:auto !important}.me-xl-0{margin-right:0 !important}.me-xl-1{margin-right:.25rem !important}.me-xl-2{margin-right:.5rem !important}.me-xl-3{margin-right:1rem !important}.me-xl-4{margin-right:1.5rem !important}.me-xl-5{margin-right:3rem !important}.me-xl-auto{margin-right:auto !important}.mb-xl-0{margin-bottom:0 !important}.mb-xl-1{margin-bottom:.25rem !important}.mb-xl-2{margin-bottom:.5rem !important}.mb-xl-3{margin-bottom:1rem !important}.mb-xl-4{margin-bottom:1.5rem !important}.mb-xl-5{margin-bottom:3rem !important}.mb-xl-auto{margin-bottom:auto !important}.ms-xl-0{margin-left:0 !important}.ms-xl-1{margin-left:.25rem !important}.ms-xl-2{margin-left:.5rem !important}.ms-xl-3{margin-left:1rem !important}.ms-xl-4{margin-left:1.5rem !important}.ms-xl-5{margin-left:3rem !important}.ms-xl-auto{margin-left:auto !important}.p-xl-0{padding:0 !important}.p-xl-1{padding:.25rem !important}.p-xl-2{padding:.5rem !important}.p-xl-3{padding:1rem !important}.p-xl-4{padding:1.5rem !important}.p-xl-5{padding:3rem !important}.px-xl-0{padding-right:0 !important;padding-left:0 !important}.px-xl-1{padding-right:.25rem !important;padding-left:.25rem !important}.px-xl-2{padding-right:.5rem !important;padding-left:.5rem !important}.px-xl-3{padding-right:1rem !important;padding-left:1rem !important}.px-xl-4{padding-right:1.5rem !important;padding-left:1.5rem !important}.px-xl-5{padding-right:3rem !important;padding-left:3rem !important}.py-xl-0{padding-top:0 !important;padding-bottom:0 !important}.py-xl-1{padding-top:.25rem !important;padding-bottom:.25rem !important}.py-xl-2{padding-top:.5rem !important;padding-bottom:.5rem !important}.py-xl-3{padding-top:1rem !important;padding-bottom:1rem !important}.py-xl-4{padding-top:1.5rem !important;padding-bottom:1.5rem !important}.py-xl-5{padding-top:3rem !important;padding-bottom:3rem !important}.pt-xl-0{padding-top:0 !important}.pt-xl-1{padding-top:.25rem !important}.pt-xl-2{padding-top:.5rem !important}.pt-xl-3{padding-top:1rem !important}.pt-xl-4{padding-top:1.5rem !important}.pt-xl-5{padding-top:3rem !important}.pe-xl-0{padding-right:0 !important}.pe-xl-1{padding-right:.25rem !important}.pe-xl-2{padding-right:.5rem !important}.pe-xl-3{padding-right:1rem !important}.pe-xl-4{padding-right:1.5rem !important}.pe-xl-5{padding-right:3rem !important}.pb-xl-0{padding-bottom:0 !important}.pb-xl-1{padding-bottom:.25rem !important}.pb-xl-2{padding-bottom:.5rem !important}.pb-xl-3{padding-bottom:1rem !important}.pb-xl-4{padding-bottom:1.5rem !important}.pb-xl-5{padding-bottom:3rem !important}.ps-xl-0{padding-left:0 !important}.ps-xl-1{padding-left:.25rem !important}.ps-xl-2{padding-left:.5rem !important}.ps-xl-3{padding-left:1rem !important}.ps-xl-4{padding-left:1.5rem !important}.ps-xl-5{padding-left:3rem !important}.gap-xl-0{gap:0 !important}.gap-xl-1{gap:.25rem !important}.gap-xl-2{gap:.5rem !important}.gap-xl-3{gap:1rem !important}.gap-xl-4{gap:1.5rem !important}.gap-xl-5{gap:3rem !important}.row-gap-xl-0{row-gap:0 !important}.row-gap-xl-1{row-gap:.25rem !important}.row-gap-xl-2{row-gap:.5rem !important}.row-gap-xl-3{row-gap:1rem !important}.row-gap-xl-4{row-gap:1.5rem !important}.row-gap-xl-5{row-gap:3rem !important}.column-gap-xl-0{column-gap:0 !important}.column-gap-xl-1{column-gap:.25rem !important}.column-gap-xl-2{column-gap:.5rem !important}.column-gap-xl-3{column-gap:1rem !important}.column-gap-xl-4{column-gap:1.5rem !important}.column-gap-xl-5{column-gap:3rem !important}.text-xl-start{text-align:left !important}.text-xl-end{text-align:right !important}.text-xl-center{text-align:center !important}}@media(min-width: 1200px){.fs-1{font-size:2.5rem !important}.fs-2{font-size:2rem !important}.fs-3{font-size:1.75rem !important}.fs-4{font-size:1.5rem !important}}@media print{.d-print-inline{display:inline !important}.d-print-inline-block{display:inline-block !important}.d-print-block{display:block !important}.d-print-grid{display:grid !important}.d-print-inline-grid{display:inline-grid !important}.d-print-table{display:table !important}.d-print-table-row{display:table-row !important}.d-print-table-cell{display:table-cell !important}.d-print-flex{display:flex !important}.d-print-inline-flex{display:inline-flex !important}.d-print-none{display:none !important}}","@mixin bsBanner($file) {\n  /*!\n   * Bootstrap #{$file} v5.3.2 (https://getbootstrap.com/)\n   * Copyright 2011-2023 The Bootstrap Authors\n   * Licensed under MIT (https://github.com/twbs/bootstrap/blob/main/LICENSE)\n   */\n}\n",":root,\n[data-bs-theme=\"light\"] {\n  // Note: Custom variable values only support SassScript inside `#{}`.\n\n  // Colors\n  //\n  // Generate palettes for full colors, grays, and theme colors.\n\n  @each $color, $value in $colors {\n    --#{$prefix}#{$color}: #{$value};\n  }\n\n  @each $color, $value in $grays {\n    --#{$prefix}gray-#{$color}: #{$value};\n  }\n\n  @each $color, $value in $theme-colors {\n    --#{$prefix}#{$color}: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-rgb {\n    --#{$prefix}#{$color}-rgb: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-text {\n    --#{$prefix}#{$color}-text-emphasis: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-bg-subtle {\n    --#{$prefix}#{$color}-bg-subtle: #{$value};\n  }\n\n  @each $color, $value in $theme-colors-border-subtle {\n    --#{$prefix}#{$color}-border-subtle: #{$value};\n  }\n\n  --#{$prefix}white-rgb: #{to-rgb($white)};\n  --#{$prefix}black-rgb: #{to-rgb($black)};\n\n  // Fonts\n\n  // Note: Use `inspect` for lists so that quoted items keep the quotes.\n  // See https://github.com/sass/sass/issues/2383#issuecomment-336349172\n  --#{$prefix}font-sans-serif: #{inspect($font-family-sans-serif)};\n  --#{$prefix}font-monospace: #{inspect($font-family-monospace)};\n  --#{$prefix}gradient: #{$gradient};\n\n  // Root and body\n  // scss-docs-start root-body-variables\n  @if $font-size-root != null {\n    --#{$prefix}root-font-size: #{$font-size-root};\n  }\n  --#{$prefix}body-font-family: #{inspect($font-family-base)};\n  @include rfs($font-size-base, --#{$prefix}body-font-size);\n  --#{$prefix}body-font-weight: #{$font-weight-base};\n  --#{$prefix}body-line-height: #{$line-height-base};\n  @if $body-text-align != null {\n    --#{$prefix}body-text-align: #{$body-text-align};\n  }\n\n  --#{$prefix}body-color: #{$body-color};\n  --#{$prefix}body-color-rgb: #{to-rgb($body-color)};\n  --#{$prefix}body-bg: #{$body-bg};\n  --#{$prefix}body-bg-rgb: #{to-rgb($body-bg)};\n\n  --#{$prefix}emphasis-color: #{$body-emphasis-color};\n  --#{$prefix}emphasis-color-rgb: #{to-rgb($body-emphasis-color)};\n\n  --#{$prefix}secondary-color: #{$body-secondary-color};\n  --#{$prefix}secondary-color-rgb: #{to-rgb($body-secondary-color)};\n  --#{$prefix}secondary-bg: #{$body-secondary-bg};\n  --#{$prefix}secondary-bg-rgb: #{to-rgb($body-secondary-bg)};\n\n  --#{$prefix}tertiary-color: #{$body-tertiary-color};\n  --#{$prefix}tertiary-color-rgb: #{to-rgb($body-tertiary-color)};\n  --#{$prefix}tertiary-bg: #{$body-tertiary-bg};\n  --#{$prefix}tertiary-bg-rgb: #{to-rgb($body-tertiary-bg)};\n  // scss-docs-end root-body-variables\n\n  --#{$prefix}heading-color: #{$headings-color};\n\n  --#{$prefix}link-color: #{$link-color};\n  --#{$prefix}link-color-rgb: #{to-rgb($link-color)};\n  --#{$prefix}link-decoration: #{$link-decoration};\n\n  --#{$prefix}link-hover-color: #{$link-hover-color};\n  --#{$prefix}link-hover-color-rgb: #{to-rgb($link-hover-color)};\n\n  @if $link-hover-decoration != null {\n    --#{$prefix}link-hover-decoration: #{$link-hover-decoration};\n  }\n\n  --#{$prefix}code-color: #{$code-color};\n  --#{$prefix}highlight-color: #{$mark-color};\n  --#{$prefix}highlight-bg: #{$mark-bg};\n\n  // scss-docs-start root-border-var\n  --#{$prefix}border-width: #{$border-width};\n  --#{$prefix}border-style: #{$border-style};\n  --#{$prefix}border-color: #{$border-color};\n  --#{$prefix}border-color-translucent: #{$border-color-translucent};\n\n  --#{$prefix}border-radius: #{$border-radius};\n  --#{$prefix}border-radius-sm: #{$border-radius-sm};\n  --#{$prefix}border-radius-lg: #{$border-radius-lg};\n  --#{$prefix}border-radius-xl: #{$border-radius-xl};\n  --#{$prefix}border-radius-xxl: #{$border-radius-xxl};\n  --#{$prefix}border-radius-2xl: var(--#{$prefix}border-radius-xxl); // Deprecated in v5.3.0 for consistency\n  --#{$prefix}border-radius-pill: #{$border-radius-pill};\n  // scss-docs-end root-border-var\n\n  --#{$prefix}box-shadow: #{$box-shadow};\n  --#{$prefix}box-shadow-sm: #{$box-shadow-sm};\n  --#{$prefix}box-shadow-lg: #{$box-shadow-lg};\n  --#{$prefix}box-shadow-inset: #{$box-shadow-inset};\n\n  // Focus styles\n  // scss-docs-start root-focus-variables\n  --#{$prefix}focus-ring-width: #{$focus-ring-width};\n  --#{$prefix}focus-ring-opacity: #{$focus-ring-opacity};\n  --#{$prefix}focus-ring-color: #{$focus-ring-color};\n  // scss-docs-end root-focus-variables\n\n  // scss-docs-start root-form-validation-variables\n  --#{$prefix}form-valid-color: #{$form-valid-color};\n  --#{$prefix}form-valid-border-color: #{$form-valid-border-color};\n  --#{$prefix}form-invalid-color: #{$form-invalid-color};\n  --#{$prefix}form-invalid-border-color: #{$form-invalid-border-color};\n  // scss-docs-end root-form-validation-variables\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark, true) {\n    color-scheme: dark;\n\n    // scss-docs-start root-dark-mode-vars\n    --#{$prefix}body-color: #{$body-color-dark};\n    --#{$prefix}body-color-rgb: #{to-rgb($body-color-dark)};\n    --#{$prefix}body-bg: #{$body-bg-dark};\n    --#{$prefix}body-bg-rgb: #{to-rgb($body-bg-dark)};\n\n    --#{$prefix}emphasis-color: #{$body-emphasis-color-dark};\n    --#{$prefix}emphasis-color-rgb: #{to-rgb($body-emphasis-color-dark)};\n\n    --#{$prefix}secondary-color: #{$body-secondary-color-dark};\n    --#{$prefix}secondary-color-rgb: #{to-rgb($body-secondary-color-dark)};\n    --#{$prefix}secondary-bg: #{$body-secondary-bg-dark};\n    --#{$prefix}secondary-bg-rgb: #{to-rgb($body-secondary-bg-dark)};\n\n    --#{$prefix}tertiary-color: #{$body-tertiary-color-dark};\n    --#{$prefix}tertiary-color-rgb: #{to-rgb($body-tertiary-color-dark)};\n    --#{$prefix}tertiary-bg: #{$body-tertiary-bg-dark};\n    --#{$prefix}tertiary-bg-rgb: #{to-rgb($body-tertiary-bg-dark)};\n\n    @each $color, $value in $theme-colors-text-dark {\n      --#{$prefix}#{$color}-text-emphasis: #{$value};\n    }\n\n    @each $color, $value in $theme-colors-bg-subtle-dark {\n      --#{$prefix}#{$color}-bg-subtle: #{$value};\n    }\n\n    @each $color, $value in $theme-colors-border-subtle-dark {\n      --#{$prefix}#{$color}-border-subtle: #{$value};\n    }\n\n    --#{$prefix}heading-color: #{$headings-color-dark};\n\n    --#{$prefix}link-color: #{$link-color-dark};\n    --#{$prefix}link-hover-color: #{$link-hover-color-dark};\n    --#{$prefix}link-color-rgb: #{to-rgb($link-color-dark)};\n    --#{$prefix}link-hover-color-rgb: #{to-rgb($link-hover-color-dark)};\n\n    --#{$prefix}code-color: #{$code-color-dark};\n    --#{$prefix}highlight-color: #{$mark-color-dark};\n    --#{$prefix}highlight-bg: #{$mark-bg-dark};\n\n    --#{$prefix}border-color: #{$border-color-dark};\n    --#{$prefix}border-color-translucent: #{$border-color-translucent-dark};\n\n    --#{$prefix}form-valid-color: #{$form-valid-color-dark};\n    --#{$prefix}form-valid-border-color: #{$form-valid-border-color-dark};\n    --#{$prefix}form-invalid-color: #{$form-invalid-color-dark};\n    --#{$prefix}form-invalid-border-color: #{$form-invalid-border-color-dark};\n    // scss-docs-end root-dark-mode-vars\n  }\n}\n","// stylelint-disable scss/dimension-no-non-numeric-values\n\n// SCSS RFS mixin\n//\n// Automated responsive values for font sizes, paddings, margins and much more\n//\n// Licensed under MIT (https://github.com/twbs/rfs/blob/main/LICENSE)\n\n// Configuration\n\n// Base value\n$rfs-base-value: 1.25rem !default;\n$rfs-unit: rem !default;\n\n@if $rfs-unit != rem and $rfs-unit != px {\n  @error \"`#{$rfs-unit}` is not a valid unit for $rfs-unit. Use `px` or `rem`.\";\n}\n\n// Breakpoint at where values start decreasing if screen width is smaller\n$rfs-breakpoint: 1200px !default;\n$rfs-breakpoint-unit: px !default;\n\n@if $rfs-breakpoint-unit != px and $rfs-breakpoint-unit != em and $rfs-breakpoint-unit != rem {\n  @error \"`#{$rfs-breakpoint-unit}` is not a valid unit for $rfs-breakpoint-unit. Use `px`, `em` or `rem`.\";\n}\n\n// Resize values based on screen height and width\n$rfs-two-dimensional: false !default;\n\n// Factor of decrease\n$rfs-factor: 10 !default;\n\n@if type-of($rfs-factor) != number or $rfs-factor <= 1 {\n  @error \"`#{$rfs-factor}` is not a valid  $rfs-factor, it must be greater than 1.\";\n}\n\n// Mode. Possibilities: \"min-media-query\", \"max-media-query\"\n$rfs-mode: min-media-query !default;\n\n// Generate enable or disable classes. Possibilities: false, \"enable\" or \"disable\"\n$rfs-class: false !default;\n\n// 1 rem = $rfs-rem-value px\n$rfs-rem-value: 16 !default;\n\n// Safari iframe resize bug: https://github.com/twbs/rfs/issues/14\n$rfs-safari-iframe-resize-bug-fix: false !default;\n\n// Disable RFS by setting $enable-rfs to false\n$enable-rfs: true !default;\n\n// Cache $rfs-base-value unit\n$rfs-base-value-unit: unit($rfs-base-value);\n\n@function divide($dividend, $divisor, $precision: 10) {\n  $sign: if($dividend > 0 and $divisor > 0 or $dividend < 0 and $divisor < 0, 1, -1);\n  $dividend: abs($dividend);\n  $divisor: abs($divisor);\n  @if $dividend == 0 {\n    @return 0;\n  }\n  @if $divisor == 0 {\n    @error \"Cannot divide by 0\";\n  }\n  $remainder: $dividend;\n  $result: 0;\n  $factor: 10;\n  @while ($remainder > 0 and $precision >= 0) {\n    $quotient: 0;\n    @while ($remainder >= $divisor) {\n      $remainder: $remainder - $divisor;\n      $quotient: $quotient + 1;\n    }\n    $result: $result * 10 + $quotient;\n    $factor: $factor * .1;\n    $remainder: $remainder * 10;\n    $precision: $precision - 1;\n    @if ($precision < 0 and $remainder >= $divisor * 5) {\n      $result: $result + 1;\n    }\n  }\n  $result: $result * $factor * $sign;\n  $dividend-unit: unit($dividend);\n  $divisor-unit: unit($divisor);\n  $unit-map: (\n    \"px\": 1px,\n    \"rem\": 1rem,\n    \"em\": 1em,\n    \"%\": 1%\n  );\n  @if ($dividend-unit != $divisor-unit and map-has-key($unit-map, $dividend-unit)) {\n    $result: $result * map-get($unit-map, $dividend-unit);\n  }\n  @return $result;\n}\n\n// Remove px-unit from $rfs-base-value for calculations\n@if $rfs-base-value-unit == px {\n  $rfs-base-value: divide($rfs-base-value, $rfs-base-value * 0 + 1);\n}\n@else if $rfs-base-value-unit == rem {\n  $rfs-base-value: divide($rfs-base-value, divide($rfs-base-value * 0 + 1, $rfs-rem-value));\n}\n\n// Cache $rfs-breakpoint unit to prevent multiple calls\n$rfs-breakpoint-unit-cache: unit($rfs-breakpoint);\n\n// Remove unit from $rfs-breakpoint for calculations\n@if $rfs-breakpoint-unit-cache == px {\n  $rfs-breakpoint: divide($rfs-breakpoint, $rfs-breakpoint * 0 + 1);\n}\n@else if $rfs-breakpoint-unit-cache == rem or $rfs-breakpoint-unit-cache == \"em\" {\n  $rfs-breakpoint: divide($rfs-breakpoint, divide($rfs-breakpoint * 0 + 1, $rfs-rem-value));\n}\n\n// Calculate the media query value\n$rfs-mq-value: if($rfs-breakpoint-unit == px, #{$rfs-breakpoint}px, #{divide($rfs-breakpoint, $rfs-rem-value)}#{$rfs-breakpoint-unit});\n$rfs-mq-property-width: if($rfs-mode == max-media-query, max-width, min-width);\n$rfs-mq-property-height: if($rfs-mode == max-media-query, max-height, min-height);\n\n// Internal mixin used to determine which media query needs to be used\n@mixin _rfs-media-query {\n  @if $rfs-two-dimensional {\n    @if $rfs-mode == max-media-query {\n      @media (#{$rfs-mq-property-width}: #{$rfs-mq-value}), (#{$rfs-mq-property-height}: #{$rfs-mq-value}) {\n        @content;\n      }\n    }\n    @else {\n      @media (#{$rfs-mq-property-width}: #{$rfs-mq-value}) and (#{$rfs-mq-property-height}: #{$rfs-mq-value}) {\n        @content;\n      }\n    }\n  }\n  @else {\n    @media (#{$rfs-mq-property-width}: #{$rfs-mq-value}) {\n      @content;\n    }\n  }\n}\n\n// Internal mixin that adds disable classes to the selector if needed.\n@mixin _rfs-rule {\n  @if $rfs-class == disable and $rfs-mode == max-media-query {\n    // Adding an extra class increases specificity, which prevents the media query to override the property\n    &,\n    .disable-rfs &,\n    &.disable-rfs {\n      @content;\n    }\n  }\n  @else if $rfs-class == enable and $rfs-mode == min-media-query {\n    .enable-rfs &,\n    &.enable-rfs {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Internal mixin that adds enable classes to the selector if needed.\n@mixin _rfs-media-query-rule {\n\n  @if $rfs-class == enable {\n    @if $rfs-mode == min-media-query {\n      @content;\n    }\n\n    @include _rfs-media-query () {\n      .enable-rfs &,\n      &.enable-rfs {\n        @content;\n      }\n    }\n  }\n  @else {\n    @if $rfs-class == disable and $rfs-mode == min-media-query {\n      .disable-rfs &,\n      &.disable-rfs {\n        @content;\n      }\n    }\n    @include _rfs-media-query () {\n      @content;\n    }\n  }\n}\n\n// Helper function to get the formatted non-responsive value\n@function rfs-value($values) {\n  // Convert to list\n  $values: if(type-of($values) != list, ($values,), $values);\n\n  $val: \"\";\n\n  // Loop over each value and calculate value\n  @each $value in $values {\n    @if $value == 0 {\n      $val: $val + \" 0\";\n    }\n    @else {\n      // Cache $value unit\n      $unit: if(type-of($value) == \"number\", unit($value), false);\n\n      @if $unit == px {\n        // Convert to rem if needed\n        $val: $val + \" \" + if($rfs-unit == rem, #{divide($value, $value * 0 + $rfs-rem-value)}rem, $value);\n      }\n      @else if $unit == rem {\n        // Convert to px if needed\n        $val: $val + \" \" + if($rfs-unit == px, #{divide($value, $value * 0 + 1) * $rfs-rem-value}px, $value);\n      } @else {\n        // If $value isn't a number (like inherit) or $value has a unit (not px or rem, like 1.5em) or $ is 0, just print the value\n        $val: $val + \" \" + $value;\n      }\n    }\n  }\n\n  // Remove first space\n  @return unquote(str-slice($val, 2));\n}\n\n// Helper function to get the responsive value calculated by RFS\n@function rfs-fluid-value($values) {\n  // Convert to list\n  $values: if(type-of($values) != list, ($values,), $values);\n\n  $val: \"\";\n\n  // Loop over each value and calculate value\n  @each $value in $values {\n    @if $value == 0 {\n      $val: $val + \" 0\";\n    } @else {\n      // Cache $value unit\n      $unit: if(type-of($value) == \"number\", unit($value), false);\n\n      // If $value isn't a number (like inherit) or $value has a unit (not px or rem, like 1.5em) or $ is 0, just print the value\n      @if not $unit or $unit != px and $unit != rem {\n        $val: $val + \" \" + $value;\n      } @else {\n        // Remove unit from $value for calculations\n        $value: divide($value, $value * 0 + if($unit == px, 1, divide(1, $rfs-rem-value)));\n\n        // Only add the media query if the value is greater than the minimum value\n        @if abs($value) <= $rfs-base-value or not $enable-rfs {\n          $val: $val + \" \" + if($rfs-unit == rem, #{divide($value, $rfs-rem-value)}rem, #{$value}px);\n        }\n        @else {\n          // Calculate the minimum value\n          $value-min: $rfs-base-value + divide(abs($value) - $rfs-base-value, $rfs-factor);\n\n          // Calculate difference between $value and the minimum value\n          $value-diff: abs($value) - $value-min;\n\n          // Base value formatting\n          $min-width: if($rfs-unit == rem, #{divide($value-min, $rfs-rem-value)}rem, #{$value-min}px);\n\n          // Use negative value if needed\n          $min-width: if($value < 0, -$min-width, $min-width);\n\n          // Use `vmin` if two-dimensional is enabled\n          $variable-unit: if($rfs-two-dimensional, vmin, vw);\n\n          // Calculate the variable width between 0 and $rfs-breakpoint\n          $variable-width: #{divide($value-diff * 100, $rfs-breakpoint)}#{$variable-unit};\n\n          // Return the calculated value\n          $val: $val + \" calc(\" + $min-width + if($value < 0, \" - \", \" + \") + $variable-width + \")\";\n        }\n      }\n    }\n  }\n\n  // Remove first space\n  @return unquote(str-slice($val, 2));\n}\n\n// RFS mixin\n@mixin rfs($values, $property: font-size) {\n  @if $values != null {\n    $val: rfs-value($values);\n    $fluid-val: rfs-fluid-value($values);\n\n    // Do not print the media query if responsive & non-responsive values are the same\n    @if $val == $fluid-val {\n      #{$property}: $val;\n    }\n    @else {\n      @include _rfs-rule () {\n        #{$property}: if($rfs-mode == max-media-query, $val, $fluid-val);\n\n        // Include safari iframe resize fix if needed\n        min-width: if($rfs-safari-iframe-resize-bug-fix, (0 * 1vw), null);\n      }\n\n      @include _rfs-media-query-rule () {\n        #{$property}: if($rfs-mode == max-media-query, $fluid-val, $val);\n      }\n    }\n  }\n}\n\n// Shorthand helper mixins\n@mixin font-size($value) {\n  @include rfs($value);\n}\n\n@mixin padding($value) {\n  @include rfs($value, padding);\n}\n\n@mixin padding-top($value) {\n  @include rfs($value, padding-top);\n}\n\n@mixin padding-right($value) {\n  @include rfs($value, padding-right);\n}\n\n@mixin padding-bottom($value) {\n  @include rfs($value, padding-bottom);\n}\n\n@mixin padding-left($value) {\n  @include rfs($value, padding-left);\n}\n\n@mixin margin($value) {\n  @include rfs($value, margin);\n}\n\n@mixin margin-top($value) {\n  @include rfs($value, margin-top);\n}\n\n@mixin margin-right($value) {\n  @include rfs($value, margin-right);\n}\n\n@mixin margin-bottom($value) {\n  @include rfs($value, margin-bottom);\n}\n\n@mixin margin-left($value) {\n  @include rfs($value, margin-left);\n}\n","// scss-docs-start color-mode-mixin\n@mixin color-mode($mode: light, $root: false) {\n  @if $color-mode-type == \"media-query\" {\n    @if $root == true {\n      @media (prefers-color-scheme: $mode) {\n        :root {\n          @content;\n        }\n      }\n    } @else {\n      @media (prefers-color-scheme: $mode) {\n        @content;\n      }\n    }\n  } @else {\n    [data-bs-theme=\"#{$mode}\"] {\n      @content;\n    }\n  }\n}\n// scss-docs-end color-mode-mixin\n","// stylelint-disable declaration-no-important, selector-no-qualifying-type, property-no-vendor-prefix\n\n\n// Reboot\n//\n// Normalization of HTML elements, manually forked from Normalize.css to remove\n// styles targeting irrelevant browsers while applying new styles.\n//\n// Normalize is licensed MIT. https://github.com/necolas/normalize.css\n\n\n// Document\n//\n// Change from `box-sizing: content-box` so that `width` is not affected by `padding` or `border`.\n\n*,\n*::before,\n*::after {\n  box-sizing: border-box;\n}\n\n\n// Root\n//\n// Ability to the value of the root font sizes, affecting the value of `rem`.\n// null by default, thus nothing is generated.\n\n:root {\n  @if $font-size-root != null {\n    @include font-size(var(--#{$prefix}root-font-size));\n  }\n\n  @if $enable-smooth-scroll {\n    @media (prefers-reduced-motion: no-preference) {\n      scroll-behavior: smooth;\n    }\n  }\n}\n\n\n// Body\n//\n// 1. Remove the margin in all browsers.\n// 2. As a best practice, apply a default `background-color`.\n// 3. Prevent adjustments of font size after orientation changes in iOS.\n// 4. Change the default tap highlight to be completely transparent in iOS.\n\n// scss-docs-start reboot-body-rules\nbody {\n  margin: 0; // 1\n  font-family: var(--#{$prefix}body-font-family);\n  @include font-size(var(--#{$prefix}body-font-size));\n  font-weight: var(--#{$prefix}body-font-weight);\n  line-height: var(--#{$prefix}body-line-height);\n  color: var(--#{$prefix}body-color);\n  text-align: var(--#{$prefix}body-text-align);\n  background-color: var(--#{$prefix}body-bg); // 2\n  -webkit-text-size-adjust: 100%; // 3\n  -webkit-tap-highlight-color: rgba($black, 0); // 4\n}\n// scss-docs-end reboot-body-rules\n\n\n// Content grouping\n//\n// 1. Reset Firefox's gray color\n\nhr {\n  margin: $hr-margin-y 0;\n  color: $hr-color; // 1\n  border: 0;\n  border-top: $hr-border-width solid $hr-border-color;\n  opacity: $hr-opacity;\n}\n\n\n// Typography\n//\n// 1. Remove top margins from headings\n//    By default, `<h1>`-`<h6>` all receive top and bottom margins. We nuke the top\n//    margin for easier control within type scales as it avoids margin collapsing.\n\n%heading {\n  margin-top: 0; // 1\n  margin-bottom: $headings-margin-bottom;\n  font-family: $headings-font-family;\n  font-style: $headings-font-style;\n  font-weight: $headings-font-weight;\n  line-height: $headings-line-height;\n  color: var(--#{$prefix}heading-color);\n}\n\nh1 {\n  @extend %heading;\n  @include font-size($h1-font-size);\n}\n\nh2 {\n  @extend %heading;\n  @include font-size($h2-font-size);\n}\n\nh3 {\n  @extend %heading;\n  @include font-size($h3-font-size);\n}\n\nh4 {\n  @extend %heading;\n  @include font-size($h4-font-size);\n}\n\nh5 {\n  @extend %heading;\n  @include font-size($h5-font-size);\n}\n\nh6 {\n  @extend %heading;\n  @include font-size($h6-font-size);\n}\n\n\n// Reset margins on paragraphs\n//\n// Similarly, the top margin on `<p>`s get reset. However, we also reset the\n// bottom margin to use `rem` units instead of `em`.\n\np {\n  margin-top: 0;\n  margin-bottom: $paragraph-margin-bottom;\n}\n\n\n// Abbreviations\n//\n// 1. Add the correct text decoration in Chrome, Edge, Opera, and Safari.\n// 2. Add explicit cursor to indicate changed behavior.\n// 3. Prevent the text-decoration to be skipped.\n\nabbr[title] {\n  text-decoration: underline dotted; // 1\n  cursor: help; // 2\n  text-decoration-skip-ink: none; // 3\n}\n\n\n// Address\n\naddress {\n  margin-bottom: 1rem;\n  font-style: normal;\n  line-height: inherit;\n}\n\n\n// Lists\n\nol,\nul {\n  padding-left: 2rem;\n}\n\nol,\nul,\ndl {\n  margin-top: 0;\n  margin-bottom: 1rem;\n}\n\nol ol,\nul ul,\nol ul,\nul ol {\n  margin-bottom: 0;\n}\n\ndt {\n  font-weight: $dt-font-weight;\n}\n\n// 1. Undo browser default\n\ndd {\n  margin-bottom: .5rem;\n  margin-left: 0; // 1\n}\n\n\n// Blockquote\n\nblockquote {\n  margin: 0 0 1rem;\n}\n\n\n// Strong\n//\n// Add the correct font weight in Chrome, Edge, and Safari\n\nb,\nstrong {\n  font-weight: $font-weight-bolder;\n}\n\n\n// Small\n//\n// Add the correct font size in all browsers\n\nsmall {\n  @include font-size($small-font-size);\n}\n\n\n// Mark\n\nmark {\n  padding: $mark-padding;\n  color: var(--#{$prefix}highlight-color);\n  background-color: var(--#{$prefix}highlight-bg);\n}\n\n\n// Sub and Sup\n//\n// Prevent `sub` and `sup` elements from affecting the line height in\n// all browsers.\n\nsub,\nsup {\n  position: relative;\n  @include font-size($sub-sup-font-size);\n  line-height: 0;\n  vertical-align: baseline;\n}\n\nsub { bottom: -.25em; }\nsup { top: -.5em; }\n\n\n// Links\n\na {\n  color: rgba(var(--#{$prefix}link-color-rgb), var(--#{$prefix}link-opacity, 1));\n  text-decoration: $link-decoration;\n\n  &:hover {\n    --#{$prefix}link-color-rgb: var(--#{$prefix}link-hover-color-rgb);\n    text-decoration: $link-hover-decoration;\n  }\n}\n\n// And undo these styles for placeholder links/named anchors (without href).\n// It would be more straightforward to just use a[href] in previous block, but that\n// causes specificity issues in many other styles that are too complex to fix.\n// See https://github.com/twbs/bootstrap/issues/19402\n\na:not([href]):not([class]) {\n  &,\n  &:hover {\n    color: inherit;\n    text-decoration: none;\n  }\n}\n\n\n// Code\n\npre,\ncode,\nkbd,\nsamp {\n  font-family: $font-family-code;\n  @include font-size(1em); // Correct the odd `em` font sizing in all browsers.\n}\n\n// 1. Remove browser default top margin\n// 2. Reset browser default of `1em` to use `rem`s\n// 3. Don't allow content to break outside\n\npre {\n  display: block;\n  margin-top: 0; // 1\n  margin-bottom: 1rem; // 2\n  overflow: auto; // 3\n  @include font-size($code-font-size);\n  color: $pre-color;\n\n  // Account for some code outputs that place code tags in pre tags\n  code {\n    @include font-size(inherit);\n    color: inherit;\n    word-break: normal;\n  }\n}\n\ncode {\n  @include font-size($code-font-size);\n  color: var(--#{$prefix}code-color);\n  word-wrap: break-word;\n\n  // Streamline the style when inside anchors to avoid broken underline and more\n  a > & {\n    color: inherit;\n  }\n}\n\nkbd {\n  padding: $kbd-padding-y $kbd-padding-x;\n  @include font-size($kbd-font-size);\n  color: $kbd-color;\n  background-color: $kbd-bg;\n  @include border-radius($border-radius-sm);\n\n  kbd {\n    padding: 0;\n    @include font-size(1em);\n    font-weight: $nested-kbd-font-weight;\n  }\n}\n\n\n// Figures\n//\n// Apply a consistent margin strategy (matches our type styles).\n\nfigure {\n  margin: 0 0 1rem;\n}\n\n\n// Images and content\n\nimg,\nsvg {\n  vertical-align: middle;\n}\n\n\n// Tables\n//\n// Prevent double borders\n\ntable {\n  caption-side: bottom;\n  border-collapse: collapse;\n}\n\ncaption {\n  padding-top: $table-cell-padding-y;\n  padding-bottom: $table-cell-padding-y;\n  color: $table-caption-color;\n  text-align: left;\n}\n\n// 1. Removes font-weight bold by inheriting\n// 2. Matches default `<td>` alignment by inheriting `text-align`.\n// 3. Fix alignment for Safari\n\nth {\n  font-weight: $table-th-font-weight; // 1\n  text-align: inherit; // 2\n  text-align: -webkit-match-parent; // 3\n}\n\nthead,\ntbody,\ntfoot,\ntr,\ntd,\nth {\n  border-color: inherit;\n  border-style: solid;\n  border-width: 0;\n}\n\n\n// Forms\n//\n// 1. Allow labels to use `margin` for spacing.\n\nlabel {\n  display: inline-block; // 1\n}\n\n// Remove the default `border-radius` that macOS Chrome adds.\n// See https://github.com/twbs/bootstrap/issues/24093\n\nbutton {\n  // stylelint-disable-next-line property-disallowed-list\n  border-radius: 0;\n}\n\n// Explicitly remove focus outline in Chromium when it shouldn't be\n// visible (e.g. as result of mouse click or touch tap). It already\n// should be doing this automatically, but seems to currently be\n// confused and applies its very visible two-tone outline anyway.\n\nbutton:focus:not(:focus-visible) {\n  outline: 0;\n}\n\n// 1. Remove the margin in Firefox and Safari\n\ninput,\nbutton,\nselect,\noptgroup,\ntextarea {\n  margin: 0; // 1\n  font-family: inherit;\n  @include font-size(inherit);\n  line-height: inherit;\n}\n\n// Remove the inheritance of text transform in Firefox\nbutton,\nselect {\n  text-transform: none;\n}\n// Set the cursor for non-`<button>` buttons\n//\n// Details at https://github.com/twbs/bootstrap/pull/30562\n[role=\"button\"] {\n  cursor: pointer;\n}\n\nselect {\n  // Remove the inheritance of word-wrap in Safari.\n  // See https://github.com/twbs/bootstrap/issues/24990\n  word-wrap: normal;\n\n  // Undo the opacity change from Chrome\n  &:disabled {\n    opacity: 1;\n  }\n}\n\n// Remove the dropdown arrow only from text type inputs built with datalists in Chrome.\n// See https://stackoverflow.com/a/54997118\n\n[list]:not([type=\"date\"]):not([type=\"datetime-local\"]):not([type=\"month\"]):not([type=\"week\"]):not([type=\"time\"])::-webkit-calendar-picker-indicator {\n  display: none !important;\n}\n\n// 1. Prevent a WebKit bug where (2) destroys native `audio` and `video`\n//    controls in Android 4.\n// 2. Correct the inability to style clickable types in iOS and Safari.\n// 3. Opinionated: add \"hand\" cursor to non-disabled button elements.\n\nbutton,\n[type=\"button\"], // 1\n[type=\"reset\"],\n[type=\"submit\"] {\n  -webkit-appearance: button; // 2\n\n  @if $enable-button-pointers {\n    &:not(:disabled) {\n      cursor: pointer; // 3\n    }\n  }\n}\n\n// Remove inner border and padding from Firefox, but don't restore the outline like Normalize.\n\n::-moz-focus-inner {\n  padding: 0;\n  border-style: none;\n}\n\n// 1. Textareas should really only resize vertically so they don't break their (horizontal) containers.\n\ntextarea {\n  resize: vertical; // 1\n}\n\n// 1. Browsers set a default `min-width: min-content;` on fieldsets,\n//    unlike e.g. `<div>`s, which have `min-width: 0;` by default.\n//    So we reset that to ensure fieldsets behave more like a standard block element.\n//    See https://github.com/twbs/bootstrap/issues/12359\n//    and https://html.spec.whatwg.org/multipage/#the-fieldset-and-legend-elements\n// 2. Reset the default outline behavior of fieldsets so they don't affect page layout.\n\nfieldset {\n  min-width: 0; // 1\n  padding: 0; // 2\n  margin: 0; // 2\n  border: 0; // 2\n}\n\n// 1. By using `float: left`, the legend will behave like a block element.\n//    This way the border of a fieldset wraps around the legend if present.\n// 2. Fix wrapping bug.\n//    See https://github.com/twbs/bootstrap/issues/29712\n\nlegend {\n  float: left; // 1\n  width: 100%;\n  padding: 0;\n  margin-bottom: $legend-margin-bottom;\n  @include font-size($legend-font-size);\n  font-weight: $legend-font-weight;\n  line-height: inherit;\n\n  + * {\n    clear: left; // 2\n  }\n}\n\n// Fix height of inputs with a type of datetime-local, date, month, week, or time\n// See https://github.com/twbs/bootstrap/issues/18842\n\n::-webkit-datetime-edit-fields-wrapper,\n::-webkit-datetime-edit-text,\n::-webkit-datetime-edit-minute,\n::-webkit-datetime-edit-hour-field,\n::-webkit-datetime-edit-day-field,\n::-webkit-datetime-edit-month-field,\n::-webkit-datetime-edit-year-field {\n  padding: 0;\n}\n\n::-webkit-inner-spin-button {\n  height: auto;\n}\n\n// 1. This overrides the extra rounded corners on search inputs in iOS so that our\n//    `.form-control` class can properly style them. Note that this cannot simply\n//    be added to `.form-control` as it's not specific enough. For details, see\n//    https://github.com/twbs/bootstrap/issues/11586.\n// 2. Correct the outline style in Safari.\n\n[type=\"search\"] {\n  -webkit-appearance: textfield; // 1\n  outline-offset: -2px; // 2\n}\n\n// 1. A few input types should stay LTR\n// See https://rtlstyling.com/posts/rtl-styling#form-inputs\n// 2. RTL only output\n// See https://rtlcss.com/learn/usage-guide/control-directives/#raw\n\n/* rtl:raw:\n[type=\"tel\"],\n[type=\"url\"],\n[type=\"email\"],\n[type=\"number\"] {\n  direction: ltr;\n}\n*/\n\n// Remove the inner padding in Chrome and Safari on macOS.\n\n::-webkit-search-decoration {\n  -webkit-appearance: none;\n}\n\n// Remove padding around color pickers in webkit browsers\n\n::-webkit-color-swatch-wrapper {\n  padding: 0;\n}\n\n\n// 1. Inherit font family and line height for file input buttons\n// 2. Correct the inability to style clickable types in iOS and Safari.\n\n::file-selector-button {\n  font: inherit; // 1\n  -webkit-appearance: button; // 2\n}\n\n// Correct element displays\n\noutput {\n  display: inline-block;\n}\n\n// Remove border from iframe\n\niframe {\n  border: 0;\n}\n\n// Summary\n//\n// 1. Add the correct display in all browsers\n\nsummary {\n  display: list-item; // 1\n  cursor: pointer;\n}\n\n\n// Progress\n//\n// Add the correct vertical alignment in Chrome, Firefox, and Opera.\n\nprogress {\n  vertical-align: baseline;\n}\n\n\n// Hidden attribute\n//\n// Always hide an element with the `hidden` HTML attribute.\n\n[hidden] {\n  display: none !important;\n}\n","// Variables\n//\n// Variables should follow the `$component-state-property-size` formula for\n// consistent naming. Ex: $nav-link-disabled-color and $modal-content-box-shadow-xs.\n\n// Color system\n\n// scss-docs-start gray-color-variables\n$white:    #fff !default;\n$gray-100: #f8f9fa !default;\n$gray-200: #e9ecef !default;\n$gray-300: #dee2e6 !default;\n$gray-400: #ced4da !default;\n$gray-500: #adb5bd !default;\n$gray-600: #6c757d !default;\n$gray-700: #495057 !default;\n$gray-800: #343a40 !default;\n$gray-900: #212529 !default;\n$black:    #000 !default;\n// scss-docs-end gray-color-variables\n\n// fusv-disable\n// scss-docs-start gray-colors-map\n$grays: (\n  \"100\": $gray-100,\n  \"200\": $gray-200,\n  \"300\": $gray-300,\n  \"400\": $gray-400,\n  \"500\": $gray-500,\n  \"600\": $gray-600,\n  \"700\": $gray-700,\n  \"800\": $gray-800,\n  \"900\": $gray-900\n) !default;\n// scss-docs-end gray-colors-map\n// fusv-enable\n\n// scss-docs-start color-variables\n$blue:    #0d6efd !default;\n$indigo:  #6610f2 !default;\n$purple:  #6f42c1 !default;\n$pink:    #d63384 !default;\n$red:     #dc3545 !default;\n$orange:  #fd7e14 !default;\n$yellow:  #ffc107 !default;\n$green:   #198754 !default;\n$teal:    #20c997 !default;\n$cyan:    #0dcaf0 !default;\n// scss-docs-end color-variables\n\n// scss-docs-start colors-map\n$colors: (\n  \"blue\":       $blue,\n  \"indigo\":     $indigo,\n  \"purple\":     $purple,\n  \"pink\":       $pink,\n  \"red\":        $red,\n  \"orange\":     $orange,\n  \"yellow\":     $yellow,\n  \"green\":      $green,\n  \"teal\":       $teal,\n  \"cyan\":       $cyan,\n  \"black\":      $black,\n  \"white\":      $white,\n  \"gray\":       $gray-600,\n  \"gray-dark\":  $gray-800\n) !default;\n// scss-docs-end colors-map\n\n// The contrast ratio to reach against white, to determine if color changes from \"light\" to \"dark\". Acceptable values for WCAG 2.0 are 3, 4.5 and 7.\n// See https://www.w3.org/TR/WCAG20/#visual-audio-contrast-contrast\n$min-contrast-ratio:   4.5 !default;\n\n// Customize the light and dark text colors for use in our color contrast function.\n$color-contrast-dark:      $black !default;\n$color-contrast-light:     $white !default;\n\n// fusv-disable\n$blue-100: tint-color($blue, 80%) !default;\n$blue-200: tint-color($blue, 60%) !default;\n$blue-300: tint-color($blue, 40%) !default;\n$blue-400: tint-color($blue, 20%) !default;\n$blue-500: $blue !default;\n$blue-600: shade-color($blue, 20%) !default;\n$blue-700: shade-color($blue, 40%) !default;\n$blue-800: shade-color($blue, 60%) !default;\n$blue-900: shade-color($blue, 80%) !default;\n\n$indigo-100: tint-color($indigo, 80%) !default;\n$indigo-200: tint-color($indigo, 60%) !default;\n$indigo-300: tint-color($indigo, 40%) !default;\n$indigo-400: tint-color($indigo, 20%) !default;\n$indigo-500: $indigo !default;\n$indigo-600: shade-color($indigo, 20%) !default;\n$indigo-700: shade-color($indigo, 40%) !default;\n$indigo-800: shade-color($indigo, 60%) !default;\n$indigo-900: shade-color($indigo, 80%) !default;\n\n$purple-100: tint-color($purple, 80%) !default;\n$purple-200: tint-color($purple, 60%) !default;\n$purple-300: tint-color($purple, 40%) !default;\n$purple-400: tint-color($purple, 20%) !default;\n$purple-500: $purple !default;\n$purple-600: shade-color($purple, 20%) !default;\n$purple-700: shade-color($purple, 40%) !default;\n$purple-800: shade-color($purple, 60%) !default;\n$purple-900: shade-color($purple, 80%) !default;\n\n$pink-100: tint-color($pink, 80%) !default;\n$pink-200: tint-color($pink, 60%) !default;\n$pink-300: tint-color($pink, 40%) !default;\n$pink-400: tint-color($pink, 20%) !default;\n$pink-500: $pink !default;\n$pink-600: shade-color($pink, 20%) !default;\n$pink-700: shade-color($pink, 40%) !default;\n$pink-800: shade-color($pink, 60%) !default;\n$pink-900: shade-color($pink, 80%) !default;\n\n$red-100: tint-color($red, 80%) !default;\n$red-200: tint-color($red, 60%) !default;\n$red-300: tint-color($red, 40%) !default;\n$red-400: tint-color($red, 20%) !default;\n$red-500: $red !default;\n$red-600: shade-color($red, 20%) !default;\n$red-700: shade-color($red, 40%) !default;\n$red-800: shade-color($red, 60%) !default;\n$red-900: shade-color($red, 80%) !default;\n\n$orange-100: tint-color($orange, 80%) !default;\n$orange-200: tint-color($orange, 60%) !default;\n$orange-300: tint-color($orange, 40%) !default;\n$orange-400: tint-color($orange, 20%) !default;\n$orange-500: $orange !default;\n$orange-600: shade-color($orange, 20%) !default;\n$orange-700: shade-color($orange, 40%) !default;\n$orange-800: shade-color($orange, 60%) !default;\n$orange-900: shade-color($orange, 80%) !default;\n\n$yellow-100: tint-color($yellow, 80%) !default;\n$yellow-200: tint-color($yellow, 60%) !default;\n$yellow-300: tint-color($yellow, 40%) !default;\n$yellow-400: tint-color($yellow, 20%) !default;\n$yellow-500: $yellow !default;\n$yellow-600: shade-color($yellow, 20%) !default;\n$yellow-700: shade-color($yellow, 40%) !default;\n$yellow-800: shade-color($yellow, 60%) !default;\n$yellow-900: shade-color($yellow, 80%) !default;\n\n$green-100: tint-color($green, 80%) !default;\n$green-200: tint-color($green, 60%) !default;\n$green-300: tint-color($green, 40%) !default;\n$green-400: tint-color($green, 20%) !default;\n$green-500: $green !default;\n$green-600: shade-color($green, 20%) !default;\n$green-700: shade-color($green, 40%) !default;\n$green-800: shade-color($green, 60%) !default;\n$green-900: shade-color($green, 80%) !default;\n\n$teal-100: tint-color($teal, 80%) !default;\n$teal-200: tint-color($teal, 60%) !default;\n$teal-300: tint-color($teal, 40%) !default;\n$teal-400: tint-color($teal, 20%) !default;\n$teal-500: $teal !default;\n$teal-600: shade-color($teal, 20%) !default;\n$teal-700: shade-color($teal, 40%) !default;\n$teal-800: shade-color($teal, 60%) !default;\n$teal-900: shade-color($teal, 80%) !default;\n\n$cyan-100: tint-color($cyan, 80%) !default;\n$cyan-200: tint-color($cyan, 60%) !default;\n$cyan-300: tint-color($cyan, 40%) !default;\n$cyan-400: tint-color($cyan, 20%) !default;\n$cyan-500: $cyan !default;\n$cyan-600: shade-color($cyan, 20%) !default;\n$cyan-700: shade-color($cyan, 40%) !default;\n$cyan-800: shade-color($cyan, 60%) !default;\n$cyan-900: shade-color($cyan, 80%) !default;\n\n$blues: (\n  \"blue-100\": $blue-100,\n  \"blue-200\": $blue-200,\n  \"blue-300\": $blue-300,\n  \"blue-400\": $blue-400,\n  \"blue-500\": $blue-500,\n  \"blue-600\": $blue-600,\n  \"blue-700\": $blue-700,\n  \"blue-800\": $blue-800,\n  \"blue-900\": $blue-900\n) !default;\n\n$indigos: (\n  \"indigo-100\": $indigo-100,\n  \"indigo-200\": $indigo-200,\n  \"indigo-300\": $indigo-300,\n  \"indigo-400\": $indigo-400,\n  \"indigo-500\": $indigo-500,\n  \"indigo-600\": $indigo-600,\n  \"indigo-700\": $indigo-700,\n  \"indigo-800\": $indigo-800,\n  \"indigo-900\": $indigo-900\n) !default;\n\n$purples: (\n  \"purple-100\": $purple-100,\n  \"purple-200\": $purple-200,\n  \"purple-300\": $purple-300,\n  \"purple-400\": $purple-400,\n  \"purple-500\": $purple-500,\n  \"purple-600\": $purple-600,\n  \"purple-700\": $purple-700,\n  \"purple-800\": $purple-800,\n  \"purple-900\": $purple-900\n) !default;\n\n$pinks: (\n  \"pink-100\": $pink-100,\n  \"pink-200\": $pink-200,\n  \"pink-300\": $pink-300,\n  \"pink-400\": $pink-400,\n  \"pink-500\": $pink-500,\n  \"pink-600\": $pink-600,\n  \"pink-700\": $pink-700,\n  \"pink-800\": $pink-800,\n  \"pink-900\": $pink-900\n) !default;\n\n$reds: (\n  \"red-100\": $red-100,\n  \"red-200\": $red-200,\n  \"red-300\": $red-300,\n  \"red-400\": $red-400,\n  \"red-500\": $red-500,\n  \"red-600\": $red-600,\n  \"red-700\": $red-700,\n  \"red-800\": $red-800,\n  \"red-900\": $red-900\n) !default;\n\n$oranges: (\n  \"orange-100\": $orange-100,\n  \"orange-200\": $orange-200,\n  \"orange-300\": $orange-300,\n  \"orange-400\": $orange-400,\n  \"orange-500\": $orange-500,\n  \"orange-600\": $orange-600,\n  \"orange-700\": $orange-700,\n  \"orange-800\": $orange-800,\n  \"orange-900\": $orange-900\n) !default;\n\n$yellows: (\n  \"yellow-100\": $yellow-100,\n  \"yellow-200\": $yellow-200,\n  \"yellow-300\": $yellow-300,\n  \"yellow-400\": $yellow-400,\n  \"yellow-500\": $yellow-500,\n  \"yellow-600\": $yellow-600,\n  \"yellow-700\": $yellow-700,\n  \"yellow-800\": $yellow-800,\n  \"yellow-900\": $yellow-900\n) !default;\n\n$greens: (\n  \"green-100\": $green-100,\n  \"green-200\": $green-200,\n  \"green-300\": $green-300,\n  \"green-400\": $green-400,\n  \"green-500\": $green-500,\n  \"green-600\": $green-600,\n  \"green-700\": $green-700,\n  \"green-800\": $green-800,\n  \"green-900\": $green-900\n) !default;\n\n$teals: (\n  \"teal-100\": $teal-100,\n  \"teal-200\": $teal-200,\n  \"teal-300\": $teal-300,\n  \"teal-400\": $teal-400,\n  \"teal-500\": $teal-500,\n  \"teal-600\": $teal-600,\n  \"teal-700\": $teal-700,\n  \"teal-800\": $teal-800,\n  \"teal-900\": $teal-900\n) !default;\n\n$cyans: (\n  \"cyan-100\": $cyan-100,\n  \"cyan-200\": $cyan-200,\n  \"cyan-300\": $cyan-300,\n  \"cyan-400\": $cyan-400,\n  \"cyan-500\": $cyan-500,\n  \"cyan-600\": $cyan-600,\n  \"cyan-700\": $cyan-700,\n  \"cyan-800\": $cyan-800,\n  \"cyan-900\": $cyan-900\n) !default;\n// fusv-enable\n\n// scss-docs-start theme-color-variables\n$primary:       $blue !default;\n$secondary:     $gray-600 !default;\n$success:       $green !default;\n$info:          $cyan !default;\n$warning:       $yellow !default;\n$danger:        $red !default;\n$light:         $gray-100 !default;\n$dark:          $gray-900 !default;\n// scss-docs-end theme-color-variables\n\n// scss-docs-start theme-colors-map\n$theme-colors: (\n  \"primary\":    $primary,\n  \"secondary\":  $secondary,\n  \"success\":    $success,\n  \"info\":       $info,\n  \"warning\":    $warning,\n  \"danger\":     $danger,\n  \"light\":      $light,\n  \"dark\":       $dark\n) !default;\n// scss-docs-end theme-colors-map\n\n// scss-docs-start theme-text-variables\n$primary-text-emphasis:   shade-color($primary, 60%) !default;\n$secondary-text-emphasis: shade-color($secondary, 60%) !default;\n$success-text-emphasis:   shade-color($success, 60%) !default;\n$info-text-emphasis:      shade-color($info, 60%) !default;\n$warning-text-emphasis:   shade-color($warning, 60%) !default;\n$danger-text-emphasis:    shade-color($danger, 60%) !default;\n$light-text-emphasis:     $gray-700 !default;\n$dark-text-emphasis:      $gray-700 !default;\n// scss-docs-end theme-text-variables\n\n// scss-docs-start theme-bg-subtle-variables\n$primary-bg-subtle:       tint-color($primary, 80%) !default;\n$secondary-bg-subtle:     tint-color($secondary, 80%) !default;\n$success-bg-subtle:       tint-color($success, 80%) !default;\n$info-bg-subtle:          tint-color($info, 80%) !default;\n$warning-bg-subtle:       tint-color($warning, 80%) !default;\n$danger-bg-subtle:        tint-color($danger, 80%) !default;\n$light-bg-subtle:         mix($gray-100, $white) !default;\n$dark-bg-subtle:          $gray-400 !default;\n// scss-docs-end theme-bg-subtle-variables\n\n// scss-docs-start theme-border-subtle-variables\n$primary-border-subtle:   tint-color($primary, 60%) !default;\n$secondary-border-subtle: tint-color($secondary, 60%) !default;\n$success-border-subtle:   tint-color($success, 60%) !default;\n$info-border-subtle:      tint-color($info, 60%) !default;\n$warning-border-subtle:   tint-color($warning, 60%) !default;\n$danger-border-subtle:    tint-color($danger, 60%) !default;\n$light-border-subtle:     $gray-200 !default;\n$dark-border-subtle:      $gray-500 !default;\n// scss-docs-end theme-border-subtle-variables\n\n// Characters which are escaped by the escape-svg function\n$escaped-characters: (\n  (\"<\", \"%3c\"),\n  (\">\", \"%3e\"),\n  (\"#\", \"%23\"),\n  (\"(\", \"%28\"),\n  (\")\", \"%29\"),\n) !default;\n\n// Options\n//\n// Quickly modify global styling by enabling or disabling optional features.\n\n$enable-caret:                true !default;\n$enable-rounded:              true !default;\n$enable-shadows:              false !default;\n$enable-gradients:            false !default;\n$enable-transitions:          true !default;\n$enable-reduced-motion:       true !default;\n$enable-smooth-scroll:        true !default;\n$enable-grid-classes:         true !default;\n$enable-container-classes:    true !default;\n$enable-cssgrid:              false !default;\n$enable-button-pointers:      true !default;\n$enable-rfs:                  true !default;\n$enable-validation-icons:     true !default;\n$enable-negative-margins:     false !default;\n$enable-deprecation-messages: true !default;\n$enable-important-utilities:  true !default;\n\n$enable-dark-mode:            true !default;\n$color-mode-type:             data !default; // `data` or `media-query`\n\n// Prefix for :root CSS variables\n\n$variable-prefix:             bs- !default; // Deprecated in v5.2.0 for the shorter `$prefix`\n$prefix:                      $variable-prefix !default;\n\n// Gradient\n//\n// The gradient which is added to components if `$enable-gradients` is `true`\n// This gradient is also added to elements with `.bg-gradient`\n// scss-docs-start variable-gradient\n$gradient: linear-gradient(180deg, rgba($white, .15), rgba($white, 0)) !default;\n// scss-docs-end variable-gradient\n\n// Spacing\n//\n// Control the default styling of most Bootstrap elements by modifying these\n// variables. Mostly focused on spacing.\n// You can add more entries to the $spacers map, should you need more variation.\n\n// scss-docs-start spacer-variables-maps\n$spacer: 1rem !default;\n$spacers: (\n  0: 0,\n  1: $spacer * .25,\n  2: $spacer * .5,\n  3: $spacer,\n  4: $spacer * 1.5,\n  5: $spacer * 3,\n) !default;\n// scss-docs-end spacer-variables-maps\n\n// Position\n//\n// Define the edge positioning anchors of the position utilities.\n\n// scss-docs-start position-map\n$position-values: (\n  0: 0,\n  50: 50%,\n  100: 100%\n) !default;\n// scss-docs-end position-map\n\n// Body\n//\n// Settings for the `<body>` element.\n\n$body-text-align:           null !default;\n$body-color:                $gray-900 !default;\n$body-bg:                   $white !default;\n\n$body-secondary-color:      rgba($body-color, .75) !default;\n$body-secondary-bg:         $gray-200 !default;\n\n$body-tertiary-color:       rgba($body-color, .5) !default;\n$body-tertiary-bg:          $gray-100 !default;\n\n$body-emphasis-color:       $black !default;\n\n// Links\n//\n// Style anchor elements.\n\n$link-color:                              $primary !default;\n$link-decoration:                         underline !default;\n$link-shade-percentage:                   20% !default;\n$link-hover-color:                        shift-color($link-color, $link-shade-percentage) !default;\n$link-hover-decoration:                   null !default;\n\n$stretched-link-pseudo-element:           after !default;\n$stretched-link-z-index:                  1 !default;\n\n// Icon links\n// scss-docs-start icon-link-variables\n$icon-link-gap:               .375rem !default;\n$icon-link-underline-offset:  .25em !default;\n$icon-link-icon-size:         1em !default;\n$icon-link-icon-transition:   .2s ease-in-out transform !default;\n$icon-link-icon-transform:    translate3d(.25em, 0, 0) !default;\n// scss-docs-end icon-link-variables\n\n// Paragraphs\n//\n// Style p element.\n\n$paragraph-margin-bottom:   1rem !default;\n\n\n// Grid breakpoints\n//\n// Define the minimum dimensions at which your layout will change,\n// adapting to different screen sizes, for use in media queries.\n\n// scss-docs-start grid-breakpoints\n$grid-breakpoints: (\n  xs: 0,\n  sm: 576px,\n  md: 768px,\n  lg: 992px,\n  xl: 1200px,\n  xxl: 1400px\n) !default;\n// scss-docs-end grid-breakpoints\n\n@include _assert-ascending($grid-breakpoints, \"$grid-breakpoints\");\n@include _assert-starts-at-zero($grid-breakpoints, \"$grid-breakpoints\");\n\n\n// Grid containers\n//\n// Define the maximum width of `.container` for different screen sizes.\n\n// scss-docs-start container-max-widths\n$container-max-widths: (\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1140px,\n  xxl: 1320px\n) !default;\n// scss-docs-end container-max-widths\n\n@include _assert-ascending($container-max-widths, \"$container-max-widths\");\n\n\n// Grid columns\n//\n// Set the number of columns and specify the width of the gutters.\n\n$grid-columns:                12 !default;\n$grid-gutter-width:           1.5rem !default;\n$grid-row-columns:            6 !default;\n\n// Container padding\n\n$container-padding-x: $grid-gutter-width !default;\n\n\n// Components\n//\n// Define common padding and border radius sizes and more.\n\n// scss-docs-start border-variables\n$border-width:                1px !default;\n$border-widths: (\n  1: 1px,\n  2: 2px,\n  3: 3px,\n  4: 4px,\n  5: 5px\n) !default;\n$border-style:                solid !default;\n$border-color:                $gray-300 !default;\n$border-color-translucent:    rgba($black, .175) !default;\n// scss-docs-end border-variables\n\n// scss-docs-start border-radius-variables\n$border-radius:               .375rem !default;\n$border-radius-sm:            .25rem !default;\n$border-radius-lg:            .5rem !default;\n$border-radius-xl:            1rem !default;\n$border-radius-xxl:           2rem !default;\n$border-radius-pill:          50rem !default;\n// scss-docs-end border-radius-variables\n// fusv-disable\n$border-radius-2xl:           $border-radius-xxl !default; // Deprecated in v5.3.0\n// fusv-enable\n\n// scss-docs-start box-shadow-variables\n$box-shadow:                  0 .5rem 1rem rgba($black, .15) !default;\n$box-shadow-sm:               0 .125rem .25rem rgba($black, .075) !default;\n$box-shadow-lg:               0 1rem 3rem rgba($black, .175) !default;\n$box-shadow-inset:            inset 0 1px 2px rgba($black, .075) !default;\n// scss-docs-end box-shadow-variables\n\n$component-active-color:      $white !default;\n$component-active-bg:         $primary !default;\n\n// scss-docs-start focus-ring-variables\n$focus-ring-width:      .25rem !default;\n$focus-ring-opacity:    .25 !default;\n$focus-ring-color:      rgba($primary, $focus-ring-opacity) !default;\n$focus-ring-blur:       0 !default;\n$focus-ring-box-shadow: 0 0 $focus-ring-blur $focus-ring-width $focus-ring-color !default;\n// scss-docs-end focus-ring-variables\n\n// scss-docs-start caret-variables\n$caret-width:                 .3em !default;\n$caret-vertical-align:        $caret-width * .85 !default;\n$caret-spacing:               $caret-width * .85 !default;\n// scss-docs-end caret-variables\n\n$transition-base:             all .2s ease-in-out !default;\n$transition-fade:             opacity .15s linear !default;\n// scss-docs-start collapse-transition\n$transition-collapse:         height .35s ease !default;\n$transition-collapse-width:   width .35s ease !default;\n// scss-docs-end collapse-transition\n\n// stylelint-disable function-disallowed-list\n// scss-docs-start aspect-ratios\n$aspect-ratios: (\n  \"1x1\": 100%,\n  \"4x3\": calc(3 / 4 * 100%),\n  \"16x9\": calc(9 / 16 * 100%),\n  \"21x9\": calc(9 / 21 * 100%)\n) !default;\n// scss-docs-end aspect-ratios\n// stylelint-enable function-disallowed-list\n\n// Typography\n//\n// Font, line-height, and color for body text, headings, and more.\n\n// scss-docs-start font-variables\n// stylelint-disable value-keyword-case\n$font-family-sans-serif:      system-ui, -apple-system, \"Segoe UI\", Roboto, \"Helvetica Neue\", \"Noto Sans\", \"Liberation Sans\", Arial, sans-serif, \"Apple Color Emoji\", \"Segoe UI Emoji\", \"Segoe UI Symbol\", \"Noto Color Emoji\" !default;\n$font-family-monospace:       SFMono-Regular, Menlo, Monaco, Consolas, \"Liberation Mono\", \"Courier New\", monospace !default;\n// stylelint-enable value-keyword-case\n$font-family-base:            var(--#{$prefix}font-sans-serif) !default;\n$font-family-code:            var(--#{$prefix}font-monospace) !default;\n\n// $font-size-root affects the value of `rem`, which is used for as well font sizes, paddings, and margins\n// $font-size-base affects the font size of the body text\n$font-size-root:              null !default;\n$font-size-base:              1rem !default; // Assumes the browser default, typically `16px`\n$font-size-sm:                $font-size-base * .875 !default;\n$font-size-lg:                $font-size-base * 1.25 !default;\n\n$font-weight-lighter:         lighter !default;\n$font-weight-light:           300 !default;\n$font-weight-normal:          400 !default;\n$font-weight-medium:          500 !default;\n$font-weight-semibold:        600 !default;\n$font-weight-bold:            700 !default;\n$font-weight-bolder:          bolder !default;\n\n$font-weight-base:            $font-weight-normal !default;\n\n$line-height-base:            1.5 !default;\n$line-height-sm:              1.25 !default;\n$line-height-lg:              2 !default;\n\n$h1-font-size:                $font-size-base * 2.5 !default;\n$h2-font-size:                $font-size-base * 2 !default;\n$h3-font-size:                $font-size-base * 1.75 !default;\n$h4-font-size:                $font-size-base * 1.5 !default;\n$h5-font-size:                $font-size-base * 1.25 !default;\n$h6-font-size:                $font-size-base !default;\n// scss-docs-end font-variables\n\n// scss-docs-start font-sizes\n$font-sizes: (\n  1: $h1-font-size,\n  2: $h2-font-size,\n  3: $h3-font-size,\n  4: $h4-font-size,\n  5: $h5-font-size,\n  6: $h6-font-size\n) !default;\n// scss-docs-end font-sizes\n\n// scss-docs-start headings-variables\n$headings-margin-bottom:      $spacer * .5 !default;\n$headings-font-family:        null !default;\n$headings-font-style:         null !default;\n$headings-font-weight:        500 !default;\n$headings-line-height:        1.2 !default;\n$headings-color:              inherit !default;\n// scss-docs-end headings-variables\n\n// scss-docs-start display-headings\n$display-font-sizes: (\n  1: 5rem,\n  2: 4.5rem,\n  3: 4rem,\n  4: 3.5rem,\n  5: 3rem,\n  6: 2.5rem\n) !default;\n\n$display-font-family: null !default;\n$display-font-style:  null !default;\n$display-font-weight: 300 !default;\n$display-line-height: $headings-line-height !default;\n// scss-docs-end display-headings\n\n// scss-docs-start type-variables\n$lead-font-size:              $font-size-base * 1.25 !default;\n$lead-font-weight:            300 !default;\n\n$small-font-size:             .875em !default;\n\n$sub-sup-font-size:           .75em !default;\n\n// fusv-disable\n$text-muted:                  var(--#{$prefix}secondary-color) !default; // Deprecated in 5.3.0\n// fusv-enable\n\n$initialism-font-size:        $small-font-size !default;\n\n$blockquote-margin-y:         $spacer !default;\n$blockquote-font-size:        $font-size-base * 1.25 !default;\n$blockquote-footer-color:     $gray-600 !default;\n$blockquote-footer-font-size: $small-font-size !default;\n\n$hr-margin-y:                 $spacer !default;\n$hr-color:                    inherit !default;\n\n// fusv-disable\n$hr-bg-color:                 null !default; // Deprecated in v5.2.0\n$hr-height:                   null !default; // Deprecated in v5.2.0\n// fusv-enable\n\n$hr-border-color:             null !default; // Allows for inherited colors\n$hr-border-width:             var(--#{$prefix}border-width) !default;\n$hr-opacity:                  .25 !default;\n\n// scss-docs-start vr-variables\n$vr-border-width:             var(--#{$prefix}border-width) !default;\n// scss-docs-end vr-variables\n\n$legend-margin-bottom:        .5rem !default;\n$legend-font-size:            1.5rem !default;\n$legend-font-weight:          null !default;\n\n$dt-font-weight:              $font-weight-bold !default;\n\n$list-inline-padding:         .5rem !default;\n\n$mark-padding:                .1875em !default;\n$mark-color:                  $body-color !default;\n$mark-bg:                     $yellow-100 !default;\n// scss-docs-end type-variables\n\n\n// Tables\n//\n// Customizes the `.table` component with basic values, each used across all table variations.\n\n// scss-docs-start table-variables\n$table-cell-padding-y:        .5rem !default;\n$table-cell-padding-x:        .5rem !default;\n$table-cell-padding-y-sm:     .25rem !default;\n$table-cell-padding-x-sm:     .25rem !default;\n\n$table-cell-vertical-align:   top !default;\n\n$table-color:                 var(--#{$prefix}emphasis-color) !default;\n$table-bg:                    var(--#{$prefix}body-bg) !default;\n$table-accent-bg:             transparent !default;\n\n$table-th-font-weight:        null !default;\n\n$table-striped-color:         $table-color !default;\n$table-striped-bg-factor:     .05 !default;\n$table-striped-bg:            rgba(var(--#{$prefix}emphasis-color-rgb), $table-striped-bg-factor) !default;\n\n$table-active-color:          $table-color !default;\n$table-active-bg-factor:      .1 !default;\n$table-active-bg:             rgba(var(--#{$prefix}emphasis-color-rgb), $table-active-bg-factor) !default;\n\n$table-hover-color:           $table-color !default;\n$table-hover-bg-factor:       .075 !default;\n$table-hover-bg:              rgba(var(--#{$prefix}emphasis-color-rgb), $table-hover-bg-factor) !default;\n\n$table-border-factor:         .2 !default;\n$table-border-width:          var(--#{$prefix}border-width) !default;\n$table-border-color:          var(--#{$prefix}border-color) !default;\n\n$table-striped-order:         odd !default;\n$table-striped-columns-order: even !default;\n\n$table-group-separator-color: currentcolor !default;\n\n$table-caption-color:         var(--#{$prefix}secondary-color) !default;\n\n$table-bg-scale:              -80% !default;\n// scss-docs-end table-variables\n\n// scss-docs-start table-loop\n$table-variants: (\n  \"primary\":    shift-color($primary, $table-bg-scale),\n  \"secondary\":  shift-color($secondary, $table-bg-scale),\n  \"success\":    shift-color($success, $table-bg-scale),\n  \"info\":       shift-color($info, $table-bg-scale),\n  \"warning\":    shift-color($warning, $table-bg-scale),\n  \"danger\":     shift-color($danger, $table-bg-scale),\n  \"light\":      $light,\n  \"dark\":       $dark,\n) !default;\n// scss-docs-end table-loop\n\n\n// Buttons + Forms\n//\n// Shared variables that are reassigned to `$input-` and `$btn-` specific variables.\n\n// scss-docs-start input-btn-variables\n$input-btn-padding-y:         .375rem !default;\n$input-btn-padding-x:         .75rem !default;\n$input-btn-font-family:       null !default;\n$input-btn-font-size:         $font-size-base !default;\n$input-btn-line-height:       $line-height-base !default;\n\n$input-btn-focus-width:         $focus-ring-width !default;\n$input-btn-focus-color-opacity: $focus-ring-opacity !default;\n$input-btn-focus-color:         $focus-ring-color !default;\n$input-btn-focus-blur:          $focus-ring-blur !default;\n$input-btn-focus-box-shadow:    $focus-ring-box-shadow !default;\n\n$input-btn-padding-y-sm:      .25rem !default;\n$input-btn-padding-x-sm:      .5rem !default;\n$input-btn-font-size-sm:      $font-size-sm !default;\n\n$input-btn-padding-y-lg:      .5rem !default;\n$input-btn-padding-x-lg:      1rem !default;\n$input-btn-font-size-lg:      $font-size-lg !default;\n\n$input-btn-border-width:      var(--#{$prefix}border-width) !default;\n// scss-docs-end input-btn-variables\n\n\n// Buttons\n//\n// For each of Bootstrap's buttons, define text, background, and border color.\n\n// scss-docs-start btn-variables\n$btn-color:                   var(--#{$prefix}body-color) !default;\n$btn-padding-y:               $input-btn-padding-y !default;\n$btn-padding-x:               $input-btn-padding-x !default;\n$btn-font-family:             $input-btn-font-family !default;\n$btn-font-size:               $input-btn-font-size !default;\n$btn-line-height:             $input-btn-line-height !default;\n$btn-white-space:             null !default; // Set to `nowrap` to prevent text wrapping\n\n$btn-padding-y-sm:            $input-btn-padding-y-sm !default;\n$btn-padding-x-sm:            $input-btn-padding-x-sm !default;\n$btn-font-size-sm:            $input-btn-font-size-sm !default;\n\n$btn-padding-y-lg:            $input-btn-padding-y-lg !default;\n$btn-padding-x-lg:            $input-btn-padding-x-lg !default;\n$btn-font-size-lg:            $input-btn-font-size-lg !default;\n\n$btn-border-width:            $input-btn-border-width !default;\n\n$btn-font-weight:             $font-weight-normal !default;\n$btn-box-shadow:              inset 0 1px 0 rgba($white, .15), 0 1px 1px rgba($black, .075) !default;\n$btn-focus-width:             $input-btn-focus-width !default;\n$btn-focus-box-shadow:        $input-btn-focus-box-shadow !default;\n$btn-disabled-opacity:        .65 !default;\n$btn-active-box-shadow:       inset 0 3px 5px rgba($black, .125) !default;\n\n$btn-link-color:              var(--#{$prefix}link-color) !default;\n$btn-link-hover-color:        var(--#{$prefix}link-hover-color) !default;\n$btn-link-disabled-color:     $gray-600 !default;\n$btn-link-focus-shadow-rgb:   to-rgb(mix(color-contrast($link-color), $link-color, 15%)) !default;\n\n// Allows for customizing button radius independently from global border radius\n$btn-border-radius:           var(--#{$prefix}border-radius) !default;\n$btn-border-radius-sm:        var(--#{$prefix}border-radius-sm) !default;\n$btn-border-radius-lg:        var(--#{$prefix}border-radius-lg) !default;\n\n$btn-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$btn-hover-bg-shade-amount:       15% !default;\n$btn-hover-bg-tint-amount:        15% !default;\n$btn-hover-border-shade-amount:   20% !default;\n$btn-hover-border-tint-amount:    10% !default;\n$btn-active-bg-shade-amount:      20% !default;\n$btn-active-bg-tint-amount:       20% !default;\n$btn-active-border-shade-amount:  25% !default;\n$btn-active-border-tint-amount:   10% !default;\n// scss-docs-end btn-variables\n\n\n// Forms\n\n// scss-docs-start form-text-variables\n$form-text-margin-top:                  .25rem !default;\n$form-text-font-size:                   $small-font-size !default;\n$form-text-font-style:                  null !default;\n$form-text-font-weight:                 null !default;\n$form-text-color:                       var(--#{$prefix}secondary-color) !default;\n// scss-docs-end form-text-variables\n\n// scss-docs-start form-label-variables\n$form-label-margin-bottom:              .5rem !default;\n$form-label-font-size:                  null !default;\n$form-label-font-style:                 null !default;\n$form-label-font-weight:                null !default;\n$form-label-color:                      null !default;\n// scss-docs-end form-label-variables\n\n// scss-docs-start form-input-variables\n$input-padding-y:                       $input-btn-padding-y !default;\n$input-padding-x:                       $input-btn-padding-x !default;\n$input-font-family:                     $input-btn-font-family !default;\n$input-font-size:                       $input-btn-font-size !default;\n$input-font-weight:                     $font-weight-base !default;\n$input-line-height:                     $input-btn-line-height !default;\n\n$input-padding-y-sm:                    $input-btn-padding-y-sm !default;\n$input-padding-x-sm:                    $input-btn-padding-x-sm !default;\n$input-font-size-sm:                    $input-btn-font-size-sm !default;\n\n$input-padding-y-lg:                    $input-btn-padding-y-lg !default;\n$input-padding-x-lg:                    $input-btn-padding-x-lg !default;\n$input-font-size-lg:                    $input-btn-font-size-lg !default;\n\n$input-bg:                              var(--#{$prefix}body-bg) !default;\n$input-disabled-color:                  null !default;\n$input-disabled-bg:                     var(--#{$prefix}secondary-bg) !default;\n$input-disabled-border-color:           null !default;\n\n$input-color:                           var(--#{$prefix}body-color) !default;\n$input-border-color:                    var(--#{$prefix}border-color) !default;\n$input-border-width:                    $input-btn-border-width !default;\n$input-box-shadow:                      var(--#{$prefix}box-shadow-inset) !default;\n\n$input-border-radius:                   var(--#{$prefix}border-radius) !default;\n$input-border-radius-sm:                var(--#{$prefix}border-radius-sm) !default;\n$input-border-radius-lg:                var(--#{$prefix}border-radius-lg) !default;\n\n$input-focus-bg:                        $input-bg !default;\n$input-focus-border-color:              tint-color($component-active-bg, 50%) !default;\n$input-focus-color:                     $input-color !default;\n$input-focus-width:                     $input-btn-focus-width !default;\n$input-focus-box-shadow:                $input-btn-focus-box-shadow !default;\n\n$input-placeholder-color:               var(--#{$prefix}secondary-color) !default;\n$input-plaintext-color:                 var(--#{$prefix}body-color) !default;\n\n$input-height-border:                   calc(#{$input-border-width} * 2) !default; // stylelint-disable-line function-disallowed-list\n\n$input-height-inner:                    add($input-line-height * 1em, $input-padding-y * 2) !default;\n$input-height-inner-half:               add($input-line-height * .5em, $input-padding-y) !default;\n$input-height-inner-quarter:            add($input-line-height * .25em, $input-padding-y * .5) !default;\n\n$input-height:                          add($input-line-height * 1em, add($input-padding-y * 2, $input-height-border, false)) !default;\n$input-height-sm:                       add($input-line-height * 1em, add($input-padding-y-sm * 2, $input-height-border, false)) !default;\n$input-height-lg:                       add($input-line-height * 1em, add($input-padding-y-lg * 2, $input-height-border, false)) !default;\n\n$input-transition:                      border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$form-color-width:                      3rem !default;\n// scss-docs-end form-input-variables\n\n// scss-docs-start form-check-variables\n$form-check-input-width:                  1em !default;\n$form-check-min-height:                   $font-size-base * $line-height-base !default;\n$form-check-padding-start:                $form-check-input-width + .5em !default;\n$form-check-margin-bottom:                .125rem !default;\n$form-check-label-color:                  null !default;\n$form-check-label-cursor:                 null !default;\n$form-check-transition:                   null !default;\n\n$form-check-input-active-filter:          brightness(90%) !default;\n\n$form-check-input-bg:                     $input-bg !default;\n$form-check-input-border:                 var(--#{$prefix}border-width) solid var(--#{$prefix}border-color) !default;\n$form-check-input-border-radius:          .25em !default;\n$form-check-radio-border-radius:          50% !default;\n$form-check-input-focus-border:           $input-focus-border-color !default;\n$form-check-input-focus-box-shadow:       $focus-ring-box-shadow !default;\n\n$form-check-input-checked-color:          $component-active-color !default;\n$form-check-input-checked-bg-color:       $component-active-bg !default;\n$form-check-input-checked-border-color:   $form-check-input-checked-bg-color !default;\n$form-check-input-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-checked-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='m6 10 3 3 6-6'/></svg>\") !default;\n$form-check-radio-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='2' fill='#{$form-check-input-checked-color}'/></svg>\") !default;\n\n$form-check-input-indeterminate-color:          $component-active-color !default;\n$form-check-input-indeterminate-bg-color:       $component-active-bg !default;\n$form-check-input-indeterminate-border-color:   $form-check-input-indeterminate-bg-color !default;\n$form-check-input-indeterminate-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-indeterminate-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='M6 10h8'/></svg>\") !default;\n\n$form-check-input-disabled-opacity:        .5 !default;\n$form-check-label-disabled-opacity:        $form-check-input-disabled-opacity !default;\n$form-check-btn-check-disabled-opacity:    $btn-disabled-opacity !default;\n\n$form-check-inline-margin-end:    1rem !default;\n// scss-docs-end form-check-variables\n\n// scss-docs-start form-switch-variables\n$form-switch-color:               rgba($black, .25) !default;\n$form-switch-width:               2em !default;\n$form-switch-padding-start:       $form-switch-width + .5em !default;\n$form-switch-bg-image:            url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-color}'/></svg>\") !default;\n$form-switch-border-radius:       $form-switch-width !default;\n$form-switch-transition:          background-position .15s ease-in-out !default;\n\n$form-switch-focus-color:         $input-focus-border-color !default;\n$form-switch-focus-bg-image:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-focus-color}'/></svg>\") !default;\n\n$form-switch-checked-color:       $component-active-color !default;\n$form-switch-checked-bg-image:    url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-checked-color}'/></svg>\") !default;\n$form-switch-checked-bg-position: right center !default;\n// scss-docs-end form-switch-variables\n\n// scss-docs-start input-group-variables\n$input-group-addon-padding-y:           $input-padding-y !default;\n$input-group-addon-padding-x:           $input-padding-x !default;\n$input-group-addon-font-weight:         $input-font-weight !default;\n$input-group-addon-color:               $input-color !default;\n$input-group-addon-bg:                  var(--#{$prefix}tertiary-bg) !default;\n$input-group-addon-border-color:        $input-border-color !default;\n// scss-docs-end input-group-variables\n\n// scss-docs-start form-select-variables\n$form-select-padding-y:             $input-padding-y !default;\n$form-select-padding-x:             $input-padding-x !default;\n$form-select-font-family:           $input-font-family !default;\n$form-select-font-size:             $input-font-size !default;\n$form-select-indicator-padding:     $form-select-padding-x * 3 !default; // Extra padding for background-image\n$form-select-font-weight:           $input-font-weight !default;\n$form-select-line-height:           $input-line-height !default;\n$form-select-color:                 $input-color !default;\n$form-select-bg:                    $input-bg !default;\n$form-select-disabled-color:        null !default;\n$form-select-disabled-bg:           $input-disabled-bg !default;\n$form-select-disabled-border-color: $input-disabled-border-color !default;\n$form-select-bg-position:           right $form-select-padding-x center !default;\n$form-select-bg-size:               16px 12px !default; // In pixels because image dimensions\n$form-select-indicator-color:       $gray-800 !default;\n$form-select-indicator:             url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16'><path fill='none' stroke='#{$form-select-indicator-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='2' d='m2 5 6 6 6-6'/></svg>\") !default;\n\n$form-select-feedback-icon-padding-end: $form-select-padding-x * 2.5 + $form-select-indicator-padding !default;\n$form-select-feedback-icon-position:    center right $form-select-indicator-padding !default;\n$form-select-feedback-icon-size:        $input-height-inner-half $input-height-inner-half !default;\n\n$form-select-border-width:        $input-border-width !default;\n$form-select-border-color:        $input-border-color !default;\n$form-select-border-radius:       $input-border-radius !default;\n$form-select-box-shadow:          var(--#{$prefix}box-shadow-inset) !default;\n\n$form-select-focus-border-color:  $input-focus-border-color !default;\n$form-select-focus-width:         $input-focus-width !default;\n$form-select-focus-box-shadow:    0 0 0 $form-select-focus-width $input-btn-focus-color !default;\n\n$form-select-padding-y-sm:        $input-padding-y-sm !default;\n$form-select-padding-x-sm:        $input-padding-x-sm !default;\n$form-select-font-size-sm:        $input-font-size-sm !default;\n$form-select-border-radius-sm:    $input-border-radius-sm !default;\n\n$form-select-padding-y-lg:        $input-padding-y-lg !default;\n$form-select-padding-x-lg:        $input-padding-x-lg !default;\n$form-select-font-size-lg:        $input-font-size-lg !default;\n$form-select-border-radius-lg:    $input-border-radius-lg !default;\n\n$form-select-transition:          $input-transition !default;\n// scss-docs-end form-select-variables\n\n// scss-docs-start form-range-variables\n$form-range-track-width:          100% !default;\n$form-range-track-height:         .5rem !default;\n$form-range-track-cursor:         pointer !default;\n$form-range-track-bg:             var(--#{$prefix}secondary-bg) !default;\n$form-range-track-border-radius:  1rem !default;\n$form-range-track-box-shadow:     var(--#{$prefix}box-shadow-inset) !default;\n\n$form-range-thumb-width:                   1rem !default;\n$form-range-thumb-height:                  $form-range-thumb-width !default;\n$form-range-thumb-bg:                      $component-active-bg !default;\n$form-range-thumb-border:                  0 !default;\n$form-range-thumb-border-radius:           1rem !default;\n$form-range-thumb-box-shadow:              0 .1rem .25rem rgba($black, .1) !default;\n$form-range-thumb-focus-box-shadow:        0 0 0 1px $body-bg, $input-focus-box-shadow !default;\n$form-range-thumb-focus-box-shadow-width:  $input-focus-width !default; // For focus box shadow issue in Edge\n$form-range-thumb-active-bg:               tint-color($component-active-bg, 70%) !default;\n$form-range-thumb-disabled-bg:             var(--#{$prefix}secondary-color) !default;\n$form-range-thumb-transition:              background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n// scss-docs-end form-range-variables\n\n// scss-docs-start form-file-variables\n$form-file-button-color:          $input-color !default;\n$form-file-button-bg:             var(--#{$prefix}tertiary-bg) !default;\n$form-file-button-hover-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end form-file-variables\n\n// scss-docs-start form-floating-variables\n$form-floating-height:                  add(3.5rem, $input-height-border) !default;\n$form-floating-line-height:             1.25 !default;\n$form-floating-padding-x:               $input-padding-x !default;\n$form-floating-padding-y:               1rem !default;\n$form-floating-input-padding-t:         1.625rem !default;\n$form-floating-input-padding-b:         .625rem !default;\n$form-floating-label-height:            1.5em !default;\n$form-floating-label-opacity:           .65 !default;\n$form-floating-label-transform:         scale(.85) translateY(-.5rem) translateX(.15rem) !default;\n$form-floating-label-disabled-color:    $gray-600 !default;\n$form-floating-transition:              opacity .1s ease-in-out, transform .1s ease-in-out !default;\n// scss-docs-end form-floating-variables\n\n// Form validation\n\n// scss-docs-start form-feedback-variables\n$form-feedback-margin-top:          $form-text-margin-top !default;\n$form-feedback-font-size:           $form-text-font-size !default;\n$form-feedback-font-style:          $form-text-font-style !default;\n$form-feedback-valid-color:         $success !default;\n$form-feedback-invalid-color:       $danger !default;\n\n$form-feedback-icon-valid-color:    $form-feedback-valid-color !default;\n$form-feedback-icon-valid:          url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 8 8'><path fill='#{$form-feedback-icon-valid-color}' d='M2.3 6.73.6 4.53c-.4-1.04.46-1.4 1.1-.8l1.1 1.4 3.4-3.8c.6-.63 1.6-.27 1.2.7l-4 4.6c-.43.5-.8.4-1.1.1z'/></svg>\") !default;\n$form-feedback-icon-invalid-color:  $form-feedback-invalid-color !default;\n$form-feedback-icon-invalid:        url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 12 12' width='12' height='12' fill='none' stroke='#{$form-feedback-icon-invalid-color}'><circle cx='6' cy='6' r='4.5'/><path stroke-linejoin='round' d='M5.8 3.6h.4L6 6.5z'/><circle cx='6' cy='8.2' r='.6' fill='#{$form-feedback-icon-invalid-color}' stroke='none'/></svg>\") !default;\n// scss-docs-end form-feedback-variables\n\n// scss-docs-start form-validation-colors\n$form-valid-color:                  $form-feedback-valid-color !default;\n$form-valid-border-color:           $form-feedback-valid-color !default;\n$form-invalid-color:                $form-feedback-invalid-color !default;\n$form-invalid-border-color:         $form-feedback-invalid-color !default;\n// scss-docs-end form-validation-colors\n\n// scss-docs-start form-validation-states\n$form-validation-states: (\n  \"valid\": (\n    \"color\": var(--#{$prefix}form-valid-color),\n    \"icon\": $form-feedback-icon-valid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}success),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}success-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-valid-border-color),\n  ),\n  \"invalid\": (\n    \"color\": var(--#{$prefix}form-invalid-color),\n    \"icon\": $form-feedback-icon-invalid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}danger),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}danger-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-invalid-border-color),\n  )\n) !default;\n// scss-docs-end form-validation-states\n\n// Z-index master list\n//\n// Warning: Avoid customizing these values. They're used for a bird's eye view\n// of components dependent on the z-axis and are designed to all work together.\n\n// scss-docs-start zindex-stack\n$zindex-dropdown:                   1000 !default;\n$zindex-sticky:                     1020 !default;\n$zindex-fixed:                      1030 !default;\n$zindex-offcanvas-backdrop:         1040 !default;\n$zindex-offcanvas:                  1045 !default;\n$zindex-modal-backdrop:             1050 !default;\n$zindex-modal:                      1055 !default;\n$zindex-popover:                    1070 !default;\n$zindex-tooltip:                    1080 !default;\n$zindex-toast:                      1090 !default;\n// scss-docs-end zindex-stack\n\n// scss-docs-start zindex-levels-map\n$zindex-levels: (\n  n1: -1,\n  0: 0,\n  1: 1,\n  2: 2,\n  3: 3\n) !default;\n// scss-docs-end zindex-levels-map\n\n\n// Navs\n\n// scss-docs-start nav-variables\n$nav-link-padding-y:                .5rem !default;\n$nav-link-padding-x:                1rem !default;\n$nav-link-font-size:                null !default;\n$nav-link-font-weight:              null !default;\n$nav-link-color:                    var(--#{$prefix}link-color) !default;\n$nav-link-hover-color:              var(--#{$prefix}link-hover-color) !default;\n$nav-link-transition:               color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out !default;\n$nav-link-disabled-color:           var(--#{$prefix}secondary-color) !default;\n$nav-link-focus-box-shadow:         $focus-ring-box-shadow !default;\n\n$nav-tabs-border-color:             var(--#{$prefix}border-color) !default;\n$nav-tabs-border-width:             var(--#{$prefix}border-width) !default;\n$nav-tabs-border-radius:            var(--#{$prefix}border-radius) !default;\n$nav-tabs-link-hover-border-color:  var(--#{$prefix}secondary-bg) var(--#{$prefix}secondary-bg) $nav-tabs-border-color !default;\n$nav-tabs-link-active-color:        var(--#{$prefix}emphasis-color) !default;\n$nav-tabs-link-active-bg:           var(--#{$prefix}body-bg) !default;\n$nav-tabs-link-active-border-color: var(--#{$prefix}border-color) var(--#{$prefix}border-color) $nav-tabs-link-active-bg !default;\n\n$nav-pills-border-radius:           var(--#{$prefix}border-radius) !default;\n$nav-pills-link-active-color:       $component-active-color !default;\n$nav-pills-link-active-bg:          $component-active-bg !default;\n\n$nav-underline-gap:                 1rem !default;\n$nav-underline-border-width:        .125rem !default;\n$nav-underline-link-active-color:   var(--#{$prefix}emphasis-color) !default;\n// scss-docs-end nav-variables\n\n\n// Navbar\n\n// scss-docs-start navbar-variables\n$navbar-padding-y:                  $spacer * .5 !default;\n$navbar-padding-x:                  null !default;\n\n$navbar-nav-link-padding-x:         .5rem !default;\n\n$navbar-brand-font-size:            $font-size-lg !default;\n// Compute the navbar-brand padding-y so the navbar-brand will have the same height as navbar-text and nav-link\n$nav-link-height:                   $font-size-base * $line-height-base + $nav-link-padding-y * 2 !default;\n$navbar-brand-height:               $navbar-brand-font-size * $line-height-base !default;\n$navbar-brand-padding-y:            ($nav-link-height - $navbar-brand-height) * .5 !default;\n$navbar-brand-margin-end:           1rem !default;\n\n$navbar-toggler-padding-y:          .25rem !default;\n$navbar-toggler-padding-x:          .75rem !default;\n$navbar-toggler-font-size:          $font-size-lg !default;\n$navbar-toggler-border-radius:      $btn-border-radius !default;\n$navbar-toggler-focus-width:        $btn-focus-width !default;\n$navbar-toggler-transition:         box-shadow .15s ease-in-out !default;\n\n$navbar-light-color:                rgba(var(--#{$prefix}emphasis-color-rgb), .65) !default;\n$navbar-light-hover-color:          rgba(var(--#{$prefix}emphasis-color-rgb), .8) !default;\n$navbar-light-active-color:         rgba(var(--#{$prefix}emphasis-color-rgb), 1) !default;\n$navbar-light-disabled-color:       rgba(var(--#{$prefix}emphasis-color-rgb), .3) !default;\n$navbar-light-icon-color:           rgba($body-color, .75) !default;\n$navbar-light-toggler-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-light-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-light-toggler-border-color: rgba(var(--#{$prefix}emphasis-color-rgb), .15) !default;\n$navbar-light-brand-color:          $navbar-light-active-color !default;\n$navbar-light-brand-hover-color:    $navbar-light-active-color !default;\n// scss-docs-end navbar-variables\n\n// scss-docs-start navbar-dark-variables\n$navbar-dark-color:                 rgba($white, .55) !default;\n$navbar-dark-hover-color:           rgba($white, .75) !default;\n$navbar-dark-active-color:          $white !default;\n$navbar-dark-disabled-color:        rgba($white, .25) !default;\n$navbar-dark-icon-color:            $navbar-dark-color !default;\n$navbar-dark-toggler-icon-bg:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-dark-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-dark-toggler-border-color:  rgba($white, .1) !default;\n$navbar-dark-brand-color:           $navbar-dark-active-color !default;\n$navbar-dark-brand-hover-color:     $navbar-dark-active-color !default;\n// scss-docs-end navbar-dark-variables\n\n\n// Dropdowns\n//\n// Dropdown menu container and contents.\n\n// scss-docs-start dropdown-variables\n$dropdown-min-width:                10rem !default;\n$dropdown-padding-x:                0 !default;\n$dropdown-padding-y:                .5rem !default;\n$dropdown-spacer:                   .125rem !default;\n$dropdown-font-size:                $font-size-base !default;\n$dropdown-color:                    var(--#{$prefix}body-color) !default;\n$dropdown-bg:                       var(--#{$prefix}body-bg) !default;\n$dropdown-border-color:             var(--#{$prefix}border-color-translucent) !default;\n$dropdown-border-radius:            var(--#{$prefix}border-radius) !default;\n$dropdown-border-width:             var(--#{$prefix}border-width) !default;\n$dropdown-inner-border-radius:      calc(#{$dropdown-border-radius} - #{$dropdown-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$dropdown-divider-bg:               $dropdown-border-color !default;\n$dropdown-divider-margin-y:         $spacer * .5 !default;\n$dropdown-box-shadow:               var(--#{$prefix}box-shadow) !default;\n\n$dropdown-link-color:               var(--#{$prefix}body-color) !default;\n$dropdown-link-hover-color:         $dropdown-link-color !default;\n$dropdown-link-hover-bg:            var(--#{$prefix}tertiary-bg) !default;\n\n$dropdown-link-active-color:        $component-active-color !default;\n$dropdown-link-active-bg:           $component-active-bg !default;\n\n$dropdown-link-disabled-color:      var(--#{$prefix}tertiary-color) !default;\n\n$dropdown-item-padding-y:           $spacer * .25 !default;\n$dropdown-item-padding-x:           $spacer !default;\n\n$dropdown-header-color:             $gray-600 !default;\n$dropdown-header-padding-x:         $dropdown-item-padding-x !default;\n$dropdown-header-padding-y:         $dropdown-padding-y !default;\n// fusv-disable\n$dropdown-header-padding:           $dropdown-header-padding-y $dropdown-header-padding-x !default; // Deprecated in v5.2.0\n// fusv-enable\n// scss-docs-end dropdown-variables\n\n// scss-docs-start dropdown-dark-variables\n$dropdown-dark-color:               $gray-300 !default;\n$dropdown-dark-bg:                  $gray-800 !default;\n$dropdown-dark-border-color:        $dropdown-border-color !default;\n$dropdown-dark-divider-bg:          $dropdown-divider-bg !default;\n$dropdown-dark-box-shadow:          null !default;\n$dropdown-dark-link-color:          $dropdown-dark-color !default;\n$dropdown-dark-link-hover-color:    $white !default;\n$dropdown-dark-link-hover-bg:       rgba($white, .15) !default;\n$dropdown-dark-link-active-color:   $dropdown-link-active-color !default;\n$dropdown-dark-link-active-bg:      $dropdown-link-active-bg !default;\n$dropdown-dark-link-disabled-color: $gray-500 !default;\n$dropdown-dark-header-color:        $gray-500 !default;\n// scss-docs-end dropdown-dark-variables\n\n\n// Pagination\n\n// scss-docs-start pagination-variables\n$pagination-padding-y:              .375rem !default;\n$pagination-padding-x:              .75rem !default;\n$pagination-padding-y-sm:           .25rem !default;\n$pagination-padding-x-sm:           .5rem !default;\n$pagination-padding-y-lg:           .75rem !default;\n$pagination-padding-x-lg:           1.5rem !default;\n\n$pagination-font-size:              $font-size-base !default;\n\n$pagination-color:                  var(--#{$prefix}link-color) !default;\n$pagination-bg:                     var(--#{$prefix}body-bg) !default;\n$pagination-border-radius:          var(--#{$prefix}border-radius) !default;\n$pagination-border-width:           var(--#{$prefix}border-width) !default;\n$pagination-margin-start:           calc(#{$pagination-border-width} * -1) !default; // stylelint-disable-line function-disallowed-list\n$pagination-border-color:           var(--#{$prefix}border-color) !default;\n\n$pagination-focus-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-focus-bg:               var(--#{$prefix}secondary-bg) !default;\n$pagination-focus-box-shadow:       $focus-ring-box-shadow !default;\n$pagination-focus-outline:          0 !default;\n\n$pagination-hover-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$pagination-hover-border-color:     var(--#{$prefix}border-color) !default; // Todo in v6: remove this?\n\n$pagination-active-color:           $component-active-color !default;\n$pagination-active-bg:              $component-active-bg !default;\n$pagination-active-border-color:    $component-active-bg !default;\n\n$pagination-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$pagination-disabled-bg:            var(--#{$prefix}secondary-bg) !default;\n$pagination-disabled-border-color:  var(--#{$prefix}border-color) !default;\n\n$pagination-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$pagination-border-radius-sm:       var(--#{$prefix}border-radius-sm) !default;\n$pagination-border-radius-lg:       var(--#{$prefix}border-radius-lg) !default;\n// scss-docs-end pagination-variables\n\n\n// Placeholders\n\n// scss-docs-start placeholders\n$placeholder-opacity-max:           .5 !default;\n$placeholder-opacity-min:           .2 !default;\n// scss-docs-end placeholders\n\n// Cards\n\n// scss-docs-start card-variables\n$card-spacer-y:                     $spacer !default;\n$card-spacer-x:                     $spacer !default;\n$card-title-spacer-y:               $spacer * .5 !default;\n$card-title-color:                  null !default;\n$card-subtitle-color:               null !default;\n$card-border-width:                 var(--#{$prefix}border-width) !default;\n$card-border-color:                 var(--#{$prefix}border-color-translucent) !default;\n$card-border-radius:                var(--#{$prefix}border-radius) !default;\n$card-box-shadow:                   null !default;\n$card-inner-border-radius:          subtract($card-border-radius, $card-border-width) !default;\n$card-cap-padding-y:                $card-spacer-y * .5 !default;\n$card-cap-padding-x:                $card-spacer-x !default;\n$card-cap-bg:                       rgba(var(--#{$prefix}body-color-rgb), .03) !default;\n$card-cap-color:                    null !default;\n$card-height:                       null !default;\n$card-color:                        null !default;\n$card-bg:                           var(--#{$prefix}body-bg) !default;\n$card-img-overlay-padding:          $spacer !default;\n$card-group-margin:                 $grid-gutter-width * .5 !default;\n// scss-docs-end card-variables\n\n// Accordion\n\n// scss-docs-start accordion-variables\n$accordion-padding-y:                     1rem !default;\n$accordion-padding-x:                     1.25rem !default;\n$accordion-color:                         var(--#{$prefix}body-color) !default;\n$accordion-bg:                            var(--#{$prefix}body-bg) !default;\n$accordion-border-width:                  var(--#{$prefix}border-width) !default;\n$accordion-border-color:                  var(--#{$prefix}border-color) !default;\n$accordion-border-radius:                 var(--#{$prefix}border-radius) !default;\n$accordion-inner-border-radius:           subtract($accordion-border-radius, $accordion-border-width) !default;\n\n$accordion-body-padding-y:                $accordion-padding-y !default;\n$accordion-body-padding-x:                $accordion-padding-x !default;\n\n$accordion-button-padding-y:              $accordion-padding-y !default;\n$accordion-button-padding-x:              $accordion-padding-x !default;\n$accordion-button-color:                  var(--#{$prefix}body-color) !default;\n$accordion-button-bg:                     var(--#{$prefix}accordion-bg) !default;\n$accordion-transition:                    $btn-transition, border-radius .15s ease !default;\n$accordion-button-active-bg:              var(--#{$prefix}primary-bg-subtle) !default;\n$accordion-button-active-color:           var(--#{$prefix}primary-text-emphasis) !default;\n\n$accordion-button-focus-border-color:     $input-focus-border-color !default;\n$accordion-button-focus-box-shadow:       $btn-focus-box-shadow !default;\n\n$accordion-icon-width:                    1.25rem !default;\n$accordion-icon-color:                    $body-color !default;\n$accordion-icon-active-color:             $primary-text-emphasis !default;\n$accordion-icon-transition:               transform .2s ease-in-out !default;\n$accordion-icon-transform:                rotate(-180deg) !default;\n\n$accordion-button-icon:         url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$accordion-icon-color}'><path fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/></svg>\") !default;\n$accordion-button-active-icon:  url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$accordion-icon-active-color}'><path fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/></svg>\") !default;\n// scss-docs-end accordion-variables\n\n// Tooltips\n\n// scss-docs-start tooltip-variables\n$tooltip-font-size:                 $font-size-sm !default;\n$tooltip-max-width:                 200px !default;\n$tooltip-color:                     var(--#{$prefix}body-bg) !default;\n$tooltip-bg:                        var(--#{$prefix}emphasis-color) !default;\n$tooltip-border-radius:             var(--#{$prefix}border-radius) !default;\n$tooltip-opacity:                   .9 !default;\n$tooltip-padding-y:                 $spacer * .25 !default;\n$tooltip-padding-x:                 $spacer * .5 !default;\n$tooltip-margin:                    null !default; // TODO: remove this in v6\n\n$tooltip-arrow-width:               .8rem !default;\n$tooltip-arrow-height:              .4rem !default;\n// fusv-disable\n$tooltip-arrow-color:               null !default; // Deprecated in Bootstrap 5.2.0 for CSS variables\n// fusv-enable\n// scss-docs-end tooltip-variables\n\n// Form tooltips must come after regular tooltips\n// scss-docs-start tooltip-feedback-variables\n$form-feedback-tooltip-padding-y:     $tooltip-padding-y !default;\n$form-feedback-tooltip-padding-x:     $tooltip-padding-x !default;\n$form-feedback-tooltip-font-size:     $tooltip-font-size !default;\n$form-feedback-tooltip-line-height:   null !default;\n$form-feedback-tooltip-opacity:       $tooltip-opacity !default;\n$form-feedback-tooltip-border-radius: $tooltip-border-radius !default;\n// scss-docs-end tooltip-feedback-variables\n\n\n// Popovers\n\n// scss-docs-start popover-variables\n$popover-font-size:                 $font-size-sm !default;\n$popover-bg:                        var(--#{$prefix}body-bg) !default;\n$popover-max-width:                 276px !default;\n$popover-border-width:              var(--#{$prefix}border-width) !default;\n$popover-border-color:              var(--#{$prefix}border-color-translucent) !default;\n$popover-border-radius:             var(--#{$prefix}border-radius-lg) !default;\n$popover-inner-border-radius:       calc(#{$popover-border-radius} - #{$popover-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$popover-box-shadow:                var(--#{$prefix}box-shadow) !default;\n\n$popover-header-font-size:          $font-size-base !default;\n$popover-header-bg:                 var(--#{$prefix}secondary-bg) !default;\n$popover-header-color:              $headings-color !default;\n$popover-header-padding-y:          .5rem !default;\n$popover-header-padding-x:          $spacer !default;\n\n$popover-body-color:                var(--#{$prefix}body-color) !default;\n$popover-body-padding-y:            $spacer !default;\n$popover-body-padding-x:            $spacer !default;\n\n$popover-arrow-width:               1rem !default;\n$popover-arrow-height:              .5rem !default;\n// scss-docs-end popover-variables\n\n// fusv-disable\n// Deprecated in Bootstrap 5.2.0 for CSS variables\n$popover-arrow-color:               $popover-bg !default;\n$popover-arrow-outer-color:         var(--#{$prefix}border-color-translucent) !default;\n// fusv-enable\n\n\n// Toasts\n\n// scss-docs-start toast-variables\n$toast-max-width:                   350px !default;\n$toast-padding-x:                   .75rem !default;\n$toast-padding-y:                   .5rem !default;\n$toast-font-size:                   .875rem !default;\n$toast-color:                       null !default;\n$toast-background-color:            rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-border-width:                var(--#{$prefix}border-width) !default;\n$toast-border-color:                var(--#{$prefix}border-color-translucent) !default;\n$toast-border-radius:               var(--#{$prefix}border-radius) !default;\n$toast-box-shadow:                  var(--#{$prefix}box-shadow) !default;\n$toast-spacing:                     $container-padding-x !default;\n\n$toast-header-color:                var(--#{$prefix}secondary-color) !default;\n$toast-header-background-color:     rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-header-border-color:         $toast-border-color !default;\n// scss-docs-end toast-variables\n\n\n// Badges\n\n// scss-docs-start badge-variables\n$badge-font-size:                   .75em !default;\n$badge-font-weight:                 $font-weight-bold !default;\n$badge-color:                       $white !default;\n$badge-padding-y:                   .35em !default;\n$badge-padding-x:                   .65em !default;\n$badge-border-radius:               var(--#{$prefix}border-radius) !default;\n// scss-docs-end badge-variables\n\n\n// Modals\n\n// scss-docs-start modal-variables\n$modal-inner-padding:               $spacer !default;\n\n$modal-footer-margin-between:       .5rem !default;\n\n$modal-dialog-margin:               .5rem !default;\n$modal-dialog-margin-y-sm-up:       1.75rem !default;\n\n$modal-title-line-height:           $line-height-base !default;\n\n$modal-content-color:               null !default;\n$modal-content-bg:                  var(--#{$prefix}body-bg) !default;\n$modal-content-border-color:        var(--#{$prefix}border-color-translucent) !default;\n$modal-content-border-width:        var(--#{$prefix}border-width) !default;\n$modal-content-border-radius:       var(--#{$prefix}border-radius-lg) !default;\n$modal-content-inner-border-radius: subtract($modal-content-border-radius, $modal-content-border-width) !default;\n$modal-content-box-shadow-xs:       var(--#{$prefix}box-shadow-sm) !default;\n$modal-content-box-shadow-sm-up:    var(--#{$prefix}box-shadow) !default;\n\n$modal-backdrop-bg:                 $black !default;\n$modal-backdrop-opacity:            .5 !default;\n\n$modal-header-border-color:         var(--#{$prefix}border-color) !default;\n$modal-header-border-width:         $modal-content-border-width !default;\n$modal-header-padding-y:            $modal-inner-padding !default;\n$modal-header-padding-x:            $modal-inner-padding !default;\n$modal-header-padding:              $modal-header-padding-y $modal-header-padding-x !default; // Keep this for backwards compatibility\n\n$modal-footer-bg:                   null !default;\n$modal-footer-border-color:         $modal-header-border-color !default;\n$modal-footer-border-width:         $modal-header-border-width !default;\n\n$modal-sm:                          300px !default;\n$modal-md:                          500px !default;\n$modal-lg:                          800px !default;\n$modal-xl:                          1140px !default;\n\n$modal-fade-transform:              translate(0, -50px) !default;\n$modal-show-transform:              none !default;\n$modal-transition:                  transform .3s ease-out !default;\n$modal-scale-transform:             scale(1.02) !default;\n// scss-docs-end modal-variables\n\n\n// Alerts\n//\n// Define alert colors, border radius, and padding.\n\n// scss-docs-start alert-variables\n$alert-padding-y:               $spacer !default;\n$alert-padding-x:               $spacer !default;\n$alert-margin-bottom:           1rem !default;\n$alert-border-radius:           var(--#{$prefix}border-radius) !default;\n$alert-link-font-weight:        $font-weight-bold !default;\n$alert-border-width:            var(--#{$prefix}border-width) !default;\n$alert-dismissible-padding-r:   $alert-padding-x * 3 !default; // 3x covers width of x plus default padding on either side\n// scss-docs-end alert-variables\n\n// fusv-disable\n$alert-bg-scale:                -80% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-border-scale:            -70% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-color-scale:             40% !default; // Deprecated in v5.2.0, to be removed in v6\n// fusv-enable\n\n// Progress bars\n\n// scss-docs-start progress-variables\n$progress-height:                   1rem !default;\n$progress-font-size:                $font-size-base * .75 !default;\n$progress-bg:                       var(--#{$prefix}secondary-bg) !default;\n$progress-border-radius:            var(--#{$prefix}border-radius) !default;\n$progress-box-shadow:               var(--#{$prefix}box-shadow-inset) !default;\n$progress-bar-color:                $white !default;\n$progress-bar-bg:                   $primary !default;\n$progress-bar-animation-timing:     1s linear infinite !default;\n$progress-bar-transition:           width .6s ease !default;\n// scss-docs-end progress-variables\n\n\n// List group\n\n// scss-docs-start list-group-variables\n$list-group-color:                  var(--#{$prefix}body-color) !default;\n$list-group-bg:                     var(--#{$prefix}body-bg) !default;\n$list-group-border-color:           var(--#{$prefix}border-color) !default;\n$list-group-border-width:           var(--#{$prefix}border-width) !default;\n$list-group-border-radius:          var(--#{$prefix}border-radius) !default;\n\n$list-group-item-padding-y:         $spacer * .5 !default;\n$list-group-item-padding-x:         $spacer !default;\n// fusv-disable\n$list-group-item-bg-scale:          -80% !default; // Deprecated in v5.3.0\n$list-group-item-color-scale:       40% !default; // Deprecated in v5.3.0\n// fusv-enable\n\n$list-group-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$list-group-active-color:           $component-active-color !default;\n$list-group-active-bg:              $component-active-bg !default;\n$list-group-active-border-color:    $list-group-active-bg !default;\n\n$list-group-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$list-group-disabled-bg:            $list-group-bg !default;\n\n$list-group-action-color:           var(--#{$prefix}secondary-color) !default;\n$list-group-action-hover-color:     var(--#{$prefix}emphasis-color) !default;\n\n$list-group-action-active-color:    var(--#{$prefix}body-color) !default;\n$list-group-action-active-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end list-group-variables\n\n\n// Image thumbnails\n\n// scss-docs-start thumbnail-variables\n$thumbnail-padding:                 .25rem !default;\n$thumbnail-bg:                      var(--#{$prefix}body-bg) !default;\n$thumbnail-border-width:            var(--#{$prefix}border-width) !default;\n$thumbnail-border-color:            var(--#{$prefix}border-color) !default;\n$thumbnail-border-radius:           var(--#{$prefix}border-radius) !default;\n$thumbnail-box-shadow:              var(--#{$prefix}box-shadow-sm) !default;\n// scss-docs-end thumbnail-variables\n\n\n// Figures\n\n// scss-docs-start figure-variables\n$figure-caption-font-size:          $small-font-size !default;\n$figure-caption-color:              var(--#{$prefix}secondary-color) !default;\n// scss-docs-end figure-variables\n\n\n// Breadcrumbs\n\n// scss-docs-start breadcrumb-variables\n$breadcrumb-font-size:              null !default;\n$breadcrumb-padding-y:              0 !default;\n$breadcrumb-padding-x:              0 !default;\n$breadcrumb-item-padding-x:         .5rem !default;\n$breadcrumb-margin-bottom:          1rem !default;\n$breadcrumb-bg:                     null !default;\n$breadcrumb-divider-color:          var(--#{$prefix}secondary-color) !default;\n$breadcrumb-active-color:           var(--#{$prefix}secondary-color) !default;\n$breadcrumb-divider:                quote(\"/\") !default;\n$breadcrumb-divider-flipped:        $breadcrumb-divider !default;\n$breadcrumb-border-radius:          null !default;\n// scss-docs-end breadcrumb-variables\n\n// Carousel\n\n// scss-docs-start carousel-variables\n$carousel-control-color:             $white !default;\n$carousel-control-width:             15% !default;\n$carousel-control-opacity:           .5 !default;\n$carousel-control-hover-opacity:     .9 !default;\n$carousel-control-transition:        opacity .15s ease !default;\n\n$carousel-indicator-width:           30px !default;\n$carousel-indicator-height:          3px !default;\n$carousel-indicator-hit-area-height: 10px !default;\n$carousel-indicator-spacer:          3px !default;\n$carousel-indicator-opacity:         .5 !default;\n$carousel-indicator-active-bg:       $white !default;\n$carousel-indicator-active-opacity:  1 !default;\n$carousel-indicator-transition:      opacity .6s ease !default;\n\n$carousel-caption-width:             70% !default;\n$carousel-caption-color:             $white !default;\n$carousel-caption-padding-y:         1.25rem !default;\n$carousel-caption-spacer:            1.25rem !default;\n\n$carousel-control-icon-width:        2rem !default;\n\n$carousel-control-prev-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M11.354 1.646a.5.5 0 0 1 0 .708L5.707 8l5.647 5.646a.5.5 0 0 1-.708.708l-6-6a.5.5 0 0 1 0-.708l6-6a.5.5 0 0 1 .708 0z'/></svg>\") !default;\n$carousel-control-next-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M4.646 1.646a.5.5 0 0 1 .708 0l6 6a.5.5 0 0 1 0 .708l-6 6a.5.5 0 0 1-.708-.708L10.293 8 4.646 2.354a.5.5 0 0 1 0-.708z'/></svg>\") !default;\n\n$carousel-transition-duration:       .6s !default;\n$carousel-transition:                transform $carousel-transition-duration ease-in-out !default; // Define transform transition first if using multiple transitions (e.g., `transform 2s ease, opacity .5s ease-out`)\n// scss-docs-end carousel-variables\n\n// scss-docs-start carousel-dark-variables\n$carousel-dark-indicator-active-bg:  $black !default;\n$carousel-dark-caption-color:        $black !default;\n$carousel-dark-control-icon-filter:  invert(1) grayscale(100) !default;\n// scss-docs-end carousel-dark-variables\n\n\n// Spinners\n\n// scss-docs-start spinner-variables\n$spinner-width:           2rem !default;\n$spinner-height:          $spinner-width !default;\n$spinner-vertical-align:  -.125em !default;\n$spinner-border-width:    .25em !default;\n$spinner-animation-speed: .75s !default;\n\n$spinner-width-sm:        1rem !default;\n$spinner-height-sm:       $spinner-width-sm !default;\n$spinner-border-width-sm: .2em !default;\n// scss-docs-end spinner-variables\n\n\n// Close\n\n// scss-docs-start close-variables\n$btn-close-width:            1em !default;\n$btn-close-height:           $btn-close-width !default;\n$btn-close-padding-x:        .25em !default;\n$btn-close-padding-y:        $btn-close-padding-x !default;\n$btn-close-color:            $black !default;\n$btn-close-bg:               url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$btn-close-color}'><path d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/></svg>\") !default;\n$btn-close-focus-shadow:     $focus-ring-box-shadow !default;\n$btn-close-opacity:          .5 !default;\n$btn-close-hover-opacity:    .75 !default;\n$btn-close-focus-opacity:    1 !default;\n$btn-close-disabled-opacity: .25 !default;\n$btn-close-white-filter:     invert(1) grayscale(100%) brightness(200%) !default;\n// scss-docs-end close-variables\n\n\n// Offcanvas\n\n// scss-docs-start offcanvas-variables\n$offcanvas-padding-y:               $modal-inner-padding !default;\n$offcanvas-padding-x:               $modal-inner-padding !default;\n$offcanvas-horizontal-width:        400px !default;\n$offcanvas-vertical-height:         30vh !default;\n$offcanvas-transition-duration:     .3s !default;\n$offcanvas-border-color:            $modal-content-border-color !default;\n$offcanvas-border-width:            $modal-content-border-width !default;\n$offcanvas-title-line-height:       $modal-title-line-height !default;\n$offcanvas-bg-color:                var(--#{$prefix}body-bg) !default;\n$offcanvas-color:                   var(--#{$prefix}body-color) !default;\n$offcanvas-box-shadow:              $modal-content-box-shadow-xs !default;\n$offcanvas-backdrop-bg:             $modal-backdrop-bg !default;\n$offcanvas-backdrop-opacity:        $modal-backdrop-opacity !default;\n// scss-docs-end offcanvas-variables\n\n// Code\n\n$code-font-size:                    $small-font-size !default;\n$code-color:                        $pink !default;\n\n$kbd-padding-y:                     .1875rem !default;\n$kbd-padding-x:                     .375rem !default;\n$kbd-font-size:                     $code-font-size !default;\n$kbd-color:                         var(--#{$prefix}body-bg) !default;\n$kbd-bg:                            var(--#{$prefix}body-color) !default;\n$nested-kbd-font-weight:            null !default; // Deprecated in v5.2.0, removing in v6\n\n$pre-color:                         null !default;\n","// stylelint-disable property-disallowed-list\n// Single side border-radius\n\n// Helper function to replace negative values with 0\n@function valid-radius($radius) {\n  $return: ();\n  @each $value in $radius {\n    @if type-of($value) == number {\n      $return: append($return, max($value, 0));\n    } @else {\n      $return: append($return, $value);\n    }\n  }\n  @return $return;\n}\n\n// scss-docs-start border-radius-mixins\n@mixin border-radius($radius: $border-radius, $fallback-border-radius: false) {\n  @if $enable-rounded {\n    border-radius: valid-radius($radius);\n  }\n  @else if $fallback-border-radius != false {\n    border-radius: $fallback-border-radius;\n  }\n}\n\n@mixin border-top-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-left-radius: valid-radius($radius);\n    border-top-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-end-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-right-radius: valid-radius($radius);\n    border-bottom-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-bottom-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-bottom-right-radius: valid-radius($radius);\n    border-bottom-left-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-start-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-left-radius: valid-radius($radius);\n    border-bottom-left-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-top-start-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-left-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-top-end-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-top-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-bottom-end-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-bottom-right-radius: valid-radius($radius);\n  }\n}\n\n@mixin border-bottom-start-radius($radius: $border-radius) {\n  @if $enable-rounded {\n    border-bottom-left-radius: valid-radius($radius);\n  }\n}\n// scss-docs-end border-radius-mixins\n","//\n// Headings\n//\n.h1 {\n  @extend h1;\n}\n\n.h2 {\n  @extend h2;\n}\n\n.h3 {\n  @extend h3;\n}\n\n.h4 {\n  @extend h4;\n}\n\n.h5 {\n  @extend h5;\n}\n\n.h6 {\n  @extend h6;\n}\n\n\n.lead {\n  @include font-size($lead-font-size);\n  font-weight: $lead-font-weight;\n}\n\n// Type display classes\n@each $display, $font-size in $display-font-sizes {\n  .display-#{$display} {\n    @include font-size($font-size);\n    font-family: $display-font-family;\n    font-style: $display-font-style;\n    font-weight: $display-font-weight;\n    line-height: $display-line-height;\n  }\n}\n\n//\n// Emphasis\n//\n.small {\n  @extend small;\n}\n\n.mark {\n  @extend mark;\n}\n\n//\n// Lists\n//\n\n.list-unstyled {\n  @include list-unstyled();\n}\n\n// Inline turns list items into inline-block\n.list-inline {\n  @include list-unstyled();\n}\n.list-inline-item {\n  display: inline-block;\n\n  &:not(:last-child) {\n    margin-right: $list-inline-padding;\n  }\n}\n\n\n//\n// Misc\n//\n\n// Builds on `abbr`\n.initialism {\n  @include font-size($initialism-font-size);\n  text-transform: uppercase;\n}\n\n// Blockquotes\n.blockquote {\n  margin-bottom: $blockquote-margin-y;\n  @include font-size($blockquote-font-size);\n\n  > :last-child {\n    margin-bottom: 0;\n  }\n}\n\n.blockquote-footer {\n  margin-top: -$blockquote-margin-y;\n  margin-bottom: $blockquote-margin-y;\n  @include font-size($blockquote-footer-font-size);\n  color: $blockquote-footer-color;\n\n  &::before {\n    content: \"\\2014\\00A0\"; // em dash, nbsp\n  }\n}\n","// Lists\n\n// Unstyled keeps list items block level, just removes default browser padding and list-style\n@mixin list-unstyled {\n  padding-left: 0;\n  list-style: none;\n}\n","// Override bootstrap variables\n$spacer: 1rem;\n\n$container-max-widths: (\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1400px,\n);\n\n$grid-breakpoints: (\n  xs: 0,\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1200px,\n);\n\n$dropdown-link-hover-bg: var(--pst-color-surface);\n// --pst-color-surface can also be assigned to the dark variant because it is\n// scoped to different values depending on light/dark theme\n$dropdown-dark-link-hover-bg: var(--pst-color-surface);\n$dropdown-link-active-bg: var(--pst-color-surface);\n$dropdown-dark-link-active-bg: var(--pst-color-surface);\n","// Responsive images (ensure images don't scale beyond their parents)\n//\n// This is purposefully opt-in via an explicit class rather than being the default for all `<img>`s.\n// We previously tried the \"images are responsive by default\" approach in Bootstrap v2,\n// and abandoned it in Bootstrap v3 because it breaks lots of third-party widgets (including Google Maps)\n// which weren't expecting the images within themselves to be involuntarily resized.\n// See also https://github.com/twbs/bootstrap/issues/18178\n.img-fluid {\n  @include img-fluid();\n}\n\n\n// Image thumbnails\n.img-thumbnail {\n  padding: $thumbnail-padding;\n  background-color: $thumbnail-bg;\n  border: $thumbnail-border-width solid $thumbnail-border-color;\n  @include border-radius($thumbnail-border-radius);\n  @include box-shadow($thumbnail-box-shadow);\n\n  // Keep them at most 100% wide\n  @include img-fluid();\n}\n\n//\n// Figures\n//\n\n.figure {\n  // Ensures the caption's text aligns with the image.\n  display: inline-block;\n}\n\n.figure-img {\n  margin-bottom: $spacer * .5;\n  line-height: 1;\n}\n\n.figure-caption {\n  @include font-size($figure-caption-font-size);\n  color: $figure-caption-color;\n}\n","// Image Mixins\n// - Responsive image\n// - Retina image\n\n\n// Responsive image\n//\n// Keep images from scaling beyond the width of their parents.\n\n@mixin img-fluid {\n  // Part 1: Set a maximum relative to the parent\n  max-width: 100%;\n  // Part 2: Override the height to auto, otherwise images will be stretched\n  // when setting a width and height attribute on the img element.\n  height: auto;\n}\n","// Container widths\n//\n// Set the container width, and override it for fixed navbars in media queries.\n\n@if $enable-container-classes {\n  // Single container class with breakpoint max-widths\n  .container,\n  // 100% wide container at all breakpoints\n  .container-fluid {\n    @include make-container();\n  }\n\n  // Responsive containers that are 100% wide until a breakpoint\n  @each $breakpoint, $container-max-width in $container-max-widths {\n    .container-#{$breakpoint} {\n      @extend .container-fluid;\n    }\n\n    @include media-breakpoint-up($breakpoint, $grid-breakpoints) {\n      %responsive-container-#{$breakpoint} {\n        max-width: $container-max-width;\n      }\n\n      // Extend each breakpoint which is smaller or equal to the current breakpoint\n      $extend-breakpoint: true;\n\n      @each $name, $width in $grid-breakpoints {\n        @if ($extend-breakpoint) {\n          .container#{breakpoint-infix($name, $grid-breakpoints)} {\n            @extend %responsive-container-#{$breakpoint};\n          }\n\n          // Once the current breakpoint is reached, stop extending\n          @if ($breakpoint == $name) {\n            $extend-breakpoint: false;\n          }\n        }\n      }\n    }\n  }\n}\n","// Container mixins\n\n@mixin make-container($gutter: $container-padding-x) {\n  --#{$prefix}gutter-x: #{$gutter};\n  --#{$prefix}gutter-y: 0;\n  width: 100%;\n  padding-right: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  padding-left: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  margin-right: auto;\n  margin-left: auto;\n}\n","// Breakpoint viewport sizes and media queries.\n//\n// Breakpoints are defined as a map of (name: minimum width), order from small to large:\n//\n//    (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px)\n//\n// The map defined in the `$grid-breakpoints` global variable is used as the `$breakpoints` argument by default.\n\n// Name of the next breakpoint, or null for the last breakpoint.\n//\n//    >> breakpoint-next(sm)\n//    md\n//    >> breakpoint-next(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    md\n//    >> breakpoint-next(sm, $breakpoint-names: (xs sm md lg xl xxl))\n//    md\n@function breakpoint-next($name, $breakpoints: $grid-breakpoints, $breakpoint-names: map-keys($breakpoints)) {\n  $n: index($breakpoint-names, $name);\n  @if not $n {\n    @error \"breakpoint `#{$name}` not found in `#{$breakpoints}`\";\n  }\n  @return if($n < length($breakpoint-names), nth($breakpoint-names, $n + 1), null);\n}\n\n// Minimum breakpoint width. Null for the smallest (first) breakpoint.\n//\n//    >> breakpoint-min(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    576px\n@function breakpoint-min($name, $breakpoints: $grid-breakpoints) {\n  $min: map-get($breakpoints, $name);\n  @return if($min != 0, $min, null);\n}\n\n// Maximum breakpoint width.\n// The maximum value is reduced by 0.02px to work around the limitations of\n// `min-` and `max-` prefixes and viewports with fractional widths.\n// See https://www.w3.org/TR/mediaqueries-4/#mq-min-max\n// Uses 0.02px rather than 0.01px to work around a current rounding bug in Safari.\n// See https://bugs.webkit.org/show_bug.cgi?id=178261\n//\n//    >> breakpoint-max(md, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    767.98px\n@function breakpoint-max($name, $breakpoints: $grid-breakpoints) {\n  $max: map-get($breakpoints, $name);\n  @return if($max and $max > 0, $max - .02, null);\n}\n\n// Returns a blank string if smallest breakpoint, otherwise returns the name with a dash in front.\n// Useful for making responsive utilities.\n//\n//    >> breakpoint-infix(xs, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"\"  (Returns a blank string)\n//    >> breakpoint-infix(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"-sm\"\n@function breakpoint-infix($name, $breakpoints: $grid-breakpoints) {\n  @return if(breakpoint-min($name, $breakpoints) == null, \"\", \"-#{$name}\");\n}\n\n// Media of at least the minimum breakpoint width. No query for the smallest breakpoint.\n// Makes the @content apply to the given breakpoint and wider.\n@mixin media-breakpoint-up($name, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($name, $breakpoints);\n  @if $min {\n    @media (min-width: $min) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media of at most the maximum breakpoint width. No query for the largest breakpoint.\n// Makes the @content apply to the given breakpoint and narrower.\n@mixin media-breakpoint-down($name, $breakpoints: $grid-breakpoints) {\n  $max: breakpoint-max($name, $breakpoints);\n  @if $max {\n    @media (max-width: $max) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media that spans multiple breakpoint widths.\n// Makes the @content apply between the min and max breakpoints\n@mixin media-breakpoint-between($lower, $upper, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($lower, $breakpoints);\n  $max: breakpoint-max($upper, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($lower, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($upper, $breakpoints) {\n      @content;\n    }\n  }\n}\n\n// Media between the breakpoint's minimum and maximum widths.\n// No minimum for the smallest breakpoint, and no maximum for the largest one.\n// Makes the @content apply only to the given breakpoint, not viewports any wider or narrower.\n@mixin media-breakpoint-only($name, $breakpoints: $grid-breakpoints) {\n  $min:  breakpoint-min($name, $breakpoints);\n  $next: breakpoint-next($name, $breakpoints);\n  $max:  breakpoint-max($next, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($name, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($next, $breakpoints) {\n      @content;\n    }\n  }\n}\n","// Row\n//\n// Rows contain your columns.\n\n:root {\n  @each $name, $value in $grid-breakpoints {\n    --#{$prefix}breakpoint-#{$name}: #{$value};\n  }\n}\n\n@if $enable-grid-classes {\n  .row {\n    @include make-row();\n\n    > * {\n      @include make-col-ready();\n    }\n  }\n}\n\n@if $enable-cssgrid {\n  .grid {\n    display: grid;\n    grid-template-rows: repeat(var(--#{$prefix}rows, 1), 1fr);\n    grid-template-columns: repeat(var(--#{$prefix}columns, #{$grid-columns}), 1fr);\n    gap: var(--#{$prefix}gap, #{$grid-gutter-width});\n\n    @include make-cssgrid();\n  }\n}\n\n\n// Columns\n//\n// Common styles for small and large grid columns\n\n@if $enable-grid-classes {\n  @include make-grid-columns();\n}\n","// Grid system\n//\n// Generate semantic grid columns with these mixins.\n\n@mixin make-row($gutter: $grid-gutter-width) {\n  --#{$prefix}gutter-x: #{$gutter};\n  --#{$prefix}gutter-y: 0;\n  display: flex;\n  flex-wrap: wrap;\n  // TODO: Revisit calc order after https://github.com/react-bootstrap/react-bootstrap/issues/6039 is fixed\n  margin-top: calc(-1 * var(--#{$prefix}gutter-y)); // stylelint-disable-line function-disallowed-list\n  margin-right: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n}\n\n@mixin make-col-ready() {\n  // Add box sizing if only the grid is loaded\n  box-sizing: if(variable-exists(include-column-box-sizing) and $include-column-box-sizing, border-box, null);\n  // Prevent columns from becoming too narrow when at smaller grid tiers by\n  // always setting `width: 100%;`. This works because we set the width\n  // later on to override this initial width.\n  flex-shrink: 0;\n  width: 100%;\n  max-width: 100%; // Prevent `.col-auto`, `.col` (& responsive variants) from breaking out the grid\n  padding-right: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  padding-left: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  margin-top: var(--#{$prefix}gutter-y);\n}\n\n@mixin make-col($size: false, $columns: $grid-columns) {\n  @if $size {\n    flex: 0 0 auto;\n    width: percentage(divide($size, $columns));\n\n  } @else {\n    flex: 1 1 0;\n    max-width: 100%;\n  }\n}\n\n@mixin make-col-auto() {\n  flex: 0 0 auto;\n  width: auto;\n}\n\n@mixin make-col-offset($size, $columns: $grid-columns) {\n  $num: divide($size, $columns);\n  margin-left: if($num == 0, 0, percentage($num));\n}\n\n// Row columns\n//\n// Specify on a parent element(e.g., .row) to force immediate children into NN\n// number of columns. Supports wrapping to new lines, but does not do a Masonry\n// style grid.\n@mixin row-cols($count) {\n  > * {\n    flex: 0 0 auto;\n    width: percentage(divide(1, $count));\n  }\n}\n\n// Framework grid generation\n//\n// Used only by Bootstrap to generate the correct number of grid classes given\n// any value of `$grid-columns`.\n\n@mixin make-grid-columns($columns: $grid-columns, $gutter: $grid-gutter-width, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      // Provide basic `.col-{bp}` classes for equal-width flexbox columns\n      .col#{$infix} {\n        flex: 1 0 0%; // Flexbugs #4: https://github.com/philipwalton/flexbugs#flexbug-4\n      }\n\n      .row-cols#{$infix}-auto > * {\n        @include make-col-auto();\n      }\n\n      @if $grid-row-columns > 0 {\n        @for $i from 1 through $grid-row-columns {\n          .row-cols#{$infix}-#{$i} {\n            @include row-cols($i);\n          }\n        }\n      }\n\n      .col#{$infix}-auto {\n        @include make-col-auto();\n      }\n\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .col#{$infix}-#{$i} {\n            @include make-col($i, $columns);\n          }\n        }\n\n        // `$columns - 1` because offsetting by the width of an entire row isn't possible\n        @for $i from 0 through ($columns - 1) {\n          @if not ($infix == \"\" and $i == 0) { // Avoid emitting useless .offset-0\n            .offset#{$infix}-#{$i} {\n              @include make-col-offset($i, $columns);\n            }\n          }\n        }\n      }\n\n      // Gutters\n      //\n      // Make use of `.g-*`, `.gx-*` or `.gy-*` utilities to change spacing between the columns.\n      @each $key, $value in $gutters {\n        .g#{$infix}-#{$key},\n        .gx#{$infix}-#{$key} {\n          --#{$prefix}gutter-x: #{$value};\n        }\n\n        .g#{$infix}-#{$key},\n        .gy#{$infix}-#{$key} {\n          --#{$prefix}gutter-y: #{$value};\n        }\n      }\n    }\n  }\n}\n\n@mixin make-cssgrid($columns: $grid-columns, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .g-col#{$infix}-#{$i} {\n            grid-column: auto / span $i;\n          }\n        }\n\n        // Start with `1` because `0` is and invalid value.\n        // Ends with `$columns - 1` because offsetting by the width of an entire row isn't possible.\n        @for $i from 1 through ($columns - 1) {\n          .g-start#{$infix}-#{$i} {\n            grid-column-start: $i;\n          }\n        }\n      }\n    }\n  }\n}\n","//\n// Basic Bootstrap table\n//\n\n.table {\n  // Reset needed for nesting tables\n  --#{$prefix}table-color-type: initial;\n  --#{$prefix}table-bg-type: initial;\n  --#{$prefix}table-color-state: initial;\n  --#{$prefix}table-bg-state: initial;\n  // End of reset\n  --#{$prefix}table-color: #{$table-color};\n  --#{$prefix}table-bg: #{$table-bg};\n  --#{$prefix}table-border-color: #{$table-border-color};\n  --#{$prefix}table-accent-bg: #{$table-accent-bg};\n  --#{$prefix}table-striped-color: #{$table-striped-color};\n  --#{$prefix}table-striped-bg: #{$table-striped-bg};\n  --#{$prefix}table-active-color: #{$table-active-color};\n  --#{$prefix}table-active-bg: #{$table-active-bg};\n  --#{$prefix}table-hover-color: #{$table-hover-color};\n  --#{$prefix}table-hover-bg: #{$table-hover-bg};\n\n  width: 100%;\n  margin-bottom: $spacer;\n  vertical-align: $table-cell-vertical-align;\n  border-color: var(--#{$prefix}table-border-color);\n\n  // Target th & td\n  // We need the child combinator to prevent styles leaking to nested tables which doesn't have a `.table` class.\n  // We use the universal selectors here to simplify the selector (else we would need 6 different selectors).\n  // Another advantage is that this generates less code and makes the selector less specific making it easier to override.\n  // stylelint-disable-next-line selector-max-universal\n  > :not(caption) > * > * {\n    padding: $table-cell-padding-y $table-cell-padding-x;\n    // Following the precept of cascades: https://codepen.io/miriamsuzanne/full/vYNgodb\n    color: var(--#{$prefix}table-color-state, var(--#{$prefix}table-color-type, var(--#{$prefix}table-color)));\n    background-color: var(--#{$prefix}table-bg);\n    border-bottom-width: $table-border-width;\n    box-shadow: inset 0 0 0 9999px var(--#{$prefix}table-bg-state, var(--#{$prefix}table-bg-type, var(--#{$prefix}table-accent-bg)));\n  }\n\n  > tbody {\n    vertical-align: inherit;\n  }\n\n  > thead {\n    vertical-align: bottom;\n  }\n}\n\n.table-group-divider {\n  border-top: calc(#{$table-border-width} * 2) solid $table-group-separator-color; // stylelint-disable-line function-disallowed-list\n}\n\n//\n// Change placement of captions with a class\n//\n\n.caption-top {\n  caption-side: top;\n}\n\n\n//\n// Condensed table w/ half padding\n//\n\n.table-sm {\n  // stylelint-disable-next-line selector-max-universal\n  > :not(caption) > * > * {\n    padding: $table-cell-padding-y-sm $table-cell-padding-x-sm;\n  }\n}\n\n\n// Border versions\n//\n// Add or remove borders all around the table and between all the columns.\n//\n// When borders are added on all sides of the cells, the corners can render odd when\n// these borders do not have the same color or if they are semi-transparent.\n// Therefor we add top and border bottoms to the `tr`s and left and right borders\n// to the `td`s or `th`s\n\n.table-bordered {\n  > :not(caption) > * {\n    border-width: $table-border-width 0;\n\n    // stylelint-disable-next-line selector-max-universal\n    > * {\n      border-width: 0 $table-border-width;\n    }\n  }\n}\n\n.table-borderless {\n  // stylelint-disable-next-line selector-max-universal\n  > :not(caption) > * > * {\n    border-bottom-width: 0;\n  }\n\n  > :not(:first-child) {\n    border-top-width: 0;\n  }\n}\n\n// Zebra-striping\n//\n// Default zebra-stripe styles (alternating gray and transparent backgrounds)\n\n// For rows\n.table-striped {\n  > tbody > tr:nth-of-type(#{$table-striped-order}) > * {\n    --#{$prefix}table-color-type: var(--#{$prefix}table-striped-color);\n    --#{$prefix}table-bg-type: var(--#{$prefix}table-striped-bg);\n  }\n}\n\n// For columns\n.table-striped-columns {\n  > :not(caption) > tr > :nth-child(#{$table-striped-columns-order}) {\n    --#{$prefix}table-color-type: var(--#{$prefix}table-striped-color);\n    --#{$prefix}table-bg-type: var(--#{$prefix}table-striped-bg);\n  }\n}\n\n// Active table\n//\n// The `.table-active` class can be added to highlight rows or cells\n\n.table-active {\n  --#{$prefix}table-color-state: var(--#{$prefix}table-active-color);\n  --#{$prefix}table-bg-state: var(--#{$prefix}table-active-bg);\n}\n\n// Hover effect\n//\n// Placed here since it has to come after the potential zebra striping\n\n.table-hover {\n  > tbody > tr:hover > * {\n    --#{$prefix}table-color-state: var(--#{$prefix}table-hover-color);\n    --#{$prefix}table-bg-state: var(--#{$prefix}table-hover-bg);\n  }\n}\n\n\n// Table variants\n//\n// Table variants set the table cell backgrounds, border colors\n// and the colors of the striped, hovered & active tables\n\n@each $color, $value in $table-variants {\n  @include table-variant($color, $value);\n}\n\n// Responsive tables\n//\n// Generate series of `.table-responsive-*` classes for configuring the screen\n// size of where your table will overflow.\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n  @include media-breakpoint-down($breakpoint) {\n    .table-responsive#{$infix} {\n      overflow-x: auto;\n      -webkit-overflow-scrolling: touch;\n    }\n  }\n}\n","// scss-docs-start table-variant\n@mixin table-variant($state, $background) {\n  .table-#{$state} {\n    $color: color-contrast(opaque($body-bg, $background));\n    $hover-bg: mix($color, $background, percentage($table-hover-bg-factor));\n    $striped-bg: mix($color, $background, percentage($table-striped-bg-factor));\n    $active-bg: mix($color, $background, percentage($table-active-bg-factor));\n    $table-border-color: mix($color, $background, percentage($table-border-factor));\n\n    --#{$prefix}table-color: #{$color};\n    --#{$prefix}table-bg: #{$background};\n    --#{$prefix}table-border-color: #{$table-border-color};\n    --#{$prefix}table-striped-bg: #{$striped-bg};\n    --#{$prefix}table-striped-color: #{color-contrast($striped-bg)};\n    --#{$prefix}table-active-bg: #{$active-bg};\n    --#{$prefix}table-active-color: #{color-contrast($active-bg)};\n    --#{$prefix}table-hover-bg: #{$hover-bg};\n    --#{$prefix}table-hover-color: #{color-contrast($hover-bg)};\n\n    color: var(--#{$prefix}table-color);\n    border-color: var(--#{$prefix}table-border-color);\n  }\n}\n// scss-docs-end table-variant\n","//\n// Labels\n//\n\n.form-label {\n  margin-bottom: $form-label-margin-bottom;\n  @include font-size($form-label-font-size);\n  font-style: $form-label-font-style;\n  font-weight: $form-label-font-weight;\n  color: $form-label-color;\n}\n\n// For use with horizontal and inline forms, when you need the label (or legend)\n// text to align with the form controls.\n.col-form-label {\n  padding-top: add($input-padding-y, $input-border-width);\n  padding-bottom: add($input-padding-y, $input-border-width);\n  margin-bottom: 0; // Override the `<legend>` default\n  @include font-size(inherit); // Override the `<legend>` default\n  font-style: $form-label-font-style;\n  font-weight: $form-label-font-weight;\n  line-height: $input-line-height;\n  color: $form-label-color;\n}\n\n.col-form-label-lg {\n  padding-top: add($input-padding-y-lg, $input-border-width);\n  padding-bottom: add($input-padding-y-lg, $input-border-width);\n  @include font-size($input-font-size-lg);\n}\n\n.col-form-label-sm {\n  padding-top: add($input-padding-y-sm, $input-border-width);\n  padding-bottom: add($input-padding-y-sm, $input-border-width);\n  @include font-size($input-font-size-sm);\n}\n","//\n// Form text\n//\n\n.form-text {\n  margin-top: $form-text-margin-top;\n  @include font-size($form-text-font-size);\n  font-style: $form-text-font-style;\n  font-weight: $form-text-font-weight;\n  color: $form-text-color;\n}\n","//\n// General form controls (plus a few specific high-level interventions)\n//\n\n.form-control {\n  display: block;\n  width: 100%;\n  padding: $input-padding-y $input-padding-x;\n  font-family: $input-font-family;\n  @include font-size($input-font-size);\n  font-weight: $input-font-weight;\n  line-height: $input-line-height;\n  color: $input-color;\n  appearance: none; // Fix appearance for date inputs in Safari\n  background-color: $input-bg;\n  background-clip: padding-box;\n  border: $input-border-width solid $input-border-color;\n\n  // Note: This has no effect on <select>s in some browsers, due to the limited stylability of `<select>`s in CSS.\n  @include border-radius($input-border-radius, 0);\n\n  @include box-shadow($input-box-shadow);\n  @include transition($input-transition);\n\n  &[type=\"file\"] {\n    overflow: hidden; // prevent pseudo element button overlap\n\n    &:not(:disabled):not([readonly]) {\n      cursor: pointer;\n    }\n  }\n\n  // Customize the `:focus` state to imitate native WebKit styles.\n  &:focus {\n    color: $input-focus-color;\n    background-color: $input-focus-bg;\n    border-color: $input-focus-border-color;\n    outline: 0;\n    @if $enable-shadows {\n      @include box-shadow($input-box-shadow, $input-focus-box-shadow);\n    } @else {\n      // Avoid using mixin so we can pass custom focus shadow properly\n      box-shadow: $input-focus-box-shadow;\n    }\n  }\n\n  &::-webkit-date-and-time-value {\n    // On Android Chrome, form-control's \"width: 100%\" makes the input width too small\n    // Tested under Android 11 / Chrome 89, Android 12 / Chrome 100, Android 13 / Chrome 109\n    //\n    // On iOS Safari, form-control's \"appearance: none\" + \"width: 100%\" makes the input width too small\n    // Tested under iOS 16.2 / Safari 16.2\n    min-width: 85px; // Seems to be a good minimum safe width\n\n    // Add some height to date inputs on iOS\n    // https://github.com/twbs/bootstrap/issues/23307\n    // TODO: we can remove this workaround once https://bugs.webkit.org/show_bug.cgi?id=198959 is resolved\n    // Multiply line-height by 1em if it has no unit\n    height: if(unit($input-line-height) == \"\", $input-line-height * 1em, $input-line-height);\n\n    // Android Chrome type=\"date\" is taller than the other inputs\n    // because of \"margin: 1px 24px 1px 4px\" inside the shadow DOM\n    // Tested under Android 11 / Chrome 89, Android 12 / Chrome 100, Android 13 / Chrome 109\n    margin: 0;\n  }\n\n  // Prevent excessive date input height in Webkit\n  // https://github.com/twbs/bootstrap/issues/34433\n  &::-webkit-datetime-edit {\n    display: block;\n    padding: 0;\n  }\n\n  // Placeholder\n  &::placeholder {\n    color: $input-placeholder-color;\n    // Override Firefox's unusual default opacity; see https://github.com/twbs/bootstrap/pull/11526.\n    opacity: 1;\n  }\n\n  // Disabled inputs\n  //\n  // HTML5 says that controls under a fieldset > legend:first-child won't be\n  // disabled if the fieldset is disabled. Due to implementation difficulty, we\n  // don't honor that edge case; we style them as disabled anyway.\n  &:disabled {\n    color: $input-disabled-color;\n    background-color: $input-disabled-bg;\n    border-color: $input-disabled-border-color;\n    // iOS fix for unreadable disabled content; see https://github.com/twbs/bootstrap/issues/11655.\n    opacity: 1;\n  }\n\n  // File input buttons theming\n  &::file-selector-button {\n    padding: $input-padding-y $input-padding-x;\n    margin: (-$input-padding-y) (-$input-padding-x);\n    margin-inline-end: $input-padding-x;\n    color: $form-file-button-color;\n    @include gradient-bg($form-file-button-bg);\n    pointer-events: none;\n    border-color: inherit;\n    border-style: solid;\n    border-width: 0;\n    border-inline-end-width: $input-border-width;\n    border-radius: 0; // stylelint-disable-line property-disallowed-list\n    @include transition($btn-transition);\n  }\n\n  &:hover:not(:disabled):not([readonly])::file-selector-button {\n    background-color: $form-file-button-hover-bg;\n  }\n}\n\n// Readonly controls as plain text\n//\n// Apply class to a readonly input to make it appear like regular plain\n// text (without any border, background color, focus indicator)\n\n.form-control-plaintext {\n  display: block;\n  width: 100%;\n  padding: $input-padding-y 0;\n  margin-bottom: 0; // match inputs if this class comes on inputs with default margins\n  line-height: $input-line-height;\n  color: $input-plaintext-color;\n  background-color: transparent;\n  border: solid transparent;\n  border-width: $input-border-width 0;\n\n  &:focus {\n    outline: 0;\n  }\n\n  &.form-control-sm,\n  &.form-control-lg {\n    padding-right: 0;\n    padding-left: 0;\n  }\n}\n\n// Form control sizing\n//\n// Build on `.form-control` with modifier classes to decrease or increase the\n// height and font-size of form controls.\n//\n// Repeated in `_input_group.scss` to avoid Sass extend issues.\n\n.form-control-sm {\n  min-height: $input-height-sm;\n  padding: $input-padding-y-sm $input-padding-x-sm;\n  @include font-size($input-font-size-sm);\n  @include border-radius($input-border-radius-sm);\n\n  &::file-selector-button {\n    padding: $input-padding-y-sm $input-padding-x-sm;\n    margin: (-$input-padding-y-sm) (-$input-padding-x-sm);\n    margin-inline-end: $input-padding-x-sm;\n  }\n}\n\n.form-control-lg {\n  min-height: $input-height-lg;\n  padding: $input-padding-y-lg $input-padding-x-lg;\n  @include font-size($input-font-size-lg);\n  @include border-radius($input-border-radius-lg);\n\n  &::file-selector-button {\n    padding: $input-padding-y-lg $input-padding-x-lg;\n    margin: (-$input-padding-y-lg) (-$input-padding-x-lg);\n    margin-inline-end: $input-padding-x-lg;\n  }\n}\n\n// Make sure textareas don't shrink too much when resized\n// https://github.com/twbs/bootstrap/pull/29124\n// stylelint-disable selector-no-qualifying-type\ntextarea {\n  &.form-control {\n    min-height: $input-height;\n  }\n\n  &.form-control-sm {\n    min-height: $input-height-sm;\n  }\n\n  &.form-control-lg {\n    min-height: $input-height-lg;\n  }\n}\n// stylelint-enable selector-no-qualifying-type\n\n.form-control-color {\n  width: $form-color-width;\n  height: $input-height;\n  padding: $input-padding-y;\n\n  &:not(:disabled):not([readonly]) {\n    cursor: pointer;\n  }\n\n  &::-moz-color-swatch {\n    border: 0 !important; // stylelint-disable-line declaration-no-important\n    @include border-radius($input-border-radius);\n  }\n\n  &::-webkit-color-swatch {\n    border: 0 !important; // stylelint-disable-line declaration-no-important\n    @include border-radius($input-border-radius);\n  }\n\n  &.form-control-sm { height: $input-height-sm; }\n  &.form-control-lg { height: $input-height-lg; }\n}\n","// stylelint-disable property-disallowed-list\n@mixin transition($transition...) {\n  @if length($transition) == 0 {\n    $transition: $transition-base;\n  }\n\n  @if length($transition) > 1 {\n    @each $value in $transition {\n      @if $value == null or $value == none {\n        @warn \"The keyword 'none' or 'null' must be used as a single argument.\";\n      }\n    }\n  }\n\n  @if $enable-transitions {\n    @if nth($transition, 1) != null {\n      transition: $transition;\n    }\n\n    @if $enable-reduced-motion and nth($transition, 1) != null and nth($transition, 1) != none {\n      @media (prefers-reduced-motion: reduce) {\n        transition: none;\n      }\n    }\n  }\n}\n","// Gradients\n\n// scss-docs-start gradient-bg-mixin\n@mixin gradient-bg($color: null) {\n  background-color: $color;\n\n  @if $enable-gradients {\n    background-image: var(--#{$prefix}gradient);\n  }\n}\n// scss-docs-end gradient-bg-mixin\n\n// scss-docs-start gradient-mixins\n// Horizontal gradient, from left to right\n//\n// Creates two color stops, start and end, by specifying a color and position for each color stop.\n@mixin gradient-x($start-color: $gray-700, $end-color: $gray-800, $start-percent: 0%, $end-percent: 100%) {\n  background-image: linear-gradient(to right, $start-color $start-percent, $end-color $end-percent);\n}\n\n// Vertical gradient, from top to bottom\n//\n// Creates two color stops, start and end, by specifying a color and position for each color stop.\n@mixin gradient-y($start-color: $gray-700, $end-color: $gray-800, $start-percent: null, $end-percent: null) {\n  background-image: linear-gradient(to bottom, $start-color $start-percent, $end-color $end-percent);\n}\n\n@mixin gradient-directional($start-color: $gray-700, $end-color: $gray-800, $deg: 45deg) {\n  background-image: linear-gradient($deg, $start-color, $end-color);\n}\n\n@mixin gradient-x-three-colors($start-color: $blue, $mid-color: $purple, $color-stop: 50%, $end-color: $red) {\n  background-image: linear-gradient(to right, $start-color, $mid-color $color-stop, $end-color);\n}\n\n@mixin gradient-y-three-colors($start-color: $blue, $mid-color: $purple, $color-stop: 50%, $end-color: $red) {\n  background-image: linear-gradient($start-color, $mid-color $color-stop, $end-color);\n}\n\n@mixin gradient-radial($inner-color: $gray-700, $outer-color: $gray-800) {\n  background-image: radial-gradient(circle, $inner-color, $outer-color);\n}\n\n@mixin gradient-striped($color: rgba($white, .15), $angle: 45deg) {\n  background-image: linear-gradient($angle, $color 25%, transparent 25%, transparent 50%, $color 50%, $color 75%, transparent 75%, transparent);\n}\n// scss-docs-end gradient-mixins\n","// Select\n//\n// Replaces the browser default select with a custom one, mostly pulled from\n// https://primer.github.io/.\n\n.form-select {\n  --#{$prefix}form-select-bg-img: #{escape-svg($form-select-indicator)};\n\n  display: block;\n  width: 100%;\n  padding: $form-select-padding-y $form-select-indicator-padding $form-select-padding-y $form-select-padding-x;\n  font-family: $form-select-font-family;\n  @include font-size($form-select-font-size);\n  font-weight: $form-select-font-weight;\n  line-height: $form-select-line-height;\n  color: $form-select-color;\n  appearance: none;\n  background-color: $form-select-bg;\n  background-image: var(--#{$prefix}form-select-bg-img), var(--#{$prefix}form-select-bg-icon, none);\n  background-repeat: no-repeat;\n  background-position: $form-select-bg-position;\n  background-size: $form-select-bg-size;\n  border: $form-select-border-width solid $form-select-border-color;\n  @include border-radius($form-select-border-radius, 0);\n  @include box-shadow($form-select-box-shadow);\n  @include transition($form-select-transition);\n\n  &:focus {\n    border-color: $form-select-focus-border-color;\n    outline: 0;\n    @if $enable-shadows {\n      @include box-shadow($form-select-box-shadow, $form-select-focus-box-shadow);\n    } @else {\n      // Avoid using mixin so we can pass custom focus shadow properly\n      box-shadow: $form-select-focus-box-shadow;\n    }\n  }\n\n  &[multiple],\n  &[size]:not([size=\"1\"]) {\n    padding-right: $form-select-padding-x;\n    background-image: none;\n  }\n\n  &:disabled {\n    color: $form-select-disabled-color;\n    background-color: $form-select-disabled-bg;\n    border-color: $form-select-disabled-border-color;\n  }\n\n  // Remove outline from select box in FF\n  &:-moz-focusring {\n    color: transparent;\n    text-shadow: 0 0 0 $form-select-color;\n  }\n}\n\n.form-select-sm {\n  padding-top: $form-select-padding-y-sm;\n  padding-bottom: $form-select-padding-y-sm;\n  padding-left: $form-select-padding-x-sm;\n  @include font-size($form-select-font-size-sm);\n  @include border-radius($form-select-border-radius-sm);\n}\n\n.form-select-lg {\n  padding-top: $form-select-padding-y-lg;\n  padding-bottom: $form-select-padding-y-lg;\n  padding-left: $form-select-padding-x-lg;\n  @include font-size($form-select-font-size-lg);\n  @include border-radius($form-select-border-radius-lg);\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .form-select {\n      --#{$prefix}form-select-bg-img: #{escape-svg($form-select-indicator-dark)};\n    }\n  }\n}\n","//\n// Check/radio\n//\n\n.form-check {\n  display: block;\n  min-height: $form-check-min-height;\n  padding-left: $form-check-padding-start;\n  margin-bottom: $form-check-margin-bottom;\n\n  .form-check-input {\n    float: left;\n    margin-left: $form-check-padding-start * -1;\n  }\n}\n\n.form-check-reverse {\n  padding-right: $form-check-padding-start;\n  padding-left: 0;\n  text-align: right;\n\n  .form-check-input {\n    float: right;\n    margin-right: $form-check-padding-start * -1;\n    margin-left: 0;\n  }\n}\n\n.form-check-input {\n  --#{$prefix}form-check-bg: #{$form-check-input-bg};\n\n  flex-shrink: 0;\n  width: $form-check-input-width;\n  height: $form-check-input-width;\n  margin-top: ($line-height-base - $form-check-input-width) * .5; // line-height minus check height\n  vertical-align: top;\n  appearance: none;\n  background-color: var(--#{$prefix}form-check-bg);\n  background-image: var(--#{$prefix}form-check-bg-image);\n  background-repeat: no-repeat;\n  background-position: center;\n  background-size: contain;\n  border: $form-check-input-border;\n  print-color-adjust: exact; // Keep themed appearance for print\n  @include transition($form-check-transition);\n\n  &[type=\"checkbox\"] {\n    @include border-radius($form-check-input-border-radius);\n  }\n\n  &[type=\"radio\"] {\n    // stylelint-disable-next-line property-disallowed-list\n    border-radius: $form-check-radio-border-radius;\n  }\n\n  &:active {\n    filter: $form-check-input-active-filter;\n  }\n\n  &:focus {\n    border-color: $form-check-input-focus-border;\n    outline: 0;\n    box-shadow: $form-check-input-focus-box-shadow;\n  }\n\n  &:checked {\n    background-color: $form-check-input-checked-bg-color;\n    border-color: $form-check-input-checked-border-color;\n\n    &[type=\"checkbox\"] {\n      @if $enable-gradients {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-checked-bg-image)}, var(--#{$prefix}gradient);\n      } @else {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-checked-bg-image)};\n      }\n    }\n\n    &[type=\"radio\"] {\n      @if $enable-gradients {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-radio-checked-bg-image)}, var(--#{$prefix}gradient);\n      } @else {\n        --#{$prefix}form-check-bg-image: #{escape-svg($form-check-radio-checked-bg-image)};\n      }\n    }\n  }\n\n  &[type=\"checkbox\"]:indeterminate {\n    background-color: $form-check-input-indeterminate-bg-color;\n    border-color: $form-check-input-indeterminate-border-color;\n\n    @if $enable-gradients {\n      --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-indeterminate-bg-image)}, var(--#{$prefix}gradient);\n    } @else {\n      --#{$prefix}form-check-bg-image: #{escape-svg($form-check-input-indeterminate-bg-image)};\n    }\n  }\n\n  &:disabled {\n    pointer-events: none;\n    filter: none;\n    opacity: $form-check-input-disabled-opacity;\n  }\n\n  // Use disabled attribute in addition of :disabled pseudo-class\n  // See: https://github.com/twbs/bootstrap/issues/28247\n  &[disabled],\n  &:disabled {\n    ~ .form-check-label {\n      cursor: default;\n      opacity: $form-check-label-disabled-opacity;\n    }\n  }\n}\n\n.form-check-label {\n  color: $form-check-label-color;\n  cursor: $form-check-label-cursor;\n}\n\n//\n// Switch\n//\n\n.form-switch {\n  padding-left: $form-switch-padding-start;\n\n  .form-check-input {\n    --#{$prefix}form-switch-bg: #{escape-svg($form-switch-bg-image)};\n\n    width: $form-switch-width;\n    margin-left: $form-switch-padding-start * -1;\n    background-image: var(--#{$prefix}form-switch-bg);\n    background-position: left center;\n    @include border-radius($form-switch-border-radius);\n    @include transition($form-switch-transition);\n\n    &:focus {\n      --#{$prefix}form-switch-bg: #{escape-svg($form-switch-focus-bg-image)};\n    }\n\n    &:checked {\n      background-position: $form-switch-checked-bg-position;\n\n      @if $enable-gradients {\n        --#{$prefix}form-switch-bg: #{escape-svg($form-switch-checked-bg-image)}, var(--#{$prefix}gradient);\n      } @else {\n        --#{$prefix}form-switch-bg: #{escape-svg($form-switch-checked-bg-image)};\n      }\n    }\n  }\n\n  &.form-check-reverse {\n    padding-right: $form-switch-padding-start;\n    padding-left: 0;\n\n    .form-check-input {\n      margin-right: $form-switch-padding-start * -1;\n      margin-left: 0;\n    }\n  }\n}\n\n.form-check-inline {\n  display: inline-block;\n  margin-right: $form-check-inline-margin-end;\n}\n\n.btn-check {\n  position: absolute;\n  clip: rect(0, 0, 0, 0);\n  pointer-events: none;\n\n  &[disabled],\n  &:disabled {\n    + .btn {\n      pointer-events: none;\n      filter: none;\n      opacity: $form-check-btn-check-disabled-opacity;\n    }\n  }\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .form-switch .form-check-input:not(:checked):not(:focus) {\n      --#{$prefix}form-switch-bg: #{escape-svg($form-switch-bg-image-dark)};\n    }\n  }\n}\n","// Range\n//\n// Style range inputs the same across browsers. Vendor-specific rules for pseudo\n// elements cannot be mixed. As such, there are no shared styles for focus or\n// active states on prefixed selectors.\n\n.form-range {\n  width: 100%;\n  height: add($form-range-thumb-height, $form-range-thumb-focus-box-shadow-width * 2);\n  padding: 0; // Need to reset padding\n  appearance: none;\n  background-color: transparent;\n\n  &:focus {\n    outline: 0;\n\n    // Pseudo-elements must be split across multiple rulesets to have an effect.\n    // No box-shadow() mixin for focus accessibility.\n    &::-webkit-slider-thumb { box-shadow: $form-range-thumb-focus-box-shadow; }\n    &::-moz-range-thumb     { box-shadow: $form-range-thumb-focus-box-shadow; }\n  }\n\n  &::-moz-focus-outer {\n    border: 0;\n  }\n\n  &::-webkit-slider-thumb {\n    width: $form-range-thumb-width;\n    height: $form-range-thumb-height;\n    margin-top: ($form-range-track-height - $form-range-thumb-height) * .5; // Webkit specific\n    appearance: none;\n    @include gradient-bg($form-range-thumb-bg);\n    border: $form-range-thumb-border;\n    @include border-radius($form-range-thumb-border-radius);\n    @include box-shadow($form-range-thumb-box-shadow);\n    @include transition($form-range-thumb-transition);\n\n    &:active {\n      @include gradient-bg($form-range-thumb-active-bg);\n    }\n  }\n\n  &::-webkit-slider-runnable-track {\n    width: $form-range-track-width;\n    height: $form-range-track-height;\n    color: transparent; // Why?\n    cursor: $form-range-track-cursor;\n    background-color: $form-range-track-bg;\n    border-color: transparent;\n    @include border-radius($form-range-track-border-radius);\n    @include box-shadow($form-range-track-box-shadow);\n  }\n\n  &::-moz-range-thumb {\n    width: $form-range-thumb-width;\n    height: $form-range-thumb-height;\n    appearance: none;\n    @include gradient-bg($form-range-thumb-bg);\n    border: $form-range-thumb-border;\n    @include border-radius($form-range-thumb-border-radius);\n    @include box-shadow($form-range-thumb-box-shadow);\n    @include transition($form-range-thumb-transition);\n\n    &:active {\n      @include gradient-bg($form-range-thumb-active-bg);\n    }\n  }\n\n  &::-moz-range-track {\n    width: $form-range-track-width;\n    height: $form-range-track-height;\n    color: transparent;\n    cursor: $form-range-track-cursor;\n    background-color: $form-range-track-bg;\n    border-color: transparent; // Firefox specific?\n    @include border-radius($form-range-track-border-radius);\n    @include box-shadow($form-range-track-box-shadow);\n  }\n\n  &:disabled {\n    pointer-events: none;\n\n    &::-webkit-slider-thumb {\n      background-color: $form-range-thumb-disabled-bg;\n    }\n\n    &::-moz-range-thumb {\n      background-color: $form-range-thumb-disabled-bg;\n    }\n  }\n}\n",".form-floating {\n  position: relative;\n\n  > .form-control,\n  > .form-control-plaintext,\n  > .form-select {\n    height: $form-floating-height;\n    min-height: $form-floating-height;\n    line-height: $form-floating-line-height;\n  }\n\n  > label {\n    position: absolute;\n    top: 0;\n    left: 0;\n    z-index: 2;\n    height: 100%; // allow textareas\n    padding: $form-floating-padding-y $form-floating-padding-x;\n    overflow: hidden;\n    text-align: start;\n    text-overflow: ellipsis;\n    white-space: nowrap;\n    pointer-events: none;\n    border: $input-border-width solid transparent; // Required for aligning label's text with the input as it affects inner box model\n    transform-origin: 0 0;\n    @include transition($form-floating-transition);\n  }\n\n  > .form-control,\n  > .form-control-plaintext {\n    padding: $form-floating-padding-y $form-floating-padding-x;\n\n    &::placeholder {\n      color: transparent;\n    }\n\n    &:focus,\n    &:not(:placeholder-shown) {\n      padding-top: $form-floating-input-padding-t;\n      padding-bottom: $form-floating-input-padding-b;\n    }\n    // Duplicated because `:-webkit-autofill` invalidates other selectors when grouped\n    &:-webkit-autofill {\n      padding-top: $form-floating-input-padding-t;\n      padding-bottom: $form-floating-input-padding-b;\n    }\n  }\n\n  > .form-select {\n    padding-top: $form-floating-input-padding-t;\n    padding-bottom: $form-floating-input-padding-b;\n  }\n\n  > .form-control:focus,\n  > .form-control:not(:placeholder-shown),\n  > .form-control-plaintext,\n  > .form-select {\n    ~ label {\n      color: rgba(var(--#{$prefix}body-color-rgb), #{$form-floating-label-opacity});\n      transform: $form-floating-label-transform;\n\n      &::after {\n        position: absolute;\n        inset: $form-floating-padding-y ($form-floating-padding-x * .5);\n        z-index: -1;\n        height: $form-floating-label-height;\n        content: \"\";\n        background-color: $input-bg;\n        @include border-radius($input-border-radius);\n      }\n    }\n  }\n  // Duplicated because `:-webkit-autofill` invalidates other selectors when grouped\n  > .form-control:-webkit-autofill {\n    ~ label {\n      color: rgba(var(--#{$prefix}body-color-rgb), #{$form-floating-label-opacity});\n      transform: $form-floating-label-transform;\n    }\n  }\n\n  > .form-control-plaintext {\n    ~ label {\n      border-width: $input-border-width 0; // Required to properly position label text - as explained above\n    }\n  }\n\n  > :disabled ~ label,\n  > .form-control:disabled ~ label { // Required for `.form-control`s because of specificity\n    color: $form-floating-label-disabled-color;\n\n    &::after {\n      background-color: $input-disabled-bg;\n    }\n  }\n}\n","//\n// Base styles\n//\n\n.input-group {\n  position: relative;\n  display: flex;\n  flex-wrap: wrap; // For form validation feedback\n  align-items: stretch;\n  width: 100%;\n\n  > .form-control,\n  > .form-select,\n  > .form-floating {\n    position: relative; // For focus state's z-index\n    flex: 1 1 auto;\n    width: 1%;\n    min-width: 0; // https://stackoverflow.com/questions/36247140/why-dont-flex-items-shrink-past-content-size\n  }\n\n  // Bring the \"active\" form control to the top of surrounding elements\n  > .form-control:focus,\n  > .form-select:focus,\n  > .form-floating:focus-within {\n    z-index: 5;\n  }\n\n  // Ensure buttons are always above inputs for more visually pleasing borders.\n  // This isn't needed for `.input-group-text` since it shares the same border-color\n  // as our inputs.\n  .btn {\n    position: relative;\n    z-index: 2;\n\n    &:focus {\n      z-index: 5;\n    }\n  }\n}\n\n\n// Textual addons\n//\n// Serves as a catch-all element for any text or radio/checkbox input you wish\n// to prepend or append to an input.\n\n.input-group-text {\n  display: flex;\n  align-items: center;\n  padding: $input-group-addon-padding-y $input-group-addon-padding-x;\n  @include font-size($input-font-size); // Match inputs\n  font-weight: $input-group-addon-font-weight;\n  line-height: $input-line-height;\n  color: $input-group-addon-color;\n  text-align: center;\n  white-space: nowrap;\n  background-color: $input-group-addon-bg;\n  border: $input-border-width solid $input-group-addon-border-color;\n  @include border-radius($input-border-radius);\n}\n\n\n// Sizing\n//\n// Remix the default form control sizing classes into new ones for easier\n// manipulation.\n\n.input-group-lg > .form-control,\n.input-group-lg > .form-select,\n.input-group-lg > .input-group-text,\n.input-group-lg > .btn {\n  padding: $input-padding-y-lg $input-padding-x-lg;\n  @include font-size($input-font-size-lg);\n  @include border-radius($input-border-radius-lg);\n}\n\n.input-group-sm > .form-control,\n.input-group-sm > .form-select,\n.input-group-sm > .input-group-text,\n.input-group-sm > .btn {\n  padding: $input-padding-y-sm $input-padding-x-sm;\n  @include font-size($input-font-size-sm);\n  @include border-radius($input-border-radius-sm);\n}\n\n.input-group-lg > .form-select,\n.input-group-sm > .form-select {\n  padding-right: $form-select-padding-x + $form-select-indicator-padding;\n}\n\n\n// Rounded corners\n//\n// These rulesets must come after the sizing ones to properly override sm and lg\n// border-radius values when extending. They're more specific than we'd like\n// with the `.input-group >` part, but without it, we cannot override the sizing.\n\n// stylelint-disable-next-line no-duplicate-selectors\n.input-group {\n  &:not(.has-validation) {\n    > :not(:last-child):not(.dropdown-toggle):not(.dropdown-menu):not(.form-floating),\n    > .dropdown-toggle:nth-last-child(n + 3),\n    > .form-floating:not(:last-child) > .form-control,\n    > .form-floating:not(:last-child) > .form-select {\n      @include border-end-radius(0);\n    }\n  }\n\n  &.has-validation {\n    > :nth-last-child(n + 3):not(.dropdown-toggle):not(.dropdown-menu):not(.form-floating),\n    > .dropdown-toggle:nth-last-child(n + 4),\n    > .form-floating:nth-last-child(n + 3) > .form-control,\n    > .form-floating:nth-last-child(n + 3) > .form-select {\n      @include border-end-radius(0);\n    }\n  }\n\n  $validation-messages: \"\";\n  @each $state in map-keys($form-validation-states) {\n    $validation-messages: $validation-messages + \":not(.\" + unquote($state) + \"-tooltip)\" + \":not(.\" + unquote($state) + \"-feedback)\";\n  }\n\n  > :not(:first-child):not(.dropdown-menu)#{$validation-messages} {\n    margin-left: calc(#{$input-border-width} * -1); // stylelint-disable-line function-disallowed-list\n    @include border-start-radius(0);\n  }\n\n  > .form-floating:not(:first-child) > .form-control,\n  > .form-floating:not(:first-child) > .form-select {\n    @include border-start-radius(0);\n  }\n}\n","// This mixin uses an `if()` technique to be compatible with Dart Sass\n// See https://github.com/sass/sass/issues/1873#issuecomment-152293725 for more details\n\n// scss-docs-start form-validation-mixins\n@mixin form-validation-state-selector($state) {\n  @if ($state == \"valid\" or $state == \"invalid\") {\n    .was-validated #{if(&, \"&\", \"\")}:#{$state},\n    #{if(&, \"&\", \"\")}.is-#{$state} {\n      @content;\n    }\n  } @else {\n    #{if(&, \"&\", \"\")}.is-#{$state} {\n      @content;\n    }\n  }\n}\n\n@mixin form-validation-state(\n  $state,\n  $color,\n  $icon,\n  $tooltip-color: color-contrast($color),\n  $tooltip-bg-color: rgba($color, $form-feedback-tooltip-opacity),\n  $focus-box-shadow: 0 0 $input-btn-focus-blur $input-focus-width rgba($color, $input-btn-focus-color-opacity),\n  $border-color: $color\n) {\n  .#{$state}-feedback {\n    display: none;\n    width: 100%;\n    margin-top: $form-feedback-margin-top;\n    @include font-size($form-feedback-font-size);\n    font-style: $form-feedback-font-style;\n    color: $color;\n  }\n\n  .#{$state}-tooltip {\n    position: absolute;\n    top: 100%;\n    z-index: 5;\n    display: none;\n    max-width: 100%; // Contain to parent when possible\n    padding: $form-feedback-tooltip-padding-y $form-feedback-tooltip-padding-x;\n    margin-top: .1rem;\n    @include font-size($form-feedback-tooltip-font-size);\n    line-height: $form-feedback-tooltip-line-height;\n    color: $tooltip-color;\n    background-color: $tooltip-bg-color;\n    @include border-radius($form-feedback-tooltip-border-radius);\n  }\n\n  @include form-validation-state-selector($state) {\n    ~ .#{$state}-feedback,\n    ~ .#{$state}-tooltip {\n      display: block;\n    }\n  }\n\n  .form-control {\n    @include form-validation-state-selector($state) {\n      border-color: $border-color;\n\n      @if $enable-validation-icons {\n        padding-right: $input-height-inner;\n        background-image: escape-svg($icon);\n        background-repeat: no-repeat;\n        background-position: right $input-height-inner-quarter center;\n        background-size: $input-height-inner-half $input-height-inner-half;\n      }\n\n      &:focus {\n        border-color: $border-color;\n        box-shadow: $focus-box-shadow;\n      }\n    }\n  }\n\n  // stylelint-disable-next-line selector-no-qualifying-type\n  textarea.form-control {\n    @include form-validation-state-selector($state) {\n      @if $enable-validation-icons {\n        padding-right: $input-height-inner;\n        background-position: top $input-height-inner-quarter right $input-height-inner-quarter;\n      }\n    }\n  }\n\n  .form-select {\n    @include form-validation-state-selector($state) {\n      border-color: $border-color;\n\n      @if $enable-validation-icons {\n        &:not([multiple]):not([size]),\n        &:not([multiple])[size=\"1\"] {\n          --#{$prefix}form-select-bg-icon: #{escape-svg($icon)};\n          padding-right: $form-select-feedback-icon-padding-end;\n          background-position: $form-select-bg-position, $form-select-feedback-icon-position;\n          background-size: $form-select-bg-size, $form-select-feedback-icon-size;\n        }\n      }\n\n      &:focus {\n        border-color: $border-color;\n        box-shadow: $focus-box-shadow;\n      }\n    }\n  }\n\n  .form-control-color {\n    @include form-validation-state-selector($state) {\n      @if $enable-validation-icons {\n        width: add($form-color-width, $input-height-inner);\n      }\n    }\n  }\n\n  .form-check-input {\n    @include form-validation-state-selector($state) {\n      border-color: $border-color;\n\n      &:checked {\n        background-color: $color;\n      }\n\n      &:focus {\n        box-shadow: $focus-box-shadow;\n      }\n\n      ~ .form-check-label {\n        color: $color;\n      }\n    }\n  }\n  .form-check-inline .form-check-input {\n    ~ .#{$state}-feedback {\n      margin-left: .5em;\n    }\n  }\n\n  .input-group {\n    > .form-control:not(:focus),\n    > .form-select:not(:focus),\n    > .form-floating:not(:focus-within) {\n      @include form-validation-state-selector($state) {\n        @if $state == \"valid\" {\n          z-index: 3;\n        } @else if $state == \"invalid\" {\n          z-index: 4;\n        }\n      }\n    }\n  }\n}\n// scss-docs-end form-validation-mixins\n","//\n// Base styles\n//\n\n.btn {\n  // scss-docs-start btn-css-vars\n  --#{$prefix}btn-padding-x: #{$btn-padding-x};\n  --#{$prefix}btn-padding-y: #{$btn-padding-y};\n  --#{$prefix}btn-font-family: #{$btn-font-family};\n  @include rfs($btn-font-size, --#{$prefix}btn-font-size);\n  --#{$prefix}btn-font-weight: #{$btn-font-weight};\n  --#{$prefix}btn-line-height: #{$btn-line-height};\n  --#{$prefix}btn-color: #{$btn-color};\n  --#{$prefix}btn-bg: transparent;\n  --#{$prefix}btn-border-width: #{$btn-border-width};\n  --#{$prefix}btn-border-color: transparent;\n  --#{$prefix}btn-border-radius: #{$btn-border-radius};\n  --#{$prefix}btn-hover-border-color: transparent;\n  --#{$prefix}btn-box-shadow: #{$btn-box-shadow};\n  --#{$prefix}btn-disabled-opacity: #{$btn-disabled-opacity};\n  --#{$prefix}btn-focus-box-shadow: 0 0 0 #{$btn-focus-width} rgba(var(--#{$prefix}btn-focus-shadow-rgb), .5);\n  // scss-docs-end btn-css-vars\n\n  display: inline-block;\n  padding: var(--#{$prefix}btn-padding-y) var(--#{$prefix}btn-padding-x);\n  font-family: var(--#{$prefix}btn-font-family);\n  @include font-size(var(--#{$prefix}btn-font-size));\n  font-weight: var(--#{$prefix}btn-font-weight);\n  line-height: var(--#{$prefix}btn-line-height);\n  color: var(--#{$prefix}btn-color);\n  text-align: center;\n  text-decoration: if($link-decoration == none, null, none);\n  white-space: $btn-white-space;\n  vertical-align: middle;\n  cursor: if($enable-button-pointers, pointer, null);\n  user-select: none;\n  border: var(--#{$prefix}btn-border-width) solid var(--#{$prefix}btn-border-color);\n  @include border-radius(var(--#{$prefix}btn-border-radius));\n  @include gradient-bg(var(--#{$prefix}btn-bg));\n  @include box-shadow(var(--#{$prefix}btn-box-shadow));\n  @include transition($btn-transition);\n\n  &:hover {\n    color: var(--#{$prefix}btn-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n    background-color: var(--#{$prefix}btn-hover-bg);\n    border-color: var(--#{$prefix}btn-hover-border-color);\n  }\n\n  .btn-check + &:hover {\n    // override for the checkbox/radio buttons\n    color: var(--#{$prefix}btn-color);\n    background-color: var(--#{$prefix}btn-bg);\n    border-color: var(--#{$prefix}btn-border-color);\n  }\n\n  &:focus-visible {\n    color: var(--#{$prefix}btn-hover-color);\n    @include gradient-bg(var(--#{$prefix}btn-hover-bg));\n    border-color: var(--#{$prefix}btn-hover-border-color);\n    outline: 0;\n    // Avoid using mixin so we can pass custom focus shadow properly\n    @if $enable-shadows {\n      box-shadow: var(--#{$prefix}btn-box-shadow), var(--#{$prefix}btn-focus-box-shadow);\n    } @else {\n      box-shadow: var(--#{$prefix}btn-focus-box-shadow);\n    }\n  }\n\n  .btn-check:focus-visible + & {\n    border-color: var(--#{$prefix}btn-hover-border-color);\n    outline: 0;\n    // Avoid using mixin so we can pass custom focus shadow properly\n    @if $enable-shadows {\n      box-shadow: var(--#{$prefix}btn-box-shadow), var(--#{$prefix}btn-focus-box-shadow);\n    } @else {\n      box-shadow: var(--#{$prefix}btn-focus-box-shadow);\n    }\n  }\n\n  .btn-check:checked + &,\n  :not(.btn-check) + &:active,\n  &:first-child:active,\n  &.active,\n  &.show {\n    color: var(--#{$prefix}btn-active-color);\n    background-color: var(--#{$prefix}btn-active-bg);\n    // Remove CSS gradients if they're enabled\n    background-image: if($enable-gradients, none, null);\n    border-color: var(--#{$prefix}btn-active-border-color);\n    @include box-shadow(var(--#{$prefix}btn-active-shadow));\n\n    &:focus-visible {\n      // Avoid using mixin so we can pass custom focus shadow properly\n      @if $enable-shadows {\n        box-shadow: var(--#{$prefix}btn-active-shadow), var(--#{$prefix}btn-focus-box-shadow);\n      } @else {\n        box-shadow: var(--#{$prefix}btn-focus-box-shadow);\n      }\n    }\n  }\n\n  &:disabled,\n  &.disabled,\n  fieldset:disabled & {\n    color: var(--#{$prefix}btn-disabled-color);\n    pointer-events: none;\n    background-color: var(--#{$prefix}btn-disabled-bg);\n    background-image: if($enable-gradients, none, null);\n    border-color: var(--#{$prefix}btn-disabled-border-color);\n    opacity: var(--#{$prefix}btn-disabled-opacity);\n    @include box-shadow(none);\n  }\n}\n\n\n//\n// Alternate buttons\n//\n\n// scss-docs-start btn-variant-loops\n@each $color, $value in $theme-colors {\n  .btn-#{$color} {\n    @if $color == \"light\" {\n      @include button-variant(\n        $value,\n        $value,\n        $hover-background: shade-color($value, $btn-hover-bg-shade-amount),\n        $hover-border: shade-color($value, $btn-hover-border-shade-amount),\n        $active-background: shade-color($value, $btn-active-bg-shade-amount),\n        $active-border: shade-color($value, $btn-active-border-shade-amount)\n      );\n    } @else if $color == \"dark\" {\n      @include button-variant(\n        $value,\n        $value,\n        $hover-background: tint-color($value, $btn-hover-bg-tint-amount),\n        $hover-border: tint-color($value, $btn-hover-border-tint-amount),\n        $active-background: tint-color($value, $btn-active-bg-tint-amount),\n        $active-border: tint-color($value, $btn-active-border-tint-amount)\n      );\n    } @else {\n      @include button-variant($value, $value);\n    }\n  }\n}\n\n@each $color, $value in $theme-colors {\n  .btn-outline-#{$color} {\n    @include button-outline-variant($value);\n  }\n}\n// scss-docs-end btn-variant-loops\n\n\n//\n// Link buttons\n//\n\n// Make a button look and behave like a link\n.btn-link {\n  --#{$prefix}btn-font-weight: #{$font-weight-normal};\n  --#{$prefix}btn-color: #{$btn-link-color};\n  --#{$prefix}btn-bg: transparent;\n  --#{$prefix}btn-border-color: transparent;\n  --#{$prefix}btn-hover-color: #{$btn-link-hover-color};\n  --#{$prefix}btn-hover-border-color: transparent;\n  --#{$prefix}btn-active-color: #{$btn-link-hover-color};\n  --#{$prefix}btn-active-border-color: transparent;\n  --#{$prefix}btn-disabled-color: #{$btn-link-disabled-color};\n  --#{$prefix}btn-disabled-border-color: transparent;\n  --#{$prefix}btn-box-shadow: 0 0 0 #000; // Can't use `none` as keyword negates all values when used with multiple shadows\n  --#{$prefix}btn-focus-shadow-rgb: #{$btn-link-focus-shadow-rgb};\n\n  text-decoration: $link-decoration;\n  @if $enable-gradients {\n    background-image: none;\n  }\n\n  &:hover,\n  &:focus-visible {\n    text-decoration: $link-hover-decoration;\n  }\n\n  &:focus-visible {\n    color: var(--#{$prefix}btn-color);\n  }\n\n  &:hover {\n    color: var(--#{$prefix}btn-hover-color);\n  }\n\n  // No need for an active state here\n}\n\n\n//\n// Button Sizes\n//\n\n.btn-lg {\n  @include button-size($btn-padding-y-lg, $btn-padding-x-lg, $btn-font-size-lg, $btn-border-radius-lg);\n}\n\n.btn-sm {\n  @include button-size($btn-padding-y-sm, $btn-padding-x-sm, $btn-font-size-sm, $btn-border-radius-sm);\n}\n","// Button variants\n//\n// Easily pump out default styles, as well as :hover, :focus, :active,\n// and disabled options for all buttons\n\n// scss-docs-start btn-variant-mixin\n@mixin button-variant(\n  $background,\n  $border,\n  $color: color-contrast($background),\n  $hover-background: if($color == $color-contrast-light, shade-color($background, $btn-hover-bg-shade-amount), tint-color($background, $btn-hover-bg-tint-amount)),\n  $hover-border: if($color == $color-contrast-light, shade-color($border, $btn-hover-border-shade-amount), tint-color($border, $btn-hover-border-tint-amount)),\n  $hover-color: color-contrast($hover-background),\n  $active-background: if($color == $color-contrast-light, shade-color($background, $btn-active-bg-shade-amount), tint-color($background, $btn-active-bg-tint-amount)),\n  $active-border: if($color == $color-contrast-light, shade-color($border, $btn-active-border-shade-amount), tint-color($border, $btn-active-border-tint-amount)),\n  $active-color: color-contrast($active-background),\n  $disabled-background: $background,\n  $disabled-border: $border,\n  $disabled-color: color-contrast($disabled-background)\n) {\n  --#{$prefix}btn-color: #{$color};\n  --#{$prefix}btn-bg: #{$background};\n  --#{$prefix}btn-border-color: #{$border};\n  --#{$prefix}btn-hover-color: #{$hover-color};\n  --#{$prefix}btn-hover-bg: #{$hover-background};\n  --#{$prefix}btn-hover-border-color: #{$hover-border};\n  --#{$prefix}btn-focus-shadow-rgb: #{to-rgb(mix($color, $border, 15%))};\n  --#{$prefix}btn-active-color: #{$active-color};\n  --#{$prefix}btn-active-bg: #{$active-background};\n  --#{$prefix}btn-active-border-color: #{$active-border};\n  --#{$prefix}btn-active-shadow: #{$btn-active-box-shadow};\n  --#{$prefix}btn-disabled-color: #{$disabled-color};\n  --#{$prefix}btn-disabled-bg: #{$disabled-background};\n  --#{$prefix}btn-disabled-border-color: #{$disabled-border};\n}\n// scss-docs-end btn-variant-mixin\n\n// scss-docs-start btn-outline-variant-mixin\n@mixin button-outline-variant(\n  $color,\n  $color-hover: color-contrast($color),\n  $active-background: $color,\n  $active-border: $color,\n  $active-color: color-contrast($active-background)\n) {\n  --#{$prefix}btn-color: #{$color};\n  --#{$prefix}btn-border-color: #{$color};\n  --#{$prefix}btn-hover-color: #{$color-hover};\n  --#{$prefix}btn-hover-bg: #{$active-background};\n  --#{$prefix}btn-hover-border-color: #{$active-border};\n  --#{$prefix}btn-focus-shadow-rgb: #{to-rgb($color)};\n  --#{$prefix}btn-active-color: #{$active-color};\n  --#{$prefix}btn-active-bg: #{$active-background};\n  --#{$prefix}btn-active-border-color: #{$active-border};\n  --#{$prefix}btn-active-shadow: #{$btn-active-box-shadow};\n  --#{$prefix}btn-disabled-color: #{$color};\n  --#{$prefix}btn-disabled-bg: transparent;\n  --#{$prefix}btn-disabled-border-color: #{$color};\n  --#{$prefix}gradient: none;\n}\n// scss-docs-end btn-outline-variant-mixin\n\n// scss-docs-start btn-size-mixin\n@mixin button-size($padding-y, $padding-x, $font-size, $border-radius) {\n  --#{$prefix}btn-padding-y: #{$padding-y};\n  --#{$prefix}btn-padding-x: #{$padding-x};\n  @include rfs($font-size, --#{$prefix}btn-font-size);\n  --#{$prefix}btn-border-radius: #{$border-radius};\n}\n// scss-docs-end btn-size-mixin\n",".fade {\n  @include transition($transition-fade);\n\n  &:not(.show) {\n    opacity: 0;\n  }\n}\n\n// scss-docs-start collapse-classes\n.collapse {\n  &:not(.show) {\n    display: none;\n  }\n}\n\n.collapsing {\n  height: 0;\n  overflow: hidden;\n  @include transition($transition-collapse);\n\n  &.collapse-horizontal {\n    width: 0;\n    height: auto;\n    @include transition($transition-collapse-width);\n  }\n}\n// scss-docs-end collapse-classes\n","// The dropdown wrapper (`<div>`)\n.dropup,\n.dropend,\n.dropdown,\n.dropstart,\n.dropup-center,\n.dropdown-center {\n  position: relative;\n}\n\n.dropdown-toggle {\n  white-space: nowrap;\n\n  // Generate the caret automatically\n  @include caret();\n}\n\n// The dropdown menu\n.dropdown-menu {\n  // scss-docs-start dropdown-css-vars\n  --#{$prefix}dropdown-zindex: #{$zindex-dropdown};\n  --#{$prefix}dropdown-min-width: #{$dropdown-min-width};\n  --#{$prefix}dropdown-padding-x: #{$dropdown-padding-x};\n  --#{$prefix}dropdown-padding-y: #{$dropdown-padding-y};\n  --#{$prefix}dropdown-spacer: #{$dropdown-spacer};\n  @include rfs($dropdown-font-size, --#{$prefix}dropdown-font-size);\n  --#{$prefix}dropdown-color: #{$dropdown-color};\n  --#{$prefix}dropdown-bg: #{$dropdown-bg};\n  --#{$prefix}dropdown-border-color: #{$dropdown-border-color};\n  --#{$prefix}dropdown-border-radius: #{$dropdown-border-radius};\n  --#{$prefix}dropdown-border-width: #{$dropdown-border-width};\n  --#{$prefix}dropdown-inner-border-radius: #{$dropdown-inner-border-radius};\n  --#{$prefix}dropdown-divider-bg: #{$dropdown-divider-bg};\n  --#{$prefix}dropdown-divider-margin-y: #{$dropdown-divider-margin-y};\n  --#{$prefix}dropdown-box-shadow: #{$dropdown-box-shadow};\n  --#{$prefix}dropdown-link-color: #{$dropdown-link-color};\n  --#{$prefix}dropdown-link-hover-color: #{$dropdown-link-hover-color};\n  --#{$prefix}dropdown-link-hover-bg: #{$dropdown-link-hover-bg};\n  --#{$prefix}dropdown-link-active-color: #{$dropdown-link-active-color};\n  --#{$prefix}dropdown-link-active-bg: #{$dropdown-link-active-bg};\n  --#{$prefix}dropdown-link-disabled-color: #{$dropdown-link-disabled-color};\n  --#{$prefix}dropdown-item-padding-x: #{$dropdown-item-padding-x};\n  --#{$prefix}dropdown-item-padding-y: #{$dropdown-item-padding-y};\n  --#{$prefix}dropdown-header-color: #{$dropdown-header-color};\n  --#{$prefix}dropdown-header-padding-x: #{$dropdown-header-padding-x};\n  --#{$prefix}dropdown-header-padding-y: #{$dropdown-header-padding-y};\n  // scss-docs-end dropdown-css-vars\n\n  position: absolute;\n  z-index: var(--#{$prefix}dropdown-zindex);\n  display: none; // none by default, but block on \"open\" of the menu\n  min-width: var(--#{$prefix}dropdown-min-width);\n  padding: var(--#{$prefix}dropdown-padding-y) var(--#{$prefix}dropdown-padding-x);\n  margin: 0; // Override default margin of ul\n  @include font-size(var(--#{$prefix}dropdown-font-size));\n  color: var(--#{$prefix}dropdown-color);\n  text-align: left; // Ensures proper alignment if parent has it changed (e.g., modal footer)\n  list-style: none;\n  background-color: var(--#{$prefix}dropdown-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}dropdown-border-width) solid var(--#{$prefix}dropdown-border-color);\n  @include border-radius(var(--#{$prefix}dropdown-border-radius));\n  @include box-shadow(var(--#{$prefix}dropdown-box-shadow));\n\n  &[data-bs-popper] {\n    top: 100%;\n    left: 0;\n    margin-top: var(--#{$prefix}dropdown-spacer);\n  }\n\n  @if $dropdown-padding-y == 0 {\n    > .dropdown-item:first-child,\n    > li:first-child .dropdown-item {\n      @include border-top-radius(var(--#{$prefix}dropdown-inner-border-radius));\n    }\n    > .dropdown-item:last-child,\n    > li:last-child .dropdown-item {\n      @include border-bottom-radius(var(--#{$prefix}dropdown-inner-border-radius));\n    }\n\n  }\n}\n\n// scss-docs-start responsive-breakpoints\n// We deliberately hardcode the `bs-` prefix because we check\n// this custom property in JS to determine Popper's positioning\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    .dropdown-menu#{$infix}-start {\n      --bs-position: start;\n\n      &[data-bs-popper] {\n        right: auto;\n        left: 0;\n      }\n    }\n\n    .dropdown-menu#{$infix}-end {\n      --bs-position: end;\n\n      &[data-bs-popper] {\n        right: 0;\n        left: auto;\n      }\n    }\n  }\n}\n// scss-docs-end responsive-breakpoints\n\n// Allow for dropdowns to go bottom up (aka, dropup-menu)\n// Just add .dropup after the standard .dropdown class and you're set.\n.dropup {\n  .dropdown-menu[data-bs-popper] {\n    top: auto;\n    bottom: 100%;\n    margin-top: 0;\n    margin-bottom: var(--#{$prefix}dropdown-spacer);\n  }\n\n  .dropdown-toggle {\n    @include caret(up);\n  }\n}\n\n.dropend {\n  .dropdown-menu[data-bs-popper] {\n    top: 0;\n    right: auto;\n    left: 100%;\n    margin-top: 0;\n    margin-left: var(--#{$prefix}dropdown-spacer);\n  }\n\n  .dropdown-toggle {\n    @include caret(end);\n    &::after {\n      vertical-align: 0;\n    }\n  }\n}\n\n.dropstart {\n  .dropdown-menu[data-bs-popper] {\n    top: 0;\n    right: 100%;\n    left: auto;\n    margin-top: 0;\n    margin-right: var(--#{$prefix}dropdown-spacer);\n  }\n\n  .dropdown-toggle {\n    @include caret(start);\n    &::before {\n      vertical-align: 0;\n    }\n  }\n}\n\n\n// Dividers (basically an `<hr>`) within the dropdown\n.dropdown-divider {\n  height: 0;\n  margin: var(--#{$prefix}dropdown-divider-margin-y) 0;\n  overflow: hidden;\n  border-top: 1px solid var(--#{$prefix}dropdown-divider-bg);\n  opacity: 1; // Revisit in v6 to de-dupe styles that conflict with <hr> element\n}\n\n// Links, buttons, and more within the dropdown menu\n//\n// `<button>`-specific styles are denoted with `// For <button>s`\n.dropdown-item {\n  display: block;\n  width: 100%; // For `<button>`s\n  padding: var(--#{$prefix}dropdown-item-padding-y) var(--#{$prefix}dropdown-item-padding-x);\n  clear: both;\n  font-weight: $font-weight-normal;\n  color: var(--#{$prefix}dropdown-link-color);\n  text-align: inherit; // For `<button>`s\n  text-decoration: if($link-decoration == none, null, none);\n  white-space: nowrap; // prevent links from randomly breaking onto new lines\n  background-color: transparent; // For `<button>`s\n  border: 0; // For `<button>`s\n  @include border-radius(var(--#{$prefix}dropdown-item-border-radius, 0));\n\n  &:hover,\n  &:focus {\n    color: var(--#{$prefix}dropdown-link-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n    @include gradient-bg(var(--#{$prefix}dropdown-link-hover-bg));\n  }\n\n  &.active,\n  &:active {\n    color: var(--#{$prefix}dropdown-link-active-color);\n    text-decoration: none;\n    @include gradient-bg(var(--#{$prefix}dropdown-link-active-bg));\n  }\n\n  &.disabled,\n  &:disabled {\n    color: var(--#{$prefix}dropdown-link-disabled-color);\n    pointer-events: none;\n    background-color: transparent;\n    // Remove CSS gradients if they're enabled\n    background-image: if($enable-gradients, none, null);\n  }\n}\n\n.dropdown-menu.show {\n  display: block;\n}\n\n// Dropdown section headers\n.dropdown-header {\n  display: block;\n  padding: var(--#{$prefix}dropdown-header-padding-y) var(--#{$prefix}dropdown-header-padding-x);\n  margin-bottom: 0; // for use with heading elements\n  @include font-size($font-size-sm);\n  color: var(--#{$prefix}dropdown-header-color);\n  white-space: nowrap; // as with > li > a\n}\n\n// Dropdown text\n.dropdown-item-text {\n  display: block;\n  padding: var(--#{$prefix}dropdown-item-padding-y) var(--#{$prefix}dropdown-item-padding-x);\n  color: var(--#{$prefix}dropdown-link-color);\n}\n\n// Dark dropdowns\n.dropdown-menu-dark {\n  // scss-docs-start dropdown-dark-css-vars\n  --#{$prefix}dropdown-color: #{$dropdown-dark-color};\n  --#{$prefix}dropdown-bg: #{$dropdown-dark-bg};\n  --#{$prefix}dropdown-border-color: #{$dropdown-dark-border-color};\n  --#{$prefix}dropdown-box-shadow: #{$dropdown-dark-box-shadow};\n  --#{$prefix}dropdown-link-color: #{$dropdown-dark-link-color};\n  --#{$prefix}dropdown-link-hover-color: #{$dropdown-dark-link-hover-color};\n  --#{$prefix}dropdown-divider-bg: #{$dropdown-dark-divider-bg};\n  --#{$prefix}dropdown-link-hover-bg: #{$dropdown-dark-link-hover-bg};\n  --#{$prefix}dropdown-link-active-color: #{$dropdown-dark-link-active-color};\n  --#{$prefix}dropdown-link-active-bg: #{$dropdown-dark-link-active-bg};\n  --#{$prefix}dropdown-link-disabled-color: #{$dropdown-dark-link-disabled-color};\n  --#{$prefix}dropdown-header-color: #{$dropdown-dark-header-color};\n  // scss-docs-end dropdown-dark-css-vars\n}\n","// scss-docs-start caret-mixins\n@mixin caret-down($width: $caret-width) {\n  border-top: $width solid;\n  border-right: $width solid transparent;\n  border-bottom: 0;\n  border-left: $width solid transparent;\n}\n\n@mixin caret-up($width: $caret-width) {\n  border-top: 0;\n  border-right: $width solid transparent;\n  border-bottom: $width solid;\n  border-left: $width solid transparent;\n}\n\n@mixin caret-end($width: $caret-width) {\n  border-top: $width solid transparent;\n  border-right: 0;\n  border-bottom: $width solid transparent;\n  border-left: $width solid;\n}\n\n@mixin caret-start($width: $caret-width) {\n  border-top: $width solid transparent;\n  border-right: $width solid;\n  border-bottom: $width solid transparent;\n}\n\n@mixin caret(\n  $direction: down,\n  $width: $caret-width,\n  $spacing: $caret-spacing,\n  $vertical-align: $caret-vertical-align\n) {\n  @if $enable-caret {\n    &::after {\n      display: inline-block;\n      margin-left: $spacing;\n      vertical-align: $vertical-align;\n      content: \"\";\n      @if $direction == down {\n        @include caret-down($width);\n      } @else if $direction == up {\n        @include caret-up($width);\n      } @else if $direction == end {\n        @include caret-end($width);\n      }\n    }\n\n    @if $direction == start {\n      &::after {\n        display: none;\n      }\n\n      &::before {\n        display: inline-block;\n        margin-right: $spacing;\n        vertical-align: $vertical-align;\n        content: \"\";\n        @include caret-start($width);\n      }\n    }\n\n    &:empty::after {\n      margin-left: 0;\n    }\n  }\n}\n// scss-docs-end caret-mixins\n","// Make the div behave like a button\n.btn-group,\n.btn-group-vertical {\n  position: relative;\n  display: inline-flex;\n  vertical-align: middle; // match .btn alignment given font-size hack above\n\n  > .btn {\n    position: relative;\n    flex: 1 1 auto;\n  }\n\n  // Bring the hover, focused, and \"active\" buttons to the front to overlay\n  // the borders properly\n  > .btn-check:checked + .btn,\n  > .btn-check:focus + .btn,\n  > .btn:hover,\n  > .btn:focus,\n  > .btn:active,\n  > .btn.active {\n    z-index: 1;\n  }\n}\n\n// Optional: Group multiple button groups together for a toolbar\n.btn-toolbar {\n  display: flex;\n  flex-wrap: wrap;\n  justify-content: flex-start;\n\n  .input-group {\n    width: auto;\n  }\n}\n\n.btn-group {\n  @include border-radius($btn-border-radius);\n\n  // Prevent double borders when buttons are next to each other\n  > :not(.btn-check:first-child) + .btn,\n  > .btn-group:not(:first-child) {\n    margin-left: calc(#{$btn-border-width} * -1); // stylelint-disable-line function-disallowed-list\n  }\n\n  // Reset rounded corners\n  > .btn:not(:last-child):not(.dropdown-toggle),\n  > .btn.dropdown-toggle-split:first-child,\n  > .btn-group:not(:last-child) > .btn {\n    @include border-end-radius(0);\n  }\n\n  // The left radius should be 0 if the button is:\n  // - the \"third or more\" child\n  // - the second child and the previous element isn't `.btn-check` (making it the first child visually)\n  // - part of a btn-group which isn't the first child\n  > .btn:nth-child(n + 3),\n  > :not(.btn-check) + .btn,\n  > .btn-group:not(:first-child) > .btn {\n    @include border-start-radius(0);\n  }\n}\n\n// Sizing\n//\n// Remix the default button sizing classes into new ones for easier manipulation.\n\n.btn-group-sm > .btn { @extend .btn-sm; }\n.btn-group-lg > .btn { @extend .btn-lg; }\n\n\n//\n// Split button dropdowns\n//\n\n.dropdown-toggle-split {\n  padding-right: $btn-padding-x * .75;\n  padding-left: $btn-padding-x * .75;\n\n  &::after,\n  .dropup &::after,\n  .dropend &::after {\n    margin-left: 0;\n  }\n\n  .dropstart &::before {\n    margin-right: 0;\n  }\n}\n\n.btn-sm + .dropdown-toggle-split {\n  padding-right: $btn-padding-x-sm * .75;\n  padding-left: $btn-padding-x-sm * .75;\n}\n\n.btn-lg + .dropdown-toggle-split {\n  padding-right: $btn-padding-x-lg * .75;\n  padding-left: $btn-padding-x-lg * .75;\n}\n\n\n// The clickable button for toggling the menu\n// Set the same inset shadow as the :active state\n.btn-group.show .dropdown-toggle {\n  @include box-shadow($btn-active-box-shadow);\n\n  // Show no shadow for `.btn-link` since it has no other button styles.\n  &.btn-link {\n    @include box-shadow(none);\n  }\n}\n\n\n//\n// Vertical button groups\n//\n\n.btn-group-vertical {\n  flex-direction: column;\n  align-items: flex-start;\n  justify-content: center;\n\n  > .btn,\n  > .btn-group {\n    width: 100%;\n  }\n\n  > .btn:not(:first-child),\n  > .btn-group:not(:first-child) {\n    margin-top: calc(#{$btn-border-width} * -1); // stylelint-disable-line function-disallowed-list\n  }\n\n  // Reset rounded corners\n  > .btn:not(:last-child):not(.dropdown-toggle),\n  > .btn-group:not(:last-child) > .btn {\n    @include border-bottom-radius(0);\n  }\n\n  > .btn ~ .btn,\n  > .btn-group:not(:first-child) > .btn {\n    @include border-top-radius(0);\n  }\n}\n","// Base class\n//\n// Kickstart any navigation component with a set of style resets. Works with\n// `<nav>`s, `<ul>`s or `<ol>`s.\n\n.nav {\n  // scss-docs-start nav-css-vars\n  --#{$prefix}nav-link-padding-x: #{$nav-link-padding-x};\n  --#{$prefix}nav-link-padding-y: #{$nav-link-padding-y};\n  @include rfs($nav-link-font-size, --#{$prefix}nav-link-font-size);\n  --#{$prefix}nav-link-font-weight: #{$nav-link-font-weight};\n  --#{$prefix}nav-link-color: #{$nav-link-color};\n  --#{$prefix}nav-link-hover-color: #{$nav-link-hover-color};\n  --#{$prefix}nav-link-disabled-color: #{$nav-link-disabled-color};\n  // scss-docs-end nav-css-vars\n\n  display: flex;\n  flex-wrap: wrap;\n  padding-left: 0;\n  margin-bottom: 0;\n  list-style: none;\n}\n\n.nav-link {\n  display: block;\n  padding: var(--#{$prefix}nav-link-padding-y) var(--#{$prefix}nav-link-padding-x);\n  @include font-size(var(--#{$prefix}nav-link-font-size));\n  font-weight: var(--#{$prefix}nav-link-font-weight);\n  color: var(--#{$prefix}nav-link-color);\n  text-decoration: if($link-decoration == none, null, none);\n  background: none;\n  border: 0;\n  @include transition($nav-link-transition);\n\n  &:hover,\n  &:focus {\n    color: var(--#{$prefix}nav-link-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n  }\n\n  &:focus-visible {\n    outline: 0;\n    box-shadow: $nav-link-focus-box-shadow;\n  }\n\n  // Disabled state lightens text\n  &.disabled,\n  &:disabled {\n    color: var(--#{$prefix}nav-link-disabled-color);\n    pointer-events: none;\n    cursor: default;\n  }\n}\n\n//\n// Tabs\n//\n\n.nav-tabs {\n  // scss-docs-start nav-tabs-css-vars\n  --#{$prefix}nav-tabs-border-width: #{$nav-tabs-border-width};\n  --#{$prefix}nav-tabs-border-color: #{$nav-tabs-border-color};\n  --#{$prefix}nav-tabs-border-radius: #{$nav-tabs-border-radius};\n  --#{$prefix}nav-tabs-link-hover-border-color: #{$nav-tabs-link-hover-border-color};\n  --#{$prefix}nav-tabs-link-active-color: #{$nav-tabs-link-active-color};\n  --#{$prefix}nav-tabs-link-active-bg: #{$nav-tabs-link-active-bg};\n  --#{$prefix}nav-tabs-link-active-border-color: #{$nav-tabs-link-active-border-color};\n  // scss-docs-end nav-tabs-css-vars\n\n  border-bottom: var(--#{$prefix}nav-tabs-border-width) solid var(--#{$prefix}nav-tabs-border-color);\n\n  .nav-link {\n    margin-bottom: calc(-1 * var(--#{$prefix}nav-tabs-border-width)); // stylelint-disable-line function-disallowed-list\n    border: var(--#{$prefix}nav-tabs-border-width) solid transparent;\n    @include border-top-radius(var(--#{$prefix}nav-tabs-border-radius));\n\n    &:hover,\n    &:focus {\n      // Prevents active .nav-link tab overlapping focus outline of previous/next .nav-link\n      isolation: isolate;\n      border-color: var(--#{$prefix}nav-tabs-link-hover-border-color);\n    }\n  }\n\n  .nav-link.active,\n  .nav-item.show .nav-link {\n    color: var(--#{$prefix}nav-tabs-link-active-color);\n    background-color: var(--#{$prefix}nav-tabs-link-active-bg);\n    border-color: var(--#{$prefix}nav-tabs-link-active-border-color);\n  }\n\n  .dropdown-menu {\n    // Make dropdown border overlap tab border\n    margin-top: calc(-1 * var(--#{$prefix}nav-tabs-border-width)); // stylelint-disable-line function-disallowed-list\n    // Remove the top rounded corners here since there is a hard edge above the menu\n    @include border-top-radius(0);\n  }\n}\n\n\n//\n// Pills\n//\n\n.nav-pills {\n  // scss-docs-start nav-pills-css-vars\n  --#{$prefix}nav-pills-border-radius: #{$nav-pills-border-radius};\n  --#{$prefix}nav-pills-link-active-color: #{$nav-pills-link-active-color};\n  --#{$prefix}nav-pills-link-active-bg: #{$nav-pills-link-active-bg};\n  // scss-docs-end nav-pills-css-vars\n\n  .nav-link {\n    @include border-radius(var(--#{$prefix}nav-pills-border-radius));\n  }\n\n  .nav-link.active,\n  .show > .nav-link {\n    color: var(--#{$prefix}nav-pills-link-active-color);\n    @include gradient-bg(var(--#{$prefix}nav-pills-link-active-bg));\n  }\n}\n\n\n//\n// Underline\n//\n\n.nav-underline {\n  // scss-docs-start nav-underline-css-vars\n  --#{$prefix}nav-underline-gap: #{$nav-underline-gap};\n  --#{$prefix}nav-underline-border-width: #{$nav-underline-border-width};\n  --#{$prefix}nav-underline-link-active-color: #{$nav-underline-link-active-color};\n  // scss-docs-end nav-underline-css-vars\n\n  gap: var(--#{$prefix}nav-underline-gap);\n\n  .nav-link {\n    padding-right: 0;\n    padding-left: 0;\n    border-bottom: var(--#{$prefix}nav-underline-border-width) solid transparent;\n\n    &:hover,\n    &:focus {\n      border-bottom-color: currentcolor;\n    }\n  }\n\n  .nav-link.active,\n  .show > .nav-link {\n    font-weight: $font-weight-bold;\n    color: var(--#{$prefix}nav-underline-link-active-color);\n    border-bottom-color: currentcolor;\n  }\n}\n\n\n//\n// Justified variants\n//\n\n.nav-fill {\n  > .nav-link,\n  .nav-item {\n    flex: 1 1 auto;\n    text-align: center;\n  }\n}\n\n.nav-justified {\n  > .nav-link,\n  .nav-item {\n    flex-basis: 0;\n    flex-grow: 1;\n    text-align: center;\n  }\n}\n\n.nav-fill,\n.nav-justified {\n  .nav-item .nav-link {\n    width: 100%; // Make sure button will grow\n  }\n}\n\n\n// Tabbable tabs\n//\n// Hide tabbable panes to start, show them when `.active`\n\n.tab-content {\n  > .tab-pane {\n    display: none;\n  }\n  > .active {\n    display: block;\n  }\n}\n","// Navbar\n//\n// Provide a static navbar from which we expand to create full-width, fixed, and\n// other navbar variations.\n\n.navbar {\n  // scss-docs-start navbar-css-vars\n  --#{$prefix}navbar-padding-x: #{if($navbar-padding-x == null, 0, $navbar-padding-x)};\n  --#{$prefix}navbar-padding-y: #{$navbar-padding-y};\n  --#{$prefix}navbar-color: #{$navbar-light-color};\n  --#{$prefix}navbar-hover-color: #{$navbar-light-hover-color};\n  --#{$prefix}navbar-disabled-color: #{$navbar-light-disabled-color};\n  --#{$prefix}navbar-active-color: #{$navbar-light-active-color};\n  --#{$prefix}navbar-brand-padding-y: #{$navbar-brand-padding-y};\n  --#{$prefix}navbar-brand-margin-end: #{$navbar-brand-margin-end};\n  --#{$prefix}navbar-brand-font-size: #{$navbar-brand-font-size};\n  --#{$prefix}navbar-brand-color: #{$navbar-light-brand-color};\n  --#{$prefix}navbar-brand-hover-color: #{$navbar-light-brand-hover-color};\n  --#{$prefix}navbar-nav-link-padding-x: #{$navbar-nav-link-padding-x};\n  --#{$prefix}navbar-toggler-padding-y: #{$navbar-toggler-padding-y};\n  --#{$prefix}navbar-toggler-padding-x: #{$navbar-toggler-padding-x};\n  --#{$prefix}navbar-toggler-font-size: #{$navbar-toggler-font-size};\n  --#{$prefix}navbar-toggler-icon-bg: #{escape-svg($navbar-light-toggler-icon-bg)};\n  --#{$prefix}navbar-toggler-border-color: #{$navbar-light-toggler-border-color};\n  --#{$prefix}navbar-toggler-border-radius: #{$navbar-toggler-border-radius};\n  --#{$prefix}navbar-toggler-focus-width: #{$navbar-toggler-focus-width};\n  --#{$prefix}navbar-toggler-transition: #{$navbar-toggler-transition};\n  // scss-docs-end navbar-css-vars\n\n  position: relative;\n  display: flex;\n  flex-wrap: wrap; // allow us to do the line break for collapsing content\n  align-items: center;\n  justify-content: space-between; // space out brand from logo\n  padding: var(--#{$prefix}navbar-padding-y) var(--#{$prefix}navbar-padding-x);\n  @include gradient-bg();\n\n  // Because flex properties aren't inherited, we need to redeclare these first\n  // few properties so that content nested within behave properly.\n  // The `flex-wrap` property is inherited to simplify the expanded navbars\n  %container-flex-properties {\n    display: flex;\n    flex-wrap: inherit;\n    align-items: center;\n    justify-content: space-between;\n  }\n\n  > .container,\n  > .container-fluid {\n    @extend %container-flex-properties;\n  }\n\n  @each $breakpoint, $container-max-width in $container-max-widths {\n    > .container#{breakpoint-infix($breakpoint, $container-max-widths)} {\n      @extend %container-flex-properties;\n    }\n  }\n}\n\n\n// Navbar brand\n//\n// Used for brand, project, or site names.\n\n.navbar-brand {\n  padding-top: var(--#{$prefix}navbar-brand-padding-y);\n  padding-bottom: var(--#{$prefix}navbar-brand-padding-y);\n  margin-right: var(--#{$prefix}navbar-brand-margin-end);\n  @include font-size(var(--#{$prefix}navbar-brand-font-size));\n  color: var(--#{$prefix}navbar-brand-color);\n  text-decoration: if($link-decoration == none, null, none);\n  white-space: nowrap;\n\n  &:hover,\n  &:focus {\n    color: var(--#{$prefix}navbar-brand-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n  }\n}\n\n\n// Navbar nav\n//\n// Custom navbar navigation (doesn't require `.nav`, but does make use of `.nav-link`).\n\n.navbar-nav {\n  // scss-docs-start navbar-nav-css-vars\n  --#{$prefix}nav-link-padding-x: 0;\n  --#{$prefix}nav-link-padding-y: #{$nav-link-padding-y};\n  @include rfs($nav-link-font-size, --#{$prefix}nav-link-font-size);\n  --#{$prefix}nav-link-font-weight: #{$nav-link-font-weight};\n  --#{$prefix}nav-link-color: var(--#{$prefix}navbar-color);\n  --#{$prefix}nav-link-hover-color: var(--#{$prefix}navbar-hover-color);\n  --#{$prefix}nav-link-disabled-color: var(--#{$prefix}navbar-disabled-color);\n  // scss-docs-end navbar-nav-css-vars\n\n  display: flex;\n  flex-direction: column; // cannot use `inherit` to get the `.navbar`s value\n  padding-left: 0;\n  margin-bottom: 0;\n  list-style: none;\n\n  .nav-link {\n    &.active,\n    &.show {\n      color: var(--#{$prefix}navbar-active-color);\n    }\n  }\n\n  .dropdown-menu {\n    position: static;\n  }\n}\n\n\n// Navbar text\n//\n//\n\n.navbar-text {\n  padding-top: $nav-link-padding-y;\n  padding-bottom: $nav-link-padding-y;\n  color: var(--#{$prefix}navbar-color);\n\n  a,\n  a:hover,\n  a:focus  {\n    color: var(--#{$prefix}navbar-active-color);\n  }\n}\n\n\n// Responsive navbar\n//\n// Custom styles for responsive collapsing and toggling of navbar contents.\n// Powered by the collapse Bootstrap JavaScript plugin.\n\n// When collapsed, prevent the toggleable navbar contents from appearing in\n// the default flexbox row orientation. Requires the use of `flex-wrap: wrap`\n// on the `.navbar` parent.\n.navbar-collapse {\n  flex-basis: 100%;\n  flex-grow: 1;\n  // For always expanded or extra full navbars, ensure content aligns itself\n  // properly vertically. Can be easily overridden with flex utilities.\n  align-items: center;\n}\n\n// Button for toggling the navbar when in its collapsed state\n.navbar-toggler {\n  padding: var(--#{$prefix}navbar-toggler-padding-y) var(--#{$prefix}navbar-toggler-padding-x);\n  @include font-size(var(--#{$prefix}navbar-toggler-font-size));\n  line-height: 1;\n  color: var(--#{$prefix}navbar-color);\n  background-color: transparent; // remove default button style\n  border: var(--#{$prefix}border-width) solid var(--#{$prefix}navbar-toggler-border-color); // remove default button style\n  @include border-radius(var(--#{$prefix}navbar-toggler-border-radius));\n  @include transition(var(--#{$prefix}navbar-toggler-transition));\n\n  &:hover {\n    text-decoration: none;\n  }\n\n  &:focus {\n    text-decoration: none;\n    outline: 0;\n    box-shadow: 0 0 0 var(--#{$prefix}navbar-toggler-focus-width);\n  }\n}\n\n// Keep as a separate element so folks can easily override it with another icon\n// or image file as needed.\n.navbar-toggler-icon {\n  display: inline-block;\n  width: 1.5em;\n  height: 1.5em;\n  vertical-align: middle;\n  background-image: var(--#{$prefix}navbar-toggler-icon-bg);\n  background-repeat: no-repeat;\n  background-position: center;\n  background-size: 100%;\n}\n\n.navbar-nav-scroll {\n  max-height: var(--#{$prefix}scroll-height, 75vh);\n  overflow-y: auto;\n}\n\n// scss-docs-start navbar-expand-loop\n// Generate series of `.navbar-expand-*` responsive classes for configuring\n// where your navbar collapses.\n.navbar-expand {\n  @each $breakpoint in map-keys($grid-breakpoints) {\n    $next: breakpoint-next($breakpoint, $grid-breakpoints);\n    $infix: breakpoint-infix($next, $grid-breakpoints);\n\n    // stylelint-disable-next-line scss/selector-no-union-class-name\n    &#{$infix} {\n      @include media-breakpoint-up($next) {\n        flex-wrap: nowrap;\n        justify-content: flex-start;\n\n        .navbar-nav {\n          flex-direction: row;\n\n          .dropdown-menu {\n            position: absolute;\n          }\n\n          .nav-link {\n            padding-right: var(--#{$prefix}navbar-nav-link-padding-x);\n            padding-left: var(--#{$prefix}navbar-nav-link-padding-x);\n          }\n        }\n\n        .navbar-nav-scroll {\n          overflow: visible;\n        }\n\n        .navbar-collapse {\n          display: flex !important; // stylelint-disable-line declaration-no-important\n          flex-basis: auto;\n        }\n\n        .navbar-toggler {\n          display: none;\n        }\n\n        .offcanvas {\n          // stylelint-disable declaration-no-important\n          position: static;\n          z-index: auto;\n          flex-grow: 1;\n          width: auto !important;\n          height: auto !important;\n          visibility: visible !important;\n          background-color: transparent !important;\n          border: 0 !important;\n          transform: none !important;\n          @include box-shadow(none);\n          @include transition(none);\n          // stylelint-enable declaration-no-important\n\n          .offcanvas-header {\n            display: none;\n          }\n\n          .offcanvas-body {\n            display: flex;\n            flex-grow: 0;\n            padding: 0;\n            overflow-y: visible;\n          }\n        }\n      }\n    }\n  }\n}\n// scss-docs-end navbar-expand-loop\n\n// Navbar themes\n//\n// Styles for switching between navbars with light or dark background.\n\n.navbar-light {\n  @include deprecate(\"`.navbar-light`\", \"v5.2.0\", \"v6.0.0\", true);\n}\n\n.navbar-dark,\n.navbar[data-bs-theme=\"dark\"] {\n  // scss-docs-start navbar-dark-css-vars\n  --#{$prefix}navbar-color: #{$navbar-dark-color};\n  --#{$prefix}navbar-hover-color: #{$navbar-dark-hover-color};\n  --#{$prefix}navbar-disabled-color: #{$navbar-dark-disabled-color};\n  --#{$prefix}navbar-active-color: #{$navbar-dark-active-color};\n  --#{$prefix}navbar-brand-color: #{$navbar-dark-brand-color};\n  --#{$prefix}navbar-brand-hover-color: #{$navbar-dark-brand-hover-color};\n  --#{$prefix}navbar-toggler-border-color: #{$navbar-dark-toggler-border-color};\n  --#{$prefix}navbar-toggler-icon-bg: #{escape-svg($navbar-dark-toggler-icon-bg)};\n  // scss-docs-end navbar-dark-css-vars\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .navbar-toggler-icon {\n      --#{$prefix}navbar-toggler-icon-bg: #{escape-svg($navbar-dark-toggler-icon-bg)};\n    }\n  }\n}\n","//\n// Base styles\n//\n\n.card {\n  // scss-docs-start card-css-vars\n  --#{$prefix}card-spacer-y: #{$card-spacer-y};\n  --#{$prefix}card-spacer-x: #{$card-spacer-x};\n  --#{$prefix}card-title-spacer-y: #{$card-title-spacer-y};\n  --#{$prefix}card-title-color: #{$card-title-color};\n  --#{$prefix}card-subtitle-color: #{$card-subtitle-color};\n  --#{$prefix}card-border-width: #{$card-border-width};\n  --#{$prefix}card-border-color: #{$card-border-color};\n  --#{$prefix}card-border-radius: #{$card-border-radius};\n  --#{$prefix}card-box-shadow: #{$card-box-shadow};\n  --#{$prefix}card-inner-border-radius: #{$card-inner-border-radius};\n  --#{$prefix}card-cap-padding-y: #{$card-cap-padding-y};\n  --#{$prefix}card-cap-padding-x: #{$card-cap-padding-x};\n  --#{$prefix}card-cap-bg: #{$card-cap-bg};\n  --#{$prefix}card-cap-color: #{$card-cap-color};\n  --#{$prefix}card-height: #{$card-height};\n  --#{$prefix}card-color: #{$card-color};\n  --#{$prefix}card-bg: #{$card-bg};\n  --#{$prefix}card-img-overlay-padding: #{$card-img-overlay-padding};\n  --#{$prefix}card-group-margin: #{$card-group-margin};\n  // scss-docs-end card-css-vars\n\n  position: relative;\n  display: flex;\n  flex-direction: column;\n  min-width: 0; // See https://github.com/twbs/bootstrap/pull/22740#issuecomment-305868106\n  height: var(--#{$prefix}card-height);\n  color: var(--#{$prefix}body-color);\n  word-wrap: break-word;\n  background-color: var(--#{$prefix}card-bg);\n  background-clip: border-box;\n  border: var(--#{$prefix}card-border-width) solid var(--#{$prefix}card-border-color);\n  @include border-radius(var(--#{$prefix}card-border-radius));\n  @include box-shadow(var(--#{$prefix}card-box-shadow));\n\n  > hr {\n    margin-right: 0;\n    margin-left: 0;\n  }\n\n  > .list-group {\n    border-top: inherit;\n    border-bottom: inherit;\n\n    &:first-child {\n      border-top-width: 0;\n      @include border-top-radius(var(--#{$prefix}card-inner-border-radius));\n    }\n\n    &:last-child  {\n      border-bottom-width: 0;\n      @include border-bottom-radius(var(--#{$prefix}card-inner-border-radius));\n    }\n  }\n\n  // Due to specificity of the above selector (`.card > .list-group`), we must\n  // use a child selector here to prevent double borders.\n  > .card-header + .list-group,\n  > .list-group + .card-footer {\n    border-top: 0;\n  }\n}\n\n.card-body {\n  // Enable `flex-grow: 1` for decks and groups so that card blocks take up\n  // as much space as possible, ensuring footers are aligned to the bottom.\n  flex: 1 1 auto;\n  padding: var(--#{$prefix}card-spacer-y) var(--#{$prefix}card-spacer-x);\n  color: var(--#{$prefix}card-color);\n}\n\n.card-title {\n  margin-bottom: var(--#{$prefix}card-title-spacer-y);\n  color: var(--#{$prefix}card-title-color);\n}\n\n.card-subtitle {\n  margin-top: calc(-.5 * var(--#{$prefix}card-title-spacer-y)); // stylelint-disable-line function-disallowed-list\n  margin-bottom: 0;\n  color: var(--#{$prefix}card-subtitle-color);\n}\n\n.card-text:last-child {\n  margin-bottom: 0;\n}\n\n.card-link {\n  &:hover {\n    text-decoration: if($link-hover-decoration == underline, none, null);\n  }\n\n  + .card-link {\n    margin-left: var(--#{$prefix}card-spacer-x);\n  }\n}\n\n//\n// Optional textual caps\n//\n\n.card-header {\n  padding: var(--#{$prefix}card-cap-padding-y) var(--#{$prefix}card-cap-padding-x);\n  margin-bottom: 0; // Removes the default margin-bottom of <hN>\n  color: var(--#{$prefix}card-cap-color);\n  background-color: var(--#{$prefix}card-cap-bg);\n  border-bottom: var(--#{$prefix}card-border-width) solid var(--#{$prefix}card-border-color);\n\n  &:first-child {\n    @include border-radius(var(--#{$prefix}card-inner-border-radius) var(--#{$prefix}card-inner-border-radius) 0 0);\n  }\n}\n\n.card-footer {\n  padding: var(--#{$prefix}card-cap-padding-y) var(--#{$prefix}card-cap-padding-x);\n  color: var(--#{$prefix}card-cap-color);\n  background-color: var(--#{$prefix}card-cap-bg);\n  border-top: var(--#{$prefix}card-border-width) solid var(--#{$prefix}card-border-color);\n\n  &:last-child {\n    @include border-radius(0 0 var(--#{$prefix}card-inner-border-radius) var(--#{$prefix}card-inner-border-radius));\n  }\n}\n\n\n//\n// Header navs\n//\n\n.card-header-tabs {\n  margin-right: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n  margin-bottom: calc(-1 * var(--#{$prefix}card-cap-padding-y)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n  border-bottom: 0;\n\n  .nav-link.active {\n    background-color: var(--#{$prefix}card-bg);\n    border-bottom-color: var(--#{$prefix}card-bg);\n  }\n}\n\n.card-header-pills {\n  margin-right: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}card-cap-padding-x)); // stylelint-disable-line function-disallowed-list\n}\n\n// Card image\n.card-img-overlay {\n  position: absolute;\n  top: 0;\n  right: 0;\n  bottom: 0;\n  left: 0;\n  padding: var(--#{$prefix}card-img-overlay-padding);\n  @include border-radius(var(--#{$prefix}card-inner-border-radius));\n}\n\n.card-img,\n.card-img-top,\n.card-img-bottom {\n  width: 100%; // Required because we use flexbox and this inherently applies align-self: stretch\n}\n\n.card-img,\n.card-img-top {\n  @include border-top-radius(var(--#{$prefix}card-inner-border-radius));\n}\n\n.card-img,\n.card-img-bottom {\n  @include border-bottom-radius(var(--#{$prefix}card-inner-border-radius));\n}\n\n\n//\n// Card groups\n//\n\n.card-group {\n  // The child selector allows nested `.card` within `.card-group`\n  // to display properly.\n  > .card {\n    margin-bottom: var(--#{$prefix}card-group-margin);\n  }\n\n  @include media-breakpoint-up(sm) {\n    display: flex;\n    flex-flow: row wrap;\n    // The child selector allows nested `.card` within `.card-group`\n    // to display properly.\n    > .card {\n      // Flexbugs #4: https://github.com/philipwalton/flexbugs#flexbug-4\n      flex: 1 0 0%;\n      margin-bottom: 0;\n\n      + .card {\n        margin-left: 0;\n        border-left: 0;\n      }\n\n      // Handle rounded corners\n      @if $enable-rounded {\n        &:not(:last-child) {\n          @include border-end-radius(0);\n\n          .card-img-top,\n          .card-header {\n            // stylelint-disable-next-line property-disallowed-list\n            border-top-right-radius: 0;\n          }\n          .card-img-bottom,\n          .card-footer {\n            // stylelint-disable-next-line property-disallowed-list\n            border-bottom-right-radius: 0;\n          }\n        }\n\n        &:not(:first-child) {\n          @include border-start-radius(0);\n\n          .card-img-top,\n          .card-header {\n            // stylelint-disable-next-line property-disallowed-list\n            border-top-left-radius: 0;\n          }\n          .card-img-bottom,\n          .card-footer {\n            // stylelint-disable-next-line property-disallowed-list\n            border-bottom-left-radius: 0;\n          }\n        }\n      }\n    }\n  }\n}\n","//\n// Base styles\n//\n\n.accordion {\n  // scss-docs-start accordion-css-vars\n  --#{$prefix}accordion-color: #{$accordion-color};\n  --#{$prefix}accordion-bg: #{$accordion-bg};\n  --#{$prefix}accordion-transition: #{$accordion-transition};\n  --#{$prefix}accordion-border-color: #{$accordion-border-color};\n  --#{$prefix}accordion-border-width: #{$accordion-border-width};\n  --#{$prefix}accordion-border-radius: #{$accordion-border-radius};\n  --#{$prefix}accordion-inner-border-radius: #{$accordion-inner-border-radius};\n  --#{$prefix}accordion-btn-padding-x: #{$accordion-button-padding-x};\n  --#{$prefix}accordion-btn-padding-y: #{$accordion-button-padding-y};\n  --#{$prefix}accordion-btn-color: #{$accordion-button-color};\n  --#{$prefix}accordion-btn-bg: #{$accordion-button-bg};\n  --#{$prefix}accordion-btn-icon: #{escape-svg($accordion-button-icon)};\n  --#{$prefix}accordion-btn-icon-width: #{$accordion-icon-width};\n  --#{$prefix}accordion-btn-icon-transform: #{$accordion-icon-transform};\n  --#{$prefix}accordion-btn-icon-transition: #{$accordion-icon-transition};\n  --#{$prefix}accordion-btn-active-icon: #{escape-svg($accordion-button-active-icon)};\n  --#{$prefix}accordion-btn-focus-border-color: #{$accordion-button-focus-border-color};\n  --#{$prefix}accordion-btn-focus-box-shadow: #{$accordion-button-focus-box-shadow};\n  --#{$prefix}accordion-body-padding-x: #{$accordion-body-padding-x};\n  --#{$prefix}accordion-body-padding-y: #{$accordion-body-padding-y};\n  --#{$prefix}accordion-active-color: #{$accordion-button-active-color};\n  --#{$prefix}accordion-active-bg: #{$accordion-button-active-bg};\n  // scss-docs-end accordion-css-vars\n}\n\n.accordion-button {\n  position: relative;\n  display: flex;\n  align-items: center;\n  width: 100%;\n  padding: var(--#{$prefix}accordion-btn-padding-y) var(--#{$prefix}accordion-btn-padding-x);\n  @include font-size($font-size-base);\n  color: var(--#{$prefix}accordion-btn-color);\n  text-align: left; // Reset button style\n  background-color: var(--#{$prefix}accordion-btn-bg);\n  border: 0;\n  @include border-radius(0);\n  overflow-anchor: none;\n  @include transition(var(--#{$prefix}accordion-transition));\n\n  &:not(.collapsed) {\n    color: var(--#{$prefix}accordion-active-color);\n    background-color: var(--#{$prefix}accordion-active-bg);\n    box-shadow: inset 0 calc(-1 * var(--#{$prefix}accordion-border-width)) 0 var(--#{$prefix}accordion-border-color); // stylelint-disable-line function-disallowed-list\n\n    &::after {\n      background-image: var(--#{$prefix}accordion-btn-active-icon);\n      transform: var(--#{$prefix}accordion-btn-icon-transform);\n    }\n  }\n\n  // Accordion icon\n  &::after {\n    flex-shrink: 0;\n    width: var(--#{$prefix}accordion-btn-icon-width);\n    height: var(--#{$prefix}accordion-btn-icon-width);\n    margin-left: auto;\n    content: \"\";\n    background-image: var(--#{$prefix}accordion-btn-icon);\n    background-repeat: no-repeat;\n    background-size: var(--#{$prefix}accordion-btn-icon-width);\n    @include transition(var(--#{$prefix}accordion-btn-icon-transition));\n  }\n\n  &:hover {\n    z-index: 2;\n  }\n\n  &:focus {\n    z-index: 3;\n    border-color: var(--#{$prefix}accordion-btn-focus-border-color);\n    outline: 0;\n    box-shadow: var(--#{$prefix}accordion-btn-focus-box-shadow);\n  }\n}\n\n.accordion-header {\n  margin-bottom: 0;\n}\n\n.accordion-item {\n  color: var(--#{$prefix}accordion-color);\n  background-color: var(--#{$prefix}accordion-bg);\n  border: var(--#{$prefix}accordion-border-width) solid var(--#{$prefix}accordion-border-color);\n\n  &:first-of-type {\n    @include border-top-radius(var(--#{$prefix}accordion-border-radius));\n\n    .accordion-button {\n      @include border-top-radius(var(--#{$prefix}accordion-inner-border-radius));\n    }\n  }\n\n  &:not(:first-of-type) {\n    border-top: 0;\n  }\n\n  // Only set a border-radius on the last item if the accordion is collapsed\n  &:last-of-type {\n    @include border-bottom-radius(var(--#{$prefix}accordion-border-radius));\n\n    .accordion-button {\n      &.collapsed {\n        @include border-bottom-radius(var(--#{$prefix}accordion-inner-border-radius));\n      }\n    }\n\n    .accordion-collapse {\n      @include border-bottom-radius(var(--#{$prefix}accordion-border-radius));\n    }\n  }\n}\n\n.accordion-body {\n  padding: var(--#{$prefix}accordion-body-padding-y) var(--#{$prefix}accordion-body-padding-x);\n}\n\n\n// Flush accordion items\n//\n// Remove borders and border-radius to keep accordion items edge-to-edge.\n\n.accordion-flush {\n  .accordion-collapse {\n    border-width: 0;\n  }\n\n  .accordion-item {\n    border-right: 0;\n    border-left: 0;\n    @include border-radius(0);\n\n    &:first-child { border-top: 0; }\n    &:last-child { border-bottom: 0; }\n\n    .accordion-button {\n      &,\n      &.collapsed {\n        @include border-radius(0);\n      }\n    }\n  }\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .accordion-button::after {\n      --#{$prefix}accordion-btn-icon: #{escape-svg($accordion-button-icon-dark)};\n      --#{$prefix}accordion-btn-active-icon: #{escape-svg($accordion-button-active-icon-dark)};\n    }\n  }\n}\n",".breadcrumb {\n  // scss-docs-start breadcrumb-css-vars\n  --#{$prefix}breadcrumb-padding-x: #{$breadcrumb-padding-x};\n  --#{$prefix}breadcrumb-padding-y: #{$breadcrumb-padding-y};\n  --#{$prefix}breadcrumb-margin-bottom: #{$breadcrumb-margin-bottom};\n  @include rfs($breadcrumb-font-size, --#{$prefix}breadcrumb-font-size);\n  --#{$prefix}breadcrumb-bg: #{$breadcrumb-bg};\n  --#{$prefix}breadcrumb-border-radius: #{$breadcrumb-border-radius};\n  --#{$prefix}breadcrumb-divider-color: #{$breadcrumb-divider-color};\n  --#{$prefix}breadcrumb-item-padding-x: #{$breadcrumb-item-padding-x};\n  --#{$prefix}breadcrumb-item-active-color: #{$breadcrumb-active-color};\n  // scss-docs-end breadcrumb-css-vars\n\n  display: flex;\n  flex-wrap: wrap;\n  padding: var(--#{$prefix}breadcrumb-padding-y) var(--#{$prefix}breadcrumb-padding-x);\n  margin-bottom: var(--#{$prefix}breadcrumb-margin-bottom);\n  @include font-size(var(--#{$prefix}breadcrumb-font-size));\n  list-style: none;\n  background-color: var(--#{$prefix}breadcrumb-bg);\n  @include border-radius(var(--#{$prefix}breadcrumb-border-radius));\n}\n\n.breadcrumb-item {\n  // The separator between breadcrumbs (by default, a forward-slash: \"/\")\n  + .breadcrumb-item {\n    padding-left: var(--#{$prefix}breadcrumb-item-padding-x);\n\n    &::before {\n      float: left; // Suppress inline spacings and underlining of the separator\n      padding-right: var(--#{$prefix}breadcrumb-item-padding-x);\n      color: var(--#{$prefix}breadcrumb-divider-color);\n      content: var(--#{$prefix}breadcrumb-divider, escape-svg($breadcrumb-divider)) #{\"/* rtl:\"} var(--#{$prefix}breadcrumb-divider, escape-svg($breadcrumb-divider-flipped)) #{\"*/\"};\n    }\n  }\n\n  &.active {\n    color: var(--#{$prefix}breadcrumb-item-active-color);\n  }\n}\n",".pagination {\n  // scss-docs-start pagination-css-vars\n  --#{$prefix}pagination-padding-x: #{$pagination-padding-x};\n  --#{$prefix}pagination-padding-y: #{$pagination-padding-y};\n  @include rfs($pagination-font-size, --#{$prefix}pagination-font-size);\n  --#{$prefix}pagination-color: #{$pagination-color};\n  --#{$prefix}pagination-bg: #{$pagination-bg};\n  --#{$prefix}pagination-border-width: #{$pagination-border-width};\n  --#{$prefix}pagination-border-color: #{$pagination-border-color};\n  --#{$prefix}pagination-border-radius: #{$pagination-border-radius};\n  --#{$prefix}pagination-hover-color: #{$pagination-hover-color};\n  --#{$prefix}pagination-hover-bg: #{$pagination-hover-bg};\n  --#{$prefix}pagination-hover-border-color: #{$pagination-hover-border-color};\n  --#{$prefix}pagination-focus-color: #{$pagination-focus-color};\n  --#{$prefix}pagination-focus-bg: #{$pagination-focus-bg};\n  --#{$prefix}pagination-focus-box-shadow: #{$pagination-focus-box-shadow};\n  --#{$prefix}pagination-active-color: #{$pagination-active-color};\n  --#{$prefix}pagination-active-bg: #{$pagination-active-bg};\n  --#{$prefix}pagination-active-border-color: #{$pagination-active-border-color};\n  --#{$prefix}pagination-disabled-color: #{$pagination-disabled-color};\n  --#{$prefix}pagination-disabled-bg: #{$pagination-disabled-bg};\n  --#{$prefix}pagination-disabled-border-color: #{$pagination-disabled-border-color};\n  // scss-docs-end pagination-css-vars\n\n  display: flex;\n  @include list-unstyled();\n}\n\n.page-link {\n  position: relative;\n  display: block;\n  padding: var(--#{$prefix}pagination-padding-y) var(--#{$prefix}pagination-padding-x);\n  @include font-size(var(--#{$prefix}pagination-font-size));\n  color: var(--#{$prefix}pagination-color);\n  text-decoration: if($link-decoration == none, null, none);\n  background-color: var(--#{$prefix}pagination-bg);\n  border: var(--#{$prefix}pagination-border-width) solid var(--#{$prefix}pagination-border-color);\n  @include transition($pagination-transition);\n\n  &:hover {\n    z-index: 2;\n    color: var(--#{$prefix}pagination-hover-color);\n    text-decoration: if($link-hover-decoration == underline, none, null);\n    background-color: var(--#{$prefix}pagination-hover-bg);\n    border-color: var(--#{$prefix}pagination-hover-border-color);\n  }\n\n  &:focus {\n    z-index: 3;\n    color: var(--#{$prefix}pagination-focus-color);\n    background-color: var(--#{$prefix}pagination-focus-bg);\n    outline: $pagination-focus-outline;\n    box-shadow: var(--#{$prefix}pagination-focus-box-shadow);\n  }\n\n  &.active,\n  .active > & {\n    z-index: 3;\n    color: var(--#{$prefix}pagination-active-color);\n    @include gradient-bg(var(--#{$prefix}pagination-active-bg));\n    border-color: var(--#{$prefix}pagination-active-border-color);\n  }\n\n  &.disabled,\n  .disabled > & {\n    color: var(--#{$prefix}pagination-disabled-color);\n    pointer-events: none;\n    background-color: var(--#{$prefix}pagination-disabled-bg);\n    border-color: var(--#{$prefix}pagination-disabled-border-color);\n  }\n}\n\n.page-item {\n  &:not(:first-child) .page-link {\n    margin-left: $pagination-margin-start;\n  }\n\n  @if $pagination-margin-start == calc(#{$pagination-border-width} * -1) {\n    &:first-child {\n      .page-link {\n        @include border-start-radius(var(--#{$prefix}pagination-border-radius));\n      }\n    }\n\n    &:last-child {\n      .page-link {\n        @include border-end-radius(var(--#{$prefix}pagination-border-radius));\n      }\n    }\n  } @else {\n    // Add border-radius to all pageLinks in case they have left margin\n    .page-link {\n      @include border-radius(var(--#{$prefix}pagination-border-radius));\n    }\n  }\n}\n\n\n//\n// Sizing\n//\n\n.pagination-lg {\n  @include pagination-size($pagination-padding-y-lg, $pagination-padding-x-lg, $font-size-lg, $pagination-border-radius-lg);\n}\n\n.pagination-sm {\n  @include pagination-size($pagination-padding-y-sm, $pagination-padding-x-sm, $font-size-sm, $pagination-border-radius-sm);\n}\n","// Pagination\n\n// scss-docs-start pagination-mixin\n@mixin pagination-size($padding-y, $padding-x, $font-size, $border-radius) {\n  --#{$prefix}pagination-padding-x: #{$padding-x};\n  --#{$prefix}pagination-padding-y: #{$padding-y};\n  @include rfs($font-size, --#{$prefix}pagination-font-size);\n  --#{$prefix}pagination-border-radius: #{$border-radius};\n}\n// scss-docs-end pagination-mixin\n","// Base class\n//\n// Requires one of the contextual, color modifier classes for `color` and\n// `background-color`.\n\n.badge {\n  // scss-docs-start badge-css-vars\n  --#{$prefix}badge-padding-x: #{$badge-padding-x};\n  --#{$prefix}badge-padding-y: #{$badge-padding-y};\n  @include rfs($badge-font-size, --#{$prefix}badge-font-size);\n  --#{$prefix}badge-font-weight: #{$badge-font-weight};\n  --#{$prefix}badge-color: #{$badge-color};\n  --#{$prefix}badge-border-radius: #{$badge-border-radius};\n  // scss-docs-end badge-css-vars\n\n  display: inline-block;\n  padding: var(--#{$prefix}badge-padding-y) var(--#{$prefix}badge-padding-x);\n  @include font-size(var(--#{$prefix}badge-font-size));\n  font-weight: var(--#{$prefix}badge-font-weight);\n  line-height: 1;\n  color: var(--#{$prefix}badge-color);\n  text-align: center;\n  white-space: nowrap;\n  vertical-align: baseline;\n  @include border-radius(var(--#{$prefix}badge-border-radius));\n  @include gradient-bg();\n\n  // Empty badges collapse automatically\n  &:empty {\n    display: none;\n  }\n}\n\n// Quick fix for badges in buttons\n.btn .badge {\n  position: relative;\n  top: -1px;\n}\n","//\n// Base styles\n//\n\n.alert {\n  // scss-docs-start alert-css-vars\n  --#{$prefix}alert-bg: transparent;\n  --#{$prefix}alert-padding-x: #{$alert-padding-x};\n  --#{$prefix}alert-padding-y: #{$alert-padding-y};\n  --#{$prefix}alert-margin-bottom: #{$alert-margin-bottom};\n  --#{$prefix}alert-color: inherit;\n  --#{$prefix}alert-border-color: transparent;\n  --#{$prefix}alert-border: #{$alert-border-width} solid var(--#{$prefix}alert-border-color);\n  --#{$prefix}alert-border-radius: #{$alert-border-radius};\n  --#{$prefix}alert-link-color: inherit;\n  // scss-docs-end alert-css-vars\n\n  position: relative;\n  padding: var(--#{$prefix}alert-padding-y) var(--#{$prefix}alert-padding-x);\n  margin-bottom: var(--#{$prefix}alert-margin-bottom);\n  color: var(--#{$prefix}alert-color);\n  background-color: var(--#{$prefix}alert-bg);\n  border: var(--#{$prefix}alert-border);\n  @include border-radius(var(--#{$prefix}alert-border-radius));\n}\n\n// Headings for larger alerts\n.alert-heading {\n  // Specified to prevent conflicts of changing $headings-color\n  color: inherit;\n}\n\n// Provide class for links that match alerts\n.alert-link {\n  font-weight: $alert-link-font-weight;\n  color: var(--#{$prefix}alert-link-color);\n}\n\n\n// Dismissible alerts\n//\n// Expand the right padding and account for the close button's positioning.\n\n.alert-dismissible {\n  padding-right: $alert-dismissible-padding-r;\n\n  // Adjust close link position\n  .btn-close {\n    position: absolute;\n    top: 0;\n    right: 0;\n    z-index: $stretched-link-z-index + 1;\n    padding: $alert-padding-y * 1.25 $alert-padding-x;\n  }\n}\n\n\n// scss-docs-start alert-modifiers\n// Generate contextual modifier classes for colorizing the alert\n@each $state in map-keys($theme-colors) {\n  .alert-#{$state} {\n    --#{$prefix}alert-color: var(--#{$prefix}#{$state}-text-emphasis);\n    --#{$prefix}alert-bg: var(--#{$prefix}#{$state}-bg-subtle);\n    --#{$prefix}alert-border-color: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}alert-link-color: var(--#{$prefix}#{$state}-text-emphasis);\n  }\n}\n// scss-docs-end alert-modifiers\n","// Disable animation if transitions are disabled\n\n// scss-docs-start progress-keyframes\n@if $enable-transitions {\n  @keyframes progress-bar-stripes {\n    0% { background-position-x: $progress-height; }\n  }\n}\n// scss-docs-end progress-keyframes\n\n.progress,\n.progress-stacked {\n  // scss-docs-start progress-css-vars\n  --#{$prefix}progress-height: #{$progress-height};\n  @include rfs($progress-font-size, --#{$prefix}progress-font-size);\n  --#{$prefix}progress-bg: #{$progress-bg};\n  --#{$prefix}progress-border-radius: #{$progress-border-radius};\n  --#{$prefix}progress-box-shadow: #{$progress-box-shadow};\n  --#{$prefix}progress-bar-color: #{$progress-bar-color};\n  --#{$prefix}progress-bar-bg: #{$progress-bar-bg};\n  --#{$prefix}progress-bar-transition: #{$progress-bar-transition};\n  // scss-docs-end progress-css-vars\n\n  display: flex;\n  height: var(--#{$prefix}progress-height);\n  overflow: hidden; // force rounded corners by cropping it\n  @include font-size(var(--#{$prefix}progress-font-size));\n  background-color: var(--#{$prefix}progress-bg);\n  @include border-radius(var(--#{$prefix}progress-border-radius));\n  @include box-shadow(var(--#{$prefix}progress-box-shadow));\n}\n\n.progress-bar {\n  display: flex;\n  flex-direction: column;\n  justify-content: center;\n  overflow: hidden;\n  color: var(--#{$prefix}progress-bar-color);\n  text-align: center;\n  white-space: nowrap;\n  background-color: var(--#{$prefix}progress-bar-bg);\n  @include transition(var(--#{$prefix}progress-bar-transition));\n}\n\n.progress-bar-striped {\n  @include gradient-striped();\n  background-size: var(--#{$prefix}progress-height) var(--#{$prefix}progress-height);\n}\n\n.progress-stacked > .progress {\n  overflow: visible;\n}\n\n.progress-stacked > .progress > .progress-bar {\n  width: 100%;\n}\n\n@if $enable-transitions {\n  .progress-bar-animated {\n    animation: $progress-bar-animation-timing progress-bar-stripes;\n\n    @if $enable-reduced-motion {\n      @media (prefers-reduced-motion: reduce) {\n        animation: none;\n      }\n    }\n  }\n}\n","// Base class\n//\n// Easily usable on <ul>, <ol>, or <div>.\n\n.list-group {\n  // scss-docs-start list-group-css-vars\n  --#{$prefix}list-group-color: #{$list-group-color};\n  --#{$prefix}list-group-bg: #{$list-group-bg};\n  --#{$prefix}list-group-border-color: #{$list-group-border-color};\n  --#{$prefix}list-group-border-width: #{$list-group-border-width};\n  --#{$prefix}list-group-border-radius: #{$list-group-border-radius};\n  --#{$prefix}list-group-item-padding-x: #{$list-group-item-padding-x};\n  --#{$prefix}list-group-item-padding-y: #{$list-group-item-padding-y};\n  --#{$prefix}list-group-action-color: #{$list-group-action-color};\n  --#{$prefix}list-group-action-hover-color: #{$list-group-action-hover-color};\n  --#{$prefix}list-group-action-hover-bg: #{$list-group-hover-bg};\n  --#{$prefix}list-group-action-active-color: #{$list-group-action-active-color};\n  --#{$prefix}list-group-action-active-bg: #{$list-group-action-active-bg};\n  --#{$prefix}list-group-disabled-color: #{$list-group-disabled-color};\n  --#{$prefix}list-group-disabled-bg: #{$list-group-disabled-bg};\n  --#{$prefix}list-group-active-color: #{$list-group-active-color};\n  --#{$prefix}list-group-active-bg: #{$list-group-active-bg};\n  --#{$prefix}list-group-active-border-color: #{$list-group-active-border-color};\n  // scss-docs-end list-group-css-vars\n\n  display: flex;\n  flex-direction: column;\n\n  // No need to set list-style: none; since .list-group-item is block level\n  padding-left: 0; // reset padding because ul and ol\n  margin-bottom: 0;\n  @include border-radius(var(--#{$prefix}list-group-border-radius));\n}\n\n.list-group-numbered {\n  list-style-type: none;\n  counter-reset: section;\n\n  > .list-group-item::before {\n    // Increments only this instance of the section counter\n    content: counters(section, \".\") \". \";\n    counter-increment: section;\n  }\n}\n\n// Interactive list items\n//\n// Use anchor or button elements instead of `li`s or `div`s to create interactive\n// list items. Includes an extra `.active` modifier class for selected items.\n\n.list-group-item-action {\n  width: 100%; // For `<button>`s (anchors become 100% by default though)\n  color: var(--#{$prefix}list-group-action-color);\n  text-align: inherit; // For `<button>`s (anchors inherit)\n\n  // Hover state\n  &:hover,\n  &:focus {\n    z-index: 1; // Place hover/focus items above their siblings for proper border styling\n    color: var(--#{$prefix}list-group-action-hover-color);\n    text-decoration: none;\n    background-color: var(--#{$prefix}list-group-action-hover-bg);\n  }\n\n  &:active {\n    color: var(--#{$prefix}list-group-action-active-color);\n    background-color: var(--#{$prefix}list-group-action-active-bg);\n  }\n}\n\n// Individual list items\n//\n// Use on `li`s or `div`s within the `.list-group` parent.\n\n.list-group-item {\n  position: relative;\n  display: block;\n  padding: var(--#{$prefix}list-group-item-padding-y) var(--#{$prefix}list-group-item-padding-x);\n  color: var(--#{$prefix}list-group-color);\n  text-decoration: if($link-decoration == none, null, none);\n  background-color: var(--#{$prefix}list-group-bg);\n  border: var(--#{$prefix}list-group-border-width) solid var(--#{$prefix}list-group-border-color);\n\n  &:first-child {\n    @include border-top-radius(inherit);\n  }\n\n  &:last-child {\n    @include border-bottom-radius(inherit);\n  }\n\n  &.disabled,\n  &:disabled {\n    color: var(--#{$prefix}list-group-disabled-color);\n    pointer-events: none;\n    background-color: var(--#{$prefix}list-group-disabled-bg);\n  }\n\n  // Include both here for `<a>`s and `<button>`s\n  &.active {\n    z-index: 2; // Place active items above their siblings for proper border styling\n    color: var(--#{$prefix}list-group-active-color);\n    background-color: var(--#{$prefix}list-group-active-bg);\n    border-color: var(--#{$prefix}list-group-active-border-color);\n  }\n\n  // stylelint-disable-next-line scss/selector-no-redundant-nesting-selector\n  & + .list-group-item {\n    border-top-width: 0;\n\n    &.active {\n      margin-top: calc(-1 * var(--#{$prefix}list-group-border-width)); // stylelint-disable-line function-disallowed-list\n      border-top-width: var(--#{$prefix}list-group-border-width);\n    }\n  }\n}\n\n// Horizontal\n//\n// Change the layout of list group items from vertical (default) to horizontal.\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    .list-group-horizontal#{$infix} {\n      flex-direction: row;\n\n      > .list-group-item {\n        &:first-child:not(:last-child) {\n          @include border-bottom-start-radius(var(--#{$prefix}list-group-border-radius));\n          @include border-top-end-radius(0);\n        }\n\n        &:last-child:not(:first-child) {\n          @include border-top-end-radius(var(--#{$prefix}list-group-border-radius));\n          @include border-bottom-start-radius(0);\n        }\n\n        &.active {\n          margin-top: 0;\n        }\n\n        + .list-group-item {\n          border-top-width: var(--#{$prefix}list-group-border-width);\n          border-left-width: 0;\n\n          &.active {\n            margin-left: calc(-1 * var(--#{$prefix}list-group-border-width)); // stylelint-disable-line function-disallowed-list\n            border-left-width: var(--#{$prefix}list-group-border-width);\n          }\n        }\n      }\n    }\n  }\n}\n\n\n// Flush list items\n//\n// Remove borders and border-radius to keep list group items edge-to-edge. Most\n// useful within other components (e.g., cards).\n\n.list-group-flush {\n  @include border-radius(0);\n\n  > .list-group-item {\n    border-width: 0 0 var(--#{$prefix}list-group-border-width);\n\n    &:last-child {\n      border-bottom-width: 0;\n    }\n  }\n}\n\n\n// scss-docs-start list-group-modifiers\n// List group contextual variants\n//\n// Add modifier classes to change text and background color on individual items.\n// Organizationally, this must come after the `:hover` states.\n\n@each $state in map-keys($theme-colors) {\n  .list-group-item-#{$state} {\n    --#{$prefix}list-group-color: var(--#{$prefix}#{$state}-text-emphasis);\n    --#{$prefix}list-group-bg: var(--#{$prefix}#{$state}-bg-subtle);\n    --#{$prefix}list-group-border-color: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}list-group-action-hover-color: var(--#{$prefix}emphasis-color);\n    --#{$prefix}list-group-action-hover-bg: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}list-group-action-active-color: var(--#{$prefix}emphasis-color);\n    --#{$prefix}list-group-action-active-bg: var(--#{$prefix}#{$state}-border-subtle);\n    --#{$prefix}list-group-active-color: var(--#{$prefix}#{$state}-bg-subtle);\n    --#{$prefix}list-group-active-bg: var(--#{$prefix}#{$state}-text-emphasis);\n    --#{$prefix}list-group-active-border-color: var(--#{$prefix}#{$state}-text-emphasis);\n  }\n}\n// scss-docs-end list-group-modifiers\n","// Transparent background and border properties included for button version.\n// iOS requires the button element instead of an anchor tag.\n// If you want the anchor version, it requires `href=\"#\"`.\n// See https://developer.mozilla.org/en-US/docs/Web/Events/click#Safari_Mobile\n\n.btn-close {\n  // scss-docs-start close-css-vars\n  --#{$prefix}btn-close-color: #{$btn-close-color};\n  --#{$prefix}btn-close-bg: #{ escape-svg($btn-close-bg) };\n  --#{$prefix}btn-close-opacity: #{$btn-close-opacity};\n  --#{$prefix}btn-close-hover-opacity: #{$btn-close-hover-opacity};\n  --#{$prefix}btn-close-focus-shadow: #{$btn-close-focus-shadow};\n  --#{$prefix}btn-close-focus-opacity: #{$btn-close-focus-opacity};\n  --#{$prefix}btn-close-disabled-opacity: #{$btn-close-disabled-opacity};\n  --#{$prefix}btn-close-white-filter: #{$btn-close-white-filter};\n  // scss-docs-end close-css-vars\n\n  box-sizing: content-box;\n  width: $btn-close-width;\n  height: $btn-close-height;\n  padding: $btn-close-padding-y $btn-close-padding-x;\n  color: var(--#{$prefix}btn-close-color);\n  background: transparent var(--#{$prefix}btn-close-bg) center / $btn-close-width auto no-repeat; // include transparent for button elements\n  border: 0; // for button elements\n  @include border-radius();\n  opacity: var(--#{$prefix}btn-close-opacity);\n\n  // Override <a>'s hover style\n  &:hover {\n    color: var(--#{$prefix}btn-close-color);\n    text-decoration: none;\n    opacity: var(--#{$prefix}btn-close-hover-opacity);\n  }\n\n  &:focus {\n    outline: 0;\n    box-shadow: var(--#{$prefix}btn-close-focus-shadow);\n    opacity: var(--#{$prefix}btn-close-focus-opacity);\n  }\n\n  &:disabled,\n  &.disabled {\n    pointer-events: none;\n    user-select: none;\n    opacity: var(--#{$prefix}btn-close-disabled-opacity);\n  }\n}\n\n@mixin btn-close-white() {\n  filter: var(--#{$prefix}btn-close-white-filter);\n}\n\n.btn-close-white {\n  @include btn-close-white();\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    .btn-close {\n      @include btn-close-white();\n    }\n  }\n}\n",".toast {\n  // scss-docs-start toast-css-vars\n  --#{$prefix}toast-zindex: #{$zindex-toast};\n  --#{$prefix}toast-padding-x: #{$toast-padding-x};\n  --#{$prefix}toast-padding-y: #{$toast-padding-y};\n  --#{$prefix}toast-spacing: #{$toast-spacing};\n  --#{$prefix}toast-max-width: #{$toast-max-width};\n  @include rfs($toast-font-size, --#{$prefix}toast-font-size);\n  --#{$prefix}toast-color: #{$toast-color};\n  --#{$prefix}toast-bg: #{$toast-background-color};\n  --#{$prefix}toast-border-width: #{$toast-border-width};\n  --#{$prefix}toast-border-color: #{$toast-border-color};\n  --#{$prefix}toast-border-radius: #{$toast-border-radius};\n  --#{$prefix}toast-box-shadow: #{$toast-box-shadow};\n  --#{$prefix}toast-header-color: #{$toast-header-color};\n  --#{$prefix}toast-header-bg: #{$toast-header-background-color};\n  --#{$prefix}toast-header-border-color: #{$toast-header-border-color};\n  // scss-docs-end toast-css-vars\n\n  width: var(--#{$prefix}toast-max-width);\n  max-width: 100%;\n  @include font-size(var(--#{$prefix}toast-font-size));\n  color: var(--#{$prefix}toast-color);\n  pointer-events: auto;\n  background-color: var(--#{$prefix}toast-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}toast-border-width) solid var(--#{$prefix}toast-border-color);\n  box-shadow: var(--#{$prefix}toast-box-shadow);\n  @include border-radius(var(--#{$prefix}toast-border-radius));\n\n  &.showing {\n    opacity: 0;\n  }\n\n  &:not(.show) {\n    display: none;\n  }\n}\n\n.toast-container {\n  --#{$prefix}toast-zindex: #{$zindex-toast};\n\n  position: absolute;\n  z-index: var(--#{$prefix}toast-zindex);\n  width: max-content;\n  max-width: 100%;\n  pointer-events: none;\n\n  > :not(:last-child) {\n    margin-bottom: var(--#{$prefix}toast-spacing);\n  }\n}\n\n.toast-header {\n  display: flex;\n  align-items: center;\n  padding: var(--#{$prefix}toast-padding-y) var(--#{$prefix}toast-padding-x);\n  color: var(--#{$prefix}toast-header-color);\n  background-color: var(--#{$prefix}toast-header-bg);\n  background-clip: padding-box;\n  border-bottom: var(--#{$prefix}toast-border-width) solid var(--#{$prefix}toast-header-border-color);\n  @include border-top-radius(calc(var(--#{$prefix}toast-border-radius) - var(--#{$prefix}toast-border-width)));\n\n  .btn-close {\n    margin-right: calc(-.5 * var(--#{$prefix}toast-padding-x)); // stylelint-disable-line function-disallowed-list\n    margin-left: var(--#{$prefix}toast-padding-x);\n  }\n}\n\n.toast-body {\n  padding: var(--#{$prefix}toast-padding-x);\n  word-wrap: break-word;\n}\n","// stylelint-disable function-disallowed-list\n\n// .modal-open      - body class for killing the scroll\n// .modal           - container to scroll within\n// .modal-dialog    - positioning shell for the actual modal\n// .modal-content   - actual modal w/ bg and corners and stuff\n\n\n// Container that the modal scrolls within\n.modal {\n  // scss-docs-start modal-css-vars\n  --#{$prefix}modal-zindex: #{$zindex-modal};\n  --#{$prefix}modal-width: #{$modal-md};\n  --#{$prefix}modal-padding: #{$modal-inner-padding};\n  --#{$prefix}modal-margin: #{$modal-dialog-margin};\n  --#{$prefix}modal-color: #{$modal-content-color};\n  --#{$prefix}modal-bg: #{$modal-content-bg};\n  --#{$prefix}modal-border-color: #{$modal-content-border-color};\n  --#{$prefix}modal-border-width: #{$modal-content-border-width};\n  --#{$prefix}modal-border-radius: #{$modal-content-border-radius};\n  --#{$prefix}modal-box-shadow: #{$modal-content-box-shadow-xs};\n  --#{$prefix}modal-inner-border-radius: #{$modal-content-inner-border-radius};\n  --#{$prefix}modal-header-padding-x: #{$modal-header-padding-x};\n  --#{$prefix}modal-header-padding-y: #{$modal-header-padding-y};\n  --#{$prefix}modal-header-padding: #{$modal-header-padding}; // Todo in v6: Split this padding into x and y\n  --#{$prefix}modal-header-border-color: #{$modal-header-border-color};\n  --#{$prefix}modal-header-border-width: #{$modal-header-border-width};\n  --#{$prefix}modal-title-line-height: #{$modal-title-line-height};\n  --#{$prefix}modal-footer-gap: #{$modal-footer-margin-between};\n  --#{$prefix}modal-footer-bg: #{$modal-footer-bg};\n  --#{$prefix}modal-footer-border-color: #{$modal-footer-border-color};\n  --#{$prefix}modal-footer-border-width: #{$modal-footer-border-width};\n  // scss-docs-end modal-css-vars\n\n  position: fixed;\n  top: 0;\n  left: 0;\n  z-index: var(--#{$prefix}modal-zindex);\n  display: none;\n  width: 100%;\n  height: 100%;\n  overflow-x: hidden;\n  overflow-y: auto;\n  // Prevent Chrome on Windows from adding a focus outline. For details, see\n  // https://github.com/twbs/bootstrap/pull/10951.\n  outline: 0;\n  // We deliberately don't use `-webkit-overflow-scrolling: touch;` due to a\n  // gnarly iOS Safari bug: https://bugs.webkit.org/show_bug.cgi?id=158342\n  // See also https://github.com/twbs/bootstrap/issues/17695\n}\n\n// Shell div to position the modal with bottom padding\n.modal-dialog {\n  position: relative;\n  width: auto;\n  margin: var(--#{$prefix}modal-margin);\n  // allow clicks to pass through for custom click handling to close modal\n  pointer-events: none;\n\n  // When fading in the modal, animate it to slide down\n  .modal.fade & {\n    @include transition($modal-transition);\n    transform: $modal-fade-transform;\n  }\n  .modal.show & {\n    transform: $modal-show-transform;\n  }\n\n  // When trying to close, animate focus to scale\n  .modal.modal-static & {\n    transform: $modal-scale-transform;\n  }\n}\n\n.modal-dialog-scrollable {\n  height: calc(100% - var(--#{$prefix}modal-margin) * 2);\n\n  .modal-content {\n    max-height: 100%;\n    overflow: hidden;\n  }\n\n  .modal-body {\n    overflow-y: auto;\n  }\n}\n\n.modal-dialog-centered {\n  display: flex;\n  align-items: center;\n  min-height: calc(100% - var(--#{$prefix}modal-margin) * 2);\n}\n\n// Actual modal\n.modal-content {\n  position: relative;\n  display: flex;\n  flex-direction: column;\n  width: 100%; // Ensure `.modal-content` extends the full width of the parent `.modal-dialog`\n  // counteract the pointer-events: none; in the .modal-dialog\n  color: var(--#{$prefix}modal-color);\n  pointer-events: auto;\n  background-color: var(--#{$prefix}modal-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}modal-border-width) solid var(--#{$prefix}modal-border-color);\n  @include border-radius(var(--#{$prefix}modal-border-radius));\n  @include box-shadow(var(--#{$prefix}modal-box-shadow));\n  // Remove focus outline from opened modal\n  outline: 0;\n}\n\n// Modal background\n.modal-backdrop {\n  // scss-docs-start modal-backdrop-css-vars\n  --#{$prefix}backdrop-zindex: #{$zindex-modal-backdrop};\n  --#{$prefix}backdrop-bg: #{$modal-backdrop-bg};\n  --#{$prefix}backdrop-opacity: #{$modal-backdrop-opacity};\n  // scss-docs-end modal-backdrop-css-vars\n\n  @include overlay-backdrop(var(--#{$prefix}backdrop-zindex), var(--#{$prefix}backdrop-bg), var(--#{$prefix}backdrop-opacity));\n}\n\n// Modal header\n// Top section of the modal w/ title and dismiss\n.modal-header {\n  display: flex;\n  flex-shrink: 0;\n  align-items: center;\n  justify-content: space-between; // Put modal header elements (title and dismiss) on opposite ends\n  padding: var(--#{$prefix}modal-header-padding);\n  border-bottom: var(--#{$prefix}modal-header-border-width) solid var(--#{$prefix}modal-header-border-color);\n  @include border-top-radius(var(--#{$prefix}modal-inner-border-radius));\n\n  .btn-close {\n    padding: calc(var(--#{$prefix}modal-header-padding-y) * .5) calc(var(--#{$prefix}modal-header-padding-x) * .5);\n    margin: calc(-.5 * var(--#{$prefix}modal-header-padding-y)) calc(-.5 * var(--#{$prefix}modal-header-padding-x)) calc(-.5 * var(--#{$prefix}modal-header-padding-y)) auto;\n  }\n}\n\n// Title text within header\n.modal-title {\n  margin-bottom: 0;\n  line-height: var(--#{$prefix}modal-title-line-height);\n}\n\n// Modal body\n// Where all modal content resides (sibling of .modal-header and .modal-footer)\n.modal-body {\n  position: relative;\n  // Enable `flex-grow: 1` so that the body take up as much space as possible\n  // when there should be a fixed height on `.modal-dialog`.\n  flex: 1 1 auto;\n  padding: var(--#{$prefix}modal-padding);\n}\n\n// Footer (for actions)\n.modal-footer {\n  display: flex;\n  flex-shrink: 0;\n  flex-wrap: wrap;\n  align-items: center; // vertically center\n  justify-content: flex-end; // Right align buttons with flex property because text-align doesn't work on flex items\n  padding: calc(var(--#{$prefix}modal-padding) - var(--#{$prefix}modal-footer-gap) * .5);\n  background-color: var(--#{$prefix}modal-footer-bg);\n  border-top: var(--#{$prefix}modal-footer-border-width) solid var(--#{$prefix}modal-footer-border-color);\n  @include border-bottom-radius(var(--#{$prefix}modal-inner-border-radius));\n\n  // Place margin between footer elements\n  // This solution is far from ideal because of the universal selector usage,\n  // but is needed to fix https://github.com/twbs/bootstrap/issues/24800\n  > * {\n    margin: calc(var(--#{$prefix}modal-footer-gap) * .5); // Todo in v6: replace with gap on parent class\n  }\n}\n\n// Scale up the modal\n@include media-breakpoint-up(sm) {\n  .modal {\n    --#{$prefix}modal-margin: #{$modal-dialog-margin-y-sm-up};\n    --#{$prefix}modal-box-shadow: #{$modal-content-box-shadow-sm-up};\n  }\n\n  // Automatically set modal's width for larger viewports\n  .modal-dialog {\n    max-width: var(--#{$prefix}modal-width);\n    margin-right: auto;\n    margin-left: auto;\n  }\n\n  .modal-sm {\n    --#{$prefix}modal-width: #{$modal-sm};\n  }\n}\n\n@include media-breakpoint-up(lg) {\n  .modal-lg,\n  .modal-xl {\n    --#{$prefix}modal-width: #{$modal-lg};\n  }\n}\n\n@include media-breakpoint-up(xl) {\n  .modal-xl {\n    --#{$prefix}modal-width: #{$modal-xl};\n  }\n}\n\n// scss-docs-start modal-fullscreen-loop\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n  $postfix: if($infix != \"\", $infix + \"-down\", \"\");\n\n  @include media-breakpoint-down($breakpoint) {\n    .modal-fullscreen#{$postfix} {\n      width: 100vw;\n      max-width: none;\n      height: 100%;\n      margin: 0;\n\n      .modal-content {\n        height: 100%;\n        border: 0;\n        @include border-radius(0);\n      }\n\n      .modal-header,\n      .modal-footer {\n        @include border-radius(0);\n      }\n\n      .modal-body {\n        overflow-y: auto;\n      }\n    }\n  }\n}\n// scss-docs-end modal-fullscreen-loop\n","// Shared between modals and offcanvases\n@mixin overlay-backdrop($zindex, $backdrop-bg, $backdrop-opacity) {\n  position: fixed;\n  top: 0;\n  left: 0;\n  z-index: $zindex;\n  width: 100vw;\n  height: 100vh;\n  background-color: $backdrop-bg;\n\n  // Fade for backdrop\n  &.fade { opacity: 0; }\n  &.show { opacity: $backdrop-opacity; }\n}\n","// Base class\n.tooltip {\n  // scss-docs-start tooltip-css-vars\n  --#{$prefix}tooltip-zindex: #{$zindex-tooltip};\n  --#{$prefix}tooltip-max-width: #{$tooltip-max-width};\n  --#{$prefix}tooltip-padding-x: #{$tooltip-padding-x};\n  --#{$prefix}tooltip-padding-y: #{$tooltip-padding-y};\n  --#{$prefix}tooltip-margin: #{$tooltip-margin};\n  @include rfs($tooltip-font-size, --#{$prefix}tooltip-font-size);\n  --#{$prefix}tooltip-color: #{$tooltip-color};\n  --#{$prefix}tooltip-bg: #{$tooltip-bg};\n  --#{$prefix}tooltip-border-radius: #{$tooltip-border-radius};\n  --#{$prefix}tooltip-opacity: #{$tooltip-opacity};\n  --#{$prefix}tooltip-arrow-width: #{$tooltip-arrow-width};\n  --#{$prefix}tooltip-arrow-height: #{$tooltip-arrow-height};\n  // scss-docs-end tooltip-css-vars\n\n  z-index: var(--#{$prefix}tooltip-zindex);\n  display: block;\n  margin: var(--#{$prefix}tooltip-margin);\n  @include deprecate(\"`$tooltip-margin`\", \"v5\", \"v5.x\", true);\n  // Our parent element can be arbitrary since tooltips are by default inserted as a sibling of their target element.\n  // So reset our font and text properties to avoid inheriting weird values.\n  @include reset-text();\n  @include font-size(var(--#{$prefix}tooltip-font-size));\n  // Allow breaking very long words so they don't overflow the tooltip's bounds\n  word-wrap: break-word;\n  opacity: 0;\n\n  &.show { opacity: var(--#{$prefix}tooltip-opacity); }\n\n  .tooltip-arrow {\n    display: block;\n    width: var(--#{$prefix}tooltip-arrow-width);\n    height: var(--#{$prefix}tooltip-arrow-height);\n\n    &::before {\n      position: absolute;\n      content: \"\";\n      border-color: transparent;\n      border-style: solid;\n    }\n  }\n}\n\n.bs-tooltip-top .tooltip-arrow {\n  bottom: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n\n  &::before {\n    top: -1px;\n    border-width: var(--#{$prefix}tooltip-arrow-height) calc(var(--#{$prefix}tooltip-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    border-top-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-tooltip-end .tooltip-arrow {\n  left: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n  width: var(--#{$prefix}tooltip-arrow-height);\n  height: var(--#{$prefix}tooltip-arrow-width);\n\n  &::before {\n    right: -1px;\n    border-width: calc(var(--#{$prefix}tooltip-arrow-width) * .5) var(--#{$prefix}tooltip-arrow-height) calc(var(--#{$prefix}tooltip-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    border-right-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-tooltip-bottom .tooltip-arrow {\n  top: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n\n  &::before {\n    bottom: -1px;\n    border-width: 0 calc(var(--#{$prefix}tooltip-arrow-width) * .5) var(--#{$prefix}tooltip-arrow-height); // stylelint-disable-line function-disallowed-list\n    border-bottom-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-tooltip-start .tooltip-arrow {\n  right: calc(-1 * var(--#{$prefix}tooltip-arrow-height)); // stylelint-disable-line function-disallowed-list\n  width: var(--#{$prefix}tooltip-arrow-height);\n  height: var(--#{$prefix}tooltip-arrow-width);\n\n  &::before {\n    left: -1px;\n    border-width: calc(var(--#{$prefix}tooltip-arrow-width) * .5) 0 calc(var(--#{$prefix}tooltip-arrow-width) * .5) var(--#{$prefix}tooltip-arrow-height); // stylelint-disable-line function-disallowed-list\n    border-left-color: var(--#{$prefix}tooltip-bg);\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-tooltip-auto {\n  &[data-popper-placement^=\"top\"] {\n    @extend .bs-tooltip-top;\n  }\n  &[data-popper-placement^=\"right\"] {\n    @extend .bs-tooltip-end;\n  }\n  &[data-popper-placement^=\"bottom\"] {\n    @extend .bs-tooltip-bottom;\n  }\n  &[data-popper-placement^=\"left\"] {\n    @extend .bs-tooltip-start;\n  }\n}\n\n// Wrapper for the tooltip content\n.tooltip-inner {\n  max-width: var(--#{$prefix}tooltip-max-width);\n  padding: var(--#{$prefix}tooltip-padding-y) var(--#{$prefix}tooltip-padding-x);\n  color: var(--#{$prefix}tooltip-color);\n  text-align: center;\n  background-color: var(--#{$prefix}tooltip-bg);\n  @include border-radius(var(--#{$prefix}tooltip-border-radius));\n}\n","@mixin reset-text {\n  font-family: $font-family-base;\n  // We deliberately do NOT reset font-size or overflow-wrap / word-wrap.\n  font-style: normal;\n  font-weight: $font-weight-normal;\n  line-height: $line-height-base;\n  text-align: left; // Fallback for where `start` is not supported\n  text-align: start;\n  text-decoration: none;\n  text-shadow: none;\n  text-transform: none;\n  letter-spacing: normal;\n  word-break: normal;\n  white-space: normal;\n  word-spacing: normal;\n  line-break: auto;\n}\n",".popover {\n  // scss-docs-start popover-css-vars\n  --#{$prefix}popover-zindex: #{$zindex-popover};\n  --#{$prefix}popover-max-width: #{$popover-max-width};\n  @include rfs($popover-font-size, --#{$prefix}popover-font-size);\n  --#{$prefix}popover-bg: #{$popover-bg};\n  --#{$prefix}popover-border-width: #{$popover-border-width};\n  --#{$prefix}popover-border-color: #{$popover-border-color};\n  --#{$prefix}popover-border-radius: #{$popover-border-radius};\n  --#{$prefix}popover-inner-border-radius: #{$popover-inner-border-radius};\n  --#{$prefix}popover-box-shadow: #{$popover-box-shadow};\n  --#{$prefix}popover-header-padding-x: #{$popover-header-padding-x};\n  --#{$prefix}popover-header-padding-y: #{$popover-header-padding-y};\n  @include rfs($popover-header-font-size, --#{$prefix}popover-header-font-size);\n  --#{$prefix}popover-header-color: #{$popover-header-color};\n  --#{$prefix}popover-header-bg: #{$popover-header-bg};\n  --#{$prefix}popover-body-padding-x: #{$popover-body-padding-x};\n  --#{$prefix}popover-body-padding-y: #{$popover-body-padding-y};\n  --#{$prefix}popover-body-color: #{$popover-body-color};\n  --#{$prefix}popover-arrow-width: #{$popover-arrow-width};\n  --#{$prefix}popover-arrow-height: #{$popover-arrow-height};\n  --#{$prefix}popover-arrow-border: var(--#{$prefix}popover-border-color);\n  // scss-docs-end popover-css-vars\n\n  z-index: var(--#{$prefix}popover-zindex);\n  display: block;\n  max-width: var(--#{$prefix}popover-max-width);\n  // Our parent element can be arbitrary since tooltips are by default inserted as a sibling of their target element.\n  // So reset our font and text properties to avoid inheriting weird values.\n  @include reset-text();\n  @include font-size(var(--#{$prefix}popover-font-size));\n  // Allow breaking very long words so they don't overflow the popover's bounds\n  word-wrap: break-word;\n  background-color: var(--#{$prefix}popover-bg);\n  background-clip: padding-box;\n  border: var(--#{$prefix}popover-border-width) solid var(--#{$prefix}popover-border-color);\n  @include border-radius(var(--#{$prefix}popover-border-radius));\n  @include box-shadow(var(--#{$prefix}popover-box-shadow));\n\n  .popover-arrow {\n    display: block;\n    width: var(--#{$prefix}popover-arrow-width);\n    height: var(--#{$prefix}popover-arrow-height);\n\n    &::before,\n    &::after {\n      position: absolute;\n      display: block;\n      content: \"\";\n      border-color: transparent;\n      border-style: solid;\n      border-width: 0;\n    }\n  }\n}\n\n.bs-popover-top {\n  > .popover-arrow {\n    bottom: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n\n    &::before,\n    &::after {\n      border-width: var(--#{$prefix}popover-arrow-height) calc(var(--#{$prefix}popover-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      bottom: 0;\n      border-top-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      bottom: var(--#{$prefix}popover-border-width);\n      border-top-color: var(--#{$prefix}popover-bg);\n    }\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-popover-end {\n  > .popover-arrow {\n    left: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n    width: var(--#{$prefix}popover-arrow-height);\n    height: var(--#{$prefix}popover-arrow-width);\n\n    &::before,\n    &::after {\n      border-width: calc(var(--#{$prefix}popover-arrow-width) * .5) var(--#{$prefix}popover-arrow-height) calc(var(--#{$prefix}popover-arrow-width) * .5) 0; // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      left: 0;\n      border-right-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      left: var(--#{$prefix}popover-border-width);\n      border-right-color: var(--#{$prefix}popover-bg);\n    }\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-popover-bottom {\n  > .popover-arrow {\n    top: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n\n    &::before,\n    &::after {\n      border-width: 0 calc(var(--#{$prefix}popover-arrow-width) * .5) var(--#{$prefix}popover-arrow-height); // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      top: 0;\n      border-bottom-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      top: var(--#{$prefix}popover-border-width);\n      border-bottom-color: var(--#{$prefix}popover-bg);\n    }\n  }\n\n  // This will remove the popover-header's border just below the arrow\n  .popover-header::before {\n    position: absolute;\n    top: 0;\n    left: 50%;\n    display: block;\n    width: var(--#{$prefix}popover-arrow-width);\n    margin-left: calc(-.5 * var(--#{$prefix}popover-arrow-width)); // stylelint-disable-line function-disallowed-list\n    content: \"\";\n    border-bottom: var(--#{$prefix}popover-border-width) solid var(--#{$prefix}popover-header-bg);\n  }\n}\n\n/* rtl:begin:ignore */\n.bs-popover-start {\n  > .popover-arrow {\n    right: calc(-1 * (var(--#{$prefix}popover-arrow-height)) - var(--#{$prefix}popover-border-width)); // stylelint-disable-line function-disallowed-list\n    width: var(--#{$prefix}popover-arrow-height);\n    height: var(--#{$prefix}popover-arrow-width);\n\n    &::before,\n    &::after {\n      border-width: calc(var(--#{$prefix}popover-arrow-width) * .5) 0 calc(var(--#{$prefix}popover-arrow-width) * .5) var(--#{$prefix}popover-arrow-height); // stylelint-disable-line function-disallowed-list\n    }\n\n    &::before {\n      right: 0;\n      border-left-color: var(--#{$prefix}popover-arrow-border);\n    }\n\n    &::after {\n      right: var(--#{$prefix}popover-border-width);\n      border-left-color: var(--#{$prefix}popover-bg);\n    }\n  }\n}\n\n/* rtl:end:ignore */\n\n.bs-popover-auto {\n  &[data-popper-placement^=\"top\"] {\n    @extend .bs-popover-top;\n  }\n  &[data-popper-placement^=\"right\"] {\n    @extend .bs-popover-end;\n  }\n  &[data-popper-placement^=\"bottom\"] {\n    @extend .bs-popover-bottom;\n  }\n  &[data-popper-placement^=\"left\"] {\n    @extend .bs-popover-start;\n  }\n}\n\n// Offset the popover to account for the popover arrow\n.popover-header {\n  padding: var(--#{$prefix}popover-header-padding-y) var(--#{$prefix}popover-header-padding-x);\n  margin-bottom: 0; // Reset the default from Reboot\n  @include font-size(var(--#{$prefix}popover-header-font-size));\n  color: var(--#{$prefix}popover-header-color);\n  background-color: var(--#{$prefix}popover-header-bg);\n  border-bottom: var(--#{$prefix}popover-border-width) solid var(--#{$prefix}popover-border-color);\n  @include border-top-radius(var(--#{$prefix}popover-inner-border-radius));\n\n  &:empty {\n    display: none;\n  }\n}\n\n.popover-body {\n  padding: var(--#{$prefix}popover-body-padding-y) var(--#{$prefix}popover-body-padding-x);\n  color: var(--#{$prefix}popover-body-color);\n}\n","// Notes on the classes:\n//\n// 1. .carousel.pointer-event should ideally be pan-y (to allow for users to scroll vertically)\n//    even when their scroll action started on a carousel, but for compatibility (with Firefox)\n//    we're preventing all actions instead\n// 2. The .carousel-item-start and .carousel-item-end is used to indicate where\n//    the active slide is heading.\n// 3. .active.carousel-item is the current slide.\n// 4. .active.carousel-item-start and .active.carousel-item-end is the current\n//    slide in its in-transition state. Only one of these occurs at a time.\n// 5. .carousel-item-next.carousel-item-start and .carousel-item-prev.carousel-item-end\n//    is the upcoming slide in transition.\n\n.carousel {\n  position: relative;\n}\n\n.carousel.pointer-event {\n  touch-action: pan-y;\n}\n\n.carousel-inner {\n  position: relative;\n  width: 100%;\n  overflow: hidden;\n  @include clearfix();\n}\n\n.carousel-item {\n  position: relative;\n  display: none;\n  float: left;\n  width: 100%;\n  margin-right: -100%;\n  backface-visibility: hidden;\n  @include transition($carousel-transition);\n}\n\n.carousel-item.active,\n.carousel-item-next,\n.carousel-item-prev {\n  display: block;\n}\n\n.carousel-item-next:not(.carousel-item-start),\n.active.carousel-item-end {\n  transform: translateX(100%);\n}\n\n.carousel-item-prev:not(.carousel-item-end),\n.active.carousel-item-start {\n  transform: translateX(-100%);\n}\n\n\n//\n// Alternate transitions\n//\n\n.carousel-fade {\n  .carousel-item {\n    opacity: 0;\n    transition-property: opacity;\n    transform: none;\n  }\n\n  .carousel-item.active,\n  .carousel-item-next.carousel-item-start,\n  .carousel-item-prev.carousel-item-end {\n    z-index: 1;\n    opacity: 1;\n  }\n\n  .active.carousel-item-start,\n  .active.carousel-item-end {\n    z-index: 0;\n    opacity: 0;\n    @include transition(opacity 0s $carousel-transition-duration);\n  }\n}\n\n\n//\n// Left/right controls for nav\n//\n\n.carousel-control-prev,\n.carousel-control-next {\n  position: absolute;\n  top: 0;\n  bottom: 0;\n  z-index: 1;\n  // Use flex for alignment (1-3)\n  display: flex; // 1. allow flex styles\n  align-items: center; // 2. vertically center contents\n  justify-content: center; // 3. horizontally center contents\n  width: $carousel-control-width;\n  padding: 0;\n  color: $carousel-control-color;\n  text-align: center;\n  background: none;\n  border: 0;\n  opacity: $carousel-control-opacity;\n  @include transition($carousel-control-transition);\n\n  // Hover/focus state\n  &:hover,\n  &:focus {\n    color: $carousel-control-color;\n    text-decoration: none;\n    outline: 0;\n    opacity: $carousel-control-hover-opacity;\n  }\n}\n.carousel-control-prev {\n  left: 0;\n  background-image: if($enable-gradients, linear-gradient(90deg, rgba($black, .25), rgba($black, .001)), null);\n}\n.carousel-control-next {\n  right: 0;\n  background-image: if($enable-gradients, linear-gradient(270deg, rgba($black, .25), rgba($black, .001)), null);\n}\n\n// Icons for within\n.carousel-control-prev-icon,\n.carousel-control-next-icon {\n  display: inline-block;\n  width: $carousel-control-icon-width;\n  height: $carousel-control-icon-width;\n  background-repeat: no-repeat;\n  background-position: 50%;\n  background-size: 100% 100%;\n}\n\n/* rtl:options: {\n  \"autoRename\": true,\n  \"stringMap\":[ {\n    \"name\"    : \"prev-next\",\n    \"search\"  : \"prev\",\n    \"replace\" : \"next\"\n  } ]\n} */\n.carousel-control-prev-icon {\n  background-image: escape-svg($carousel-control-prev-icon-bg);\n}\n.carousel-control-next-icon {\n  background-image: escape-svg($carousel-control-next-icon-bg);\n}\n\n// Optional indicator pips/controls\n//\n// Add a container (such as a list) with the following class and add an item (ideally a focusable control,\n// like a button) with data-bs-target for each slide your carousel holds.\n\n.carousel-indicators {\n  position: absolute;\n  right: 0;\n  bottom: 0;\n  left: 0;\n  z-index: 2;\n  display: flex;\n  justify-content: center;\n  padding: 0;\n  // Use the .carousel-control's width as margin so we don't overlay those\n  margin-right: $carousel-control-width;\n  margin-bottom: 1rem;\n  margin-left: $carousel-control-width;\n\n  [data-bs-target] {\n    box-sizing: content-box;\n    flex: 0 1 auto;\n    width: $carousel-indicator-width;\n    height: $carousel-indicator-height;\n    padding: 0;\n    margin-right: $carousel-indicator-spacer;\n    margin-left: $carousel-indicator-spacer;\n    text-indent: -999px;\n    cursor: pointer;\n    background-color: $carousel-indicator-active-bg;\n    background-clip: padding-box;\n    border: 0;\n    // Use transparent borders to increase the hit area by 10px on top and bottom.\n    border-top: $carousel-indicator-hit-area-height solid transparent;\n    border-bottom: $carousel-indicator-hit-area-height solid transparent;\n    opacity: $carousel-indicator-opacity;\n    @include transition($carousel-indicator-transition);\n  }\n\n  .active {\n    opacity: $carousel-indicator-active-opacity;\n  }\n}\n\n\n// Optional captions\n//\n//\n\n.carousel-caption {\n  position: absolute;\n  right: (100% - $carousel-caption-width) * .5;\n  bottom: $carousel-caption-spacer;\n  left: (100% - $carousel-caption-width) * .5;\n  padding-top: $carousel-caption-padding-y;\n  padding-bottom: $carousel-caption-padding-y;\n  color: $carousel-caption-color;\n  text-align: center;\n}\n\n// Dark mode carousel\n\n@mixin carousel-dark() {\n  .carousel-control-prev-icon,\n  .carousel-control-next-icon {\n    filter: $carousel-dark-control-icon-filter;\n  }\n\n  .carousel-indicators [data-bs-target] {\n    background-color: $carousel-dark-indicator-active-bg;\n  }\n\n  .carousel-caption {\n    color: $carousel-dark-caption-color;\n  }\n}\n\n.carousel-dark {\n  @include carousel-dark();\n}\n\n@if $enable-dark-mode {\n  @include color-mode(dark) {\n    @if $color-mode-type == \"media-query\" {\n      .carousel {\n        @include carousel-dark();\n      }\n    } @else {\n      .carousel,\n      &.carousel {\n        @include carousel-dark();\n      }\n    }\n  }\n}\n","// scss-docs-start clearfix\n@mixin clearfix() {\n  &::after {\n    display: block;\n    clear: both;\n    content: \"\";\n  }\n}\n// scss-docs-end clearfix\n","//\n// Rotating border\n//\n\n.spinner-grow,\n.spinner-border {\n  display: inline-block;\n  width: var(--#{$prefix}spinner-width);\n  height: var(--#{$prefix}spinner-height);\n  vertical-align: var(--#{$prefix}spinner-vertical-align);\n  // stylelint-disable-next-line property-disallowed-list\n  border-radius: 50%;\n  animation: var(--#{$prefix}spinner-animation-speed) linear infinite var(--#{$prefix}spinner-animation-name);\n}\n\n// scss-docs-start spinner-border-keyframes\n@keyframes spinner-border {\n  to { transform: rotate(360deg) #{\"/* rtl:ignore */\"}; }\n}\n// scss-docs-end spinner-border-keyframes\n\n.spinner-border {\n  // scss-docs-start spinner-border-css-vars\n  --#{$prefix}spinner-width: #{$spinner-width};\n  --#{$prefix}spinner-height: #{$spinner-height};\n  --#{$prefix}spinner-vertical-align: #{$spinner-vertical-align};\n  --#{$prefix}spinner-border-width: #{$spinner-border-width};\n  --#{$prefix}spinner-animation-speed: #{$spinner-animation-speed};\n  --#{$prefix}spinner-animation-name: spinner-border;\n  // scss-docs-end spinner-border-css-vars\n\n  border: var(--#{$prefix}spinner-border-width) solid currentcolor;\n  border-right-color: transparent;\n}\n\n.spinner-border-sm {\n  // scss-docs-start spinner-border-sm-css-vars\n  --#{$prefix}spinner-width: #{$spinner-width-sm};\n  --#{$prefix}spinner-height: #{$spinner-height-sm};\n  --#{$prefix}spinner-border-width: #{$spinner-border-width-sm};\n  // scss-docs-end spinner-border-sm-css-vars\n}\n\n//\n// Growing circle\n//\n\n// scss-docs-start spinner-grow-keyframes\n@keyframes spinner-grow {\n  0% {\n    transform: scale(0);\n  }\n  50% {\n    opacity: 1;\n    transform: none;\n  }\n}\n// scss-docs-end spinner-grow-keyframes\n\n.spinner-grow {\n  // scss-docs-start spinner-grow-css-vars\n  --#{$prefix}spinner-width: #{$spinner-width};\n  --#{$prefix}spinner-height: #{$spinner-height};\n  --#{$prefix}spinner-vertical-align: #{$spinner-vertical-align};\n  --#{$prefix}spinner-animation-speed: #{$spinner-animation-speed};\n  --#{$prefix}spinner-animation-name: spinner-grow;\n  // scss-docs-end spinner-grow-css-vars\n\n  background-color: currentcolor;\n  opacity: 0;\n}\n\n.spinner-grow-sm {\n  --#{$prefix}spinner-width: #{$spinner-width-sm};\n  --#{$prefix}spinner-height: #{$spinner-height-sm};\n}\n\n@if $enable-reduced-motion {\n  @media (prefers-reduced-motion: reduce) {\n    .spinner-border,\n    .spinner-grow {\n      --#{$prefix}spinner-animation-speed: #{$spinner-animation-speed * 2};\n    }\n  }\n}\n","// stylelint-disable function-disallowed-list\n\n%offcanvas-css-vars {\n  // scss-docs-start offcanvas-css-vars\n  --#{$prefix}offcanvas-zindex: #{$zindex-offcanvas};\n  --#{$prefix}offcanvas-width: #{$offcanvas-horizontal-width};\n  --#{$prefix}offcanvas-height: #{$offcanvas-vertical-height};\n  --#{$prefix}offcanvas-padding-x: #{$offcanvas-padding-x};\n  --#{$prefix}offcanvas-padding-y: #{$offcanvas-padding-y};\n  --#{$prefix}offcanvas-color: #{$offcanvas-color};\n  --#{$prefix}offcanvas-bg: #{$offcanvas-bg-color};\n  --#{$prefix}offcanvas-border-width: #{$offcanvas-border-width};\n  --#{$prefix}offcanvas-border-color: #{$offcanvas-border-color};\n  --#{$prefix}offcanvas-box-shadow: #{$offcanvas-box-shadow};\n  --#{$prefix}offcanvas-transition: #{transform $offcanvas-transition-duration ease-in-out};\n  --#{$prefix}offcanvas-title-line-height: #{$offcanvas-title-line-height};\n  // scss-docs-end offcanvas-css-vars\n}\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $next: breakpoint-next($breakpoint, $grid-breakpoints);\n  $infix: breakpoint-infix($next, $grid-breakpoints);\n\n  .offcanvas#{$infix} {\n    @extend %offcanvas-css-vars;\n  }\n}\n\n@each $breakpoint in map-keys($grid-breakpoints) {\n  $next: breakpoint-next($breakpoint, $grid-breakpoints);\n  $infix: breakpoint-infix($next, $grid-breakpoints);\n\n  .offcanvas#{$infix} {\n    @include media-breakpoint-down($next) {\n      position: fixed;\n      bottom: 0;\n      z-index: var(--#{$prefix}offcanvas-zindex);\n      display: flex;\n      flex-direction: column;\n      max-width: 100%;\n      color: var(--#{$prefix}offcanvas-color);\n      visibility: hidden;\n      background-color: var(--#{$prefix}offcanvas-bg);\n      background-clip: padding-box;\n      outline: 0;\n      @include box-shadow(var(--#{$prefix}offcanvas-box-shadow));\n      @include transition(var(--#{$prefix}offcanvas-transition));\n\n      &.offcanvas-start {\n        top: 0;\n        left: 0;\n        width: var(--#{$prefix}offcanvas-width);\n        border-right: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateX(-100%);\n      }\n\n      &.offcanvas-end {\n        top: 0;\n        right: 0;\n        width: var(--#{$prefix}offcanvas-width);\n        border-left: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateX(100%);\n      }\n\n      &.offcanvas-top {\n        top: 0;\n        right: 0;\n        left: 0;\n        height: var(--#{$prefix}offcanvas-height);\n        max-height: 100%;\n        border-bottom: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateY(-100%);\n      }\n\n      &.offcanvas-bottom {\n        right: 0;\n        left: 0;\n        height: var(--#{$prefix}offcanvas-height);\n        max-height: 100%;\n        border-top: var(--#{$prefix}offcanvas-border-width) solid var(--#{$prefix}offcanvas-border-color);\n        transform: translateY(100%);\n      }\n\n      &.showing,\n      &.show:not(.hiding) {\n        transform: none;\n      }\n\n      &.showing,\n      &.hiding,\n      &.show {\n        visibility: visible;\n      }\n    }\n\n    @if not ($infix == \"\") {\n      @include media-breakpoint-up($next) {\n        --#{$prefix}offcanvas-height: auto;\n        --#{$prefix}offcanvas-border-width: 0;\n        background-color: transparent !important; // stylelint-disable-line declaration-no-important\n\n        .offcanvas-header {\n          display: none;\n        }\n\n        .offcanvas-body {\n          display: flex;\n          flex-grow: 0;\n          padding: 0;\n          overflow-y: visible;\n          // Reset `background-color` in case `.bg-*` classes are used in offcanvas\n          background-color: transparent !important; // stylelint-disable-line declaration-no-important\n        }\n      }\n    }\n  }\n}\n\n.offcanvas-backdrop {\n  @include overlay-backdrop($zindex-offcanvas-backdrop, $offcanvas-backdrop-bg, $offcanvas-backdrop-opacity);\n}\n\n.offcanvas-header {\n  display: flex;\n  align-items: center;\n  justify-content: space-between;\n  padding: var(--#{$prefix}offcanvas-padding-y) var(--#{$prefix}offcanvas-padding-x);\n\n  .btn-close {\n    padding: calc(var(--#{$prefix}offcanvas-padding-y) * .5) calc(var(--#{$prefix}offcanvas-padding-x) * .5);\n    margin-top: calc(-.5 * var(--#{$prefix}offcanvas-padding-y));\n    margin-right: calc(-.5 * var(--#{$prefix}offcanvas-padding-x));\n    margin-bottom: calc(-.5 * var(--#{$prefix}offcanvas-padding-y));\n  }\n}\n\n.offcanvas-title {\n  margin-bottom: 0;\n  line-height: var(--#{$prefix}offcanvas-title-line-height);\n}\n\n.offcanvas-body {\n  flex-grow: 1;\n  padding: var(--#{$prefix}offcanvas-padding-y) var(--#{$prefix}offcanvas-padding-x);\n  overflow-y: auto;\n}\n",".placeholder {\n  display: inline-block;\n  min-height: 1em;\n  vertical-align: middle;\n  cursor: wait;\n  background-color: currentcolor;\n  opacity: $placeholder-opacity-max;\n\n  &.btn::before {\n    display: inline-block;\n    content: \"\";\n  }\n}\n\n// Sizing\n.placeholder-xs {\n  min-height: .6em;\n}\n\n.placeholder-sm {\n  min-height: .8em;\n}\n\n.placeholder-lg {\n  min-height: 1.2em;\n}\n\n// Animation\n.placeholder-glow {\n  .placeholder {\n    animation: placeholder-glow 2s ease-in-out infinite;\n  }\n}\n\n@keyframes placeholder-glow {\n  50% {\n    opacity: $placeholder-opacity-min;\n  }\n}\n\n.placeholder-wave {\n  mask-image: linear-gradient(130deg, $black 55%, rgba(0, 0, 0, (1 - $placeholder-opacity-min)) 75%, $black 95%);\n  mask-size: 200% 100%;\n  animation: placeholder-wave 2s linear infinite;\n}\n\n@keyframes placeholder-wave {\n  100% {\n    mask-position: -200% 0%;\n  }\n}\n","// All-caps `RGBA()` function used because of this Sass bug: https://github.com/sass/node-sass/issues/2251\n@each $color, $value in $theme-colors {\n  .text-bg-#{$color} {\n    color: color-contrast($value) if($enable-important-utilities, !important, null);\n    background-color: RGBA(var(--#{$prefix}#{$color}-rgb), var(--#{$prefix}bg-opacity, 1)) if($enable-important-utilities, !important, null);\n  }\n}\n","// All-caps `RGBA()` function used because of this Sass bug: https://github.com/sass/node-sass/issues/2251\n@each $color, $value in $theme-colors {\n  .link-#{$color} {\n    color: RGBA(var(--#{$prefix}#{$color}-rgb), var(--#{$prefix}link-opacity, 1)) if($enable-important-utilities, !important, null);\n    text-decoration-color: RGBA(var(--#{$prefix}#{$color}-rgb), var(--#{$prefix}link-underline-opacity, 1)) if($enable-important-utilities, !important, null);\n\n    @if $link-shade-percentage != 0 {\n      &:hover,\n      &:focus {\n        $hover-color: if(color-contrast($value) == $color-contrast-light, shade-color($value, $link-shade-percentage), tint-color($value, $link-shade-percentage));\n        color: RGBA(#{to-rgb($hover-color)}, var(--#{$prefix}link-opacity, 1)) if($enable-important-utilities, !important, null);\n        text-decoration-color: RGBA(to-rgb($hover-color), var(--#{$prefix}link-underline-opacity, 1)) if($enable-important-utilities, !important, null);\n      }\n    }\n  }\n}\n\n// One-off special link helper as a bridge until v6\n.link-body-emphasis {\n  color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-opacity, 1)) if($enable-important-utilities, !important, null);\n  text-decoration-color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-underline-opacity, 1)) if($enable-important-utilities, !important, null);\n\n  @if $link-shade-percentage != 0 {\n    &:hover,\n    &:focus {\n      color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-opacity, .75)) if($enable-important-utilities, !important, null);\n      text-decoration-color: RGBA(var(--#{$prefix}emphasis-color-rgb), var(--#{$prefix}link-underline-opacity, .75)) if($enable-important-utilities, !important, null);\n    }\n  }\n}\n",".focus-ring:focus {\n  outline: 0;\n  // By default, there is no `--bs-focus-ring-x`, `--bs-focus-ring-y`, or `--bs-focus-ring-blur`, but we provide CSS variables with fallbacks to initial `0` values\n  box-shadow: var(--#{$prefix}focus-ring-x, 0) var(--#{$prefix}focus-ring-y, 0) var(--#{$prefix}focus-ring-blur, 0) var(--#{$prefix}focus-ring-width) var(--#{$prefix}focus-ring-color);\n}\n",".icon-link {\n  display: inline-flex;\n  gap: $icon-link-gap;\n  align-items: center;\n  text-decoration-color: rgba(var(--#{$prefix}link-color-rgb), var(--#{$prefix}link-opacity, .5));\n  text-underline-offset: $icon-link-underline-offset;\n  backface-visibility: hidden;\n\n  > .bi {\n    flex-shrink: 0;\n    width: $icon-link-icon-size;\n    height: $icon-link-icon-size;\n    fill: currentcolor;\n    @include transition($icon-link-icon-transition);\n  }\n}\n\n.icon-link-hover {\n  &:hover,\n  &:focus-visible {\n    > .bi {\n      transform: var(--#{$prefix}icon-link-transform, $icon-link-icon-transform);\n    }\n  }\n}\n","// Credit: Nicolas Gallagher and SUIT CSS.\n\n.ratio {\n  position: relative;\n  width: 100%;\n\n  &::before {\n    display: block;\n    padding-top: var(--#{$prefix}aspect-ratio);\n    content: \"\";\n  }\n\n  > * {\n    position: absolute;\n    top: 0;\n    left: 0;\n    width: 100%;\n    height: 100%;\n  }\n}\n\n@each $key, $ratio in $aspect-ratios {\n  .ratio-#{$key} {\n    --#{$prefix}aspect-ratio: #{$ratio};\n  }\n}\n","// Shorthand\n\n.fixed-top {\n  position: fixed;\n  top: 0;\n  right: 0;\n  left: 0;\n  z-index: $zindex-fixed;\n}\n\n.fixed-bottom {\n  position: fixed;\n  right: 0;\n  bottom: 0;\n  left: 0;\n  z-index: $zindex-fixed;\n}\n\n// Responsive sticky top and bottom\n@each $breakpoint in map-keys($grid-breakpoints) {\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    .sticky#{$infix}-top {\n      position: sticky;\n      top: 0;\n      z-index: $zindex-sticky;\n    }\n\n    .sticky#{$infix}-bottom {\n      position: sticky;\n      bottom: 0;\n      z-index: $zindex-sticky;\n    }\n  }\n}\n","// scss-docs-start stacks\n.hstack {\n  display: flex;\n  flex-direction: row;\n  align-items: center;\n  align-self: stretch;\n}\n\n.vstack {\n  display: flex;\n  flex: 1 1 auto;\n  flex-direction: column;\n  align-self: stretch;\n}\n// scss-docs-end stacks\n","//\n// Visually hidden\n//\n\n.visually-hidden,\n.visually-hidden-focusable:not(:focus):not(:focus-within) {\n  @include visually-hidden();\n}\n","// stylelint-disable declaration-no-important\n\n// Hide content visually while keeping it accessible to assistive technologies\n//\n// See: https://www.a11yproject.com/posts/2013-01-11-how-to-hide-content/\n// See: https://kittygiraudel.com/2016/10/13/css-hide-and-seek/\n\n@mixin visually-hidden() {\n  width: 1px !important;\n  height: 1px !important;\n  padding: 0 !important;\n  margin: -1px !important; // Fix for https://github.com/twbs/bootstrap/issues/25686\n  overflow: hidden !important;\n  clip: rect(0, 0, 0, 0) !important;\n  white-space: nowrap !important;\n  border: 0 !important;\n\n  // Fix for positioned table caption that could become anonymous cells\n  &:not(caption) {\n    position: absolute !important;\n  }\n}\n\n// Use to only display content when it's focused, or one of its child elements is focused\n// (i.e. when focus is within the element/container that the class was applied to)\n//\n// Useful for \"Skip to main content\" links; see https://www.w3.org/TR/2013/NOTE-WCAG20-TECHS-20130905/G1\n\n@mixin visually-hidden-focusable() {\n  &:not(:focus):not(:focus-within) {\n    @include visually-hidden();\n  }\n}\n","//\n// Stretched link\n//\n\n.stretched-link {\n  &::#{$stretched-link-pseudo-element} {\n    position: absolute;\n    top: 0;\n    right: 0;\n    bottom: 0;\n    left: 0;\n    z-index: $stretched-link-z-index;\n    content: \"\";\n  }\n}\n","//\n// Text truncation\n//\n\n.text-truncate {\n  @include text-truncate();\n}\n","// Text truncate\n// Requires inline-block or block for proper styling\n\n@mixin text-truncate() {\n  overflow: hidden;\n  text-overflow: ellipsis;\n  white-space: nowrap;\n}\n",".vr {\n  display: inline-block;\n  align-self: stretch;\n  width: $vr-border-width;\n  min-height: 1em;\n  background-color: currentcolor;\n  opacity: $hr-opacity;\n}\n","// Utility generator\n// Used to generate utilities & print utilities\n@mixin generate-utility($utility, $infix: \"\", $is-rfs-media-query: false) {\n  $values: map-get($utility, values);\n\n  // If the values are a list or string, convert it into a map\n  @if type-of($values) == \"string\" or type-of(nth($values, 1)) != \"list\" {\n    $values: zip($values, $values);\n  }\n\n  @each $key, $value in $values {\n    $properties: map-get($utility, property);\n\n    // Multiple properties are possible, for example with vertical or horizontal margins or paddings\n    @if type-of($properties) == \"string\" {\n      $properties: append((), $properties);\n    }\n\n    // Use custom class if present\n    $property-class: if(map-has-key($utility, class), map-get($utility, class), nth($properties, 1));\n    $property-class: if($property-class == null, \"\", $property-class);\n\n    // Use custom CSS variable name if present, otherwise default to `class`\n    $css-variable-name: if(map-has-key($utility, css-variable-name), map-get($utility, css-variable-name), map-get($utility, class));\n\n    // State params to generate pseudo-classes\n    $state: if(map-has-key($utility, state), map-get($utility, state), ());\n\n    $infix: if($property-class == \"\" and str-slice($infix, 1, 1) == \"-\", str-slice($infix, 2), $infix);\n\n    // Don't prefix if value key is null (e.g. with shadow class)\n    $property-class-modifier: if($key, if($property-class == \"\" and $infix == \"\", \"\", \"-\") + $key, \"\");\n\n    @if map-get($utility, rfs) {\n      // Inside the media query\n      @if $is-rfs-media-query {\n        $val: rfs-value($value);\n\n        // Do not render anything if fluid and non fluid values are the same\n        $value: if($val == rfs-fluid-value($value), null, $val);\n      }\n      @else {\n        $value: rfs-fluid-value($value);\n      }\n    }\n\n    $is-css-var: map-get($utility, css-var);\n    $is-local-vars: map-get($utility, local-vars);\n    $is-rtl: map-get($utility, rtl);\n\n    @if $value != null {\n      @if $is-rtl == false {\n        /* rtl:begin:remove */\n      }\n\n      @if $is-css-var {\n        .#{$property-class + $infix + $property-class-modifier} {\n          --#{$prefix}#{$css-variable-name}: #{$value};\n        }\n\n        @each $pseudo in $state {\n          .#{$property-class + $infix + $property-class-modifier}-#{$pseudo}:#{$pseudo} {\n            --#{$prefix}#{$css-variable-name}: #{$value};\n          }\n        }\n      } @else {\n        .#{$property-class + $infix + $property-class-modifier} {\n          @each $property in $properties {\n            @if $is-local-vars {\n              @each $local-var, $variable in $is-local-vars {\n                --#{$prefix}#{$local-var}: #{$variable};\n              }\n            }\n            #{$property}: $value if($enable-important-utilities, !important, null);\n          }\n        }\n\n        @each $pseudo in $state {\n          .#{$property-class + $infix + $property-class-modifier}-#{$pseudo}:#{$pseudo} {\n            @each $property in $properties {\n              @if $is-local-vars {\n                @each $local-var, $variable in $is-local-vars {\n                  --#{$prefix}#{$local-var}: #{$variable};\n                }\n              }\n              #{$property}: $value if($enable-important-utilities, !important, null);\n            }\n          }\n        }\n      }\n\n      @if $is-rtl == false {\n        /* rtl:end:remove */\n      }\n    }\n  }\n}\n","// Loop over each breakpoint\n@each $breakpoint in map-keys($grid-breakpoints) {\n\n  // Generate media query if needed\n  @include media-breakpoint-up($breakpoint) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    // Loop over each utility property\n    @each $key, $utility in $utilities {\n      // The utility can be disabled with `false`, thus check if the utility is a map first\n      // Only proceed if responsive media queries are enabled or if it's the base media query\n      @if type-of($utility) == \"map\" and (map-get($utility, responsive) or $infix == \"\") {\n        @include generate-utility($utility, $infix);\n      }\n    }\n  }\n}\n\n// RFS rescaling\n@media (min-width: $rfs-mq-value) {\n  @each $breakpoint in map-keys($grid-breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $grid-breakpoints);\n\n    @if (map-get($grid-breakpoints, $breakpoint) < $rfs-breakpoint) {\n      // Loop over each utility property\n      @each $key, $utility in $utilities {\n        // The utility can be disabled with `false`, thus check if the utility is a map first\n        // Only proceed if responsive media queries are enabled or if it's the base media query\n        @if type-of($utility) == \"map\" and map-get($utility, rfs) and (map-get($utility, responsive) or $infix == \"\") {\n          @include generate-utility($utility, $infix, true);\n        }\n      }\n    }\n  }\n}\n\n\n// Print utilities\n@media print {\n  @each $key, $utility in $utilities {\n    // The utility can be disabled with `false`, thus check if the utility is a map first\n    // Then check if the utility needs print styles\n    @if type-of($utility) == \"map\" and map-get($utility, print) == true {\n      @include generate-utility($utility, \"-print\");\n    }\n  }\n}\n"],"names":[],"sourceRoot":""}
\ No newline at end of file
diff --git a/_static/styles/pydata-sphinx-theme.css b/_static/styles/pydata-sphinx-theme.css
new file mode 100644
index 0000000..b81ac47
--- /dev/null
+++ b/_static/styles/pydata-sphinx-theme.css
@@ -0,0 +1,2 @@
+html{--pst-header-height:4rem;--pst-header-article-height:calc(var(--pst-header-height)*2/3);--pst-sidebar-secondary:17rem;--pst-font-size-base:1rem;--pst-font-size-h1:2.5rem;--pst-font-size-h2:2rem;--pst-font-size-h3:1.75rem;--pst-font-size-h4:1.5rem;--pst-font-size-h5:1.25rem;--pst-font-size-h6:1.1rem;--pst-font-size-milli:0.9rem;--pst-sidebar-font-size:0.9rem;--pst-sidebar-font-size-mobile:1.1rem;--pst-sidebar-header-font-size:1.2rem;--pst-sidebar-header-font-weight:600;--pst-admonition-font-weight-heading:600;--pst-font-weight-caption:300;--pst-font-weight-heading:400;--pst-font-family-base-system:-apple-system,BlinkMacSystemFont,Segoe UI,"Helvetica Neue",Arial,sans-serif,Apple Color Emoji,Segoe UI Emoji,Segoe UI Symbol;--pst-font-family-monospace-system:"SFMono-Regular",Menlo,Consolas,Monaco,Liberation Mono,Lucida 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+ 1rem)}body{background-color:var(--pst-color-background);color:var(--pst-color-text-base);display:flex;flex-direction:column;font-family:var(--pst-font-family-base);font-weight:400;line-height:1.65;min-height:100vh}body::-webkit-scrollbar-track{background:var(--pst-color-background)}p{color:var(--pst-color-text-base);font-size:1em;margin-bottom:1.15rem}p.rubric{border-bottom:1px solid var(--pst-color-border)}p.centered{text-align:center}a{word-wrap:break-word;color:var(--pst-color-link);text-decoration:underline;text-decoration-thickness:max(1px,.0625rem);text-underline-offset:.1578em}a:hover{text-decoration-skip:none;color:var(--pst-color-link-hover);text-decoration-skip-ink:none;text-decoration-thickness:max(3px,.1875rem,.12em)}a:active,a:visited{color:var(--pst-color-link)}a:visited:hover{color:var(--pst-color-link-hover)}a:focus-visible{outline:2px solid var(--pst-color-accent)}a.headerlink{color:var(--pst-color-secondary);font-size:.8em;margin-left:.2em;opacity:.7;padding:0 4px;text-decoration:none;transition:all .2s ease-out;user-select:none}a.headerlink:hover{opacity:1}a.github:before,a.gitlab:before{color:var(--pst-color-text-muted);font:var(--fa-font-brands);margin-right:.25rem}a.github:before{content:var(--pst-icon-github)}a.gitlab:before{content:var(--pst-icon-gitlab)}.heading-style,h1,h2,h3,h4,h5,h6{font-family:var(--pst-font-family-heading);font-weight:var(--pst-font-weight-heading);line-height:1.15;margin:2.75rem 0 1.05rem}h1{font-size:var(--pst-font-size-h1);margin-top:0}h1,h2{color:var(--pst-heading-color)}h2{font-size:var(--pst-font-size-h2)}h3{font-size:var(--pst-font-size-h3)}h3,h4{color:var(--pst-heading-color)}h4{font-size:var(--pst-font-size-h4)}h5{font-size:var(--pst-font-size-h5)}h5,h6{color:var(--pst-color-text-base)}h6{font-size:var(--pst-font-size-h6)}.text_small,small{font-size:var(--pst-font-size-milli)}hr{border:0;border-top:1px solid 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.bd-footer__inner{display:flex;flex-grow:1;margin:auto;padding:1rem}.bd-footer .footer-items__center,.bd-footer .footer-items__end,.bd-footer .footer-items__start{display:flex;flex-direction:column;flex-grow:1;gap:.5rem;justify-content:center}.bd-footer .footer-items__center{text-align:center}.bd-footer .footer-items__end{text-align:end}.bd-footer .footer-item p{margin-bottom:0}.bd-footer-article{margin-top:auto}.bd-footer-article .footer-article-items{display:flex;flex-direction:column}.bd-footer-content .footer-content-items{display:flex;flex-direction:column;margin-top:auto}.bd-header{background:var(--pst-color-on-background)!important;box-shadow:0 .125rem .25rem 0 var(--pst-color-shadow);justify-content:center;max-width:100vw;padding:0;position:sticky;top:0;width:100%;z-index:1030}.bd-header .bd-header__inner{align-items:center;display:flex;height:fit-content;padding-left:1rem;padding-right:1rem}.bd-header .navbar-item{align-items:center;display:flex;height:var(--pst-header-height);max-height:var(--pst-header-height)}.bd-header .navbar-header-items{flex-shrink:1}@media(min-width:960px){.bd-header .navbar-header-items{display:flex;flex-grow:1;padding:0 0 0 .5rem}}.bd-header .navbar-header-items__center,.bd-header .navbar-header-items__end,.bd-header .navbar-header-items__start{align-items:center;display:flex;flex-flow:wrap;row-gap:0}.bd-header .navbar-header-items__center,.bd-header .navbar-header-items__end{column-gap:1rem}.bd-header .navbar-header-items__start{flex-shrink:0;gap:.5rem;margin-right:auto}.bd-header .navbar-header-items__end{justify-content:end}.bd-header .navbar-nav{display:flex}@media(min-width:960px){.bd-header .navbar-nav{align-items:center}}.bd-header .navbar-nav li a.nav-link{color:var(--pst-color-text-muted);text-decoration:none}.bd-header .navbar-nav li 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details.toggle-details summary{border-left:3px solid var(--pst-color-primary)}html div.rendered_html html .jp-RenderedHTMLCommon table{table-layout:auto}html[data-theme=dark] .bd-content .nboutput .output_area.rendered_html{color:var(--pst-color-on-background);background-color:var(--pst-color-text-base);border-radius:.25rem;padding:.5rem}html[data-theme=dark] .bd-content .nboutput .output_area.stderr{background:var(--pst-color-danger)}div.nblast.container{margin-bottom:1rem}div.cell_output .output{max-width:100%;overflow-x:auto}html[data-theme=dark] .bd-content div.cell_output img,html[data-theme=dark] .bd-content div.cell_output .text_html{color:var(--pst-color-on-background);background-color:var(--pst-color-text-base);border-radius:.25rem;padding:.5rem}.bd-content div.cell_input{display:flex;flex-direction:column;justify-content:stretch}.bd-content div.cell_input,.bd-content div.output{border-radius:.25rem}.bd-content div.output table{table-layout:auto}html[data-theme=dark] .bd-content img.leaflet-tile.leaflet-tile-loaded{border-radius:0;padding:0}.bd-search-container div#search-results>h2{font-size:var(--pst-font-size-icon);margin-top:0}.bd-search-container div#search-results p.search-summary{color:var(--pst-color-text-muted)}.bd-search-container ul.search{margin:0;list-style:none}.bd-search-container ul.search li{background-image:none;padding:1rem 0;margin:1rem 0;border-top:1px solid var(--pst-color-text-muted)}.bd-search-container ul.search li>a{font-size:1.2em}.bd-search-container ul.search li div.context,.bd-search-container ul.search li p.context{color:var(--pst-color-text-base);margin:.5em 0 0 0}.bd-search-container ul.search li div.context a:before,.bd-search-container ul.search li p.context a:before{content:\"#\";padding-right:.2em;color:var(--pst-color-text-muted)}","html {\n  /*****************************************************************************\n  * Overall Layout Variables\n  */\n\n  // Header height will impact the top offset for many sections\n  // Article header is 66% of Header\n  --pst-header-height: 4rem;\n  --pst-header-article-height: calc(var(--pst-header-height) * 2 / 3);\n  --pst-sidebar-secondary: 17rem;\n}\n\n/*******************************************************************************\n* Breakpoints that trigger UI changes\n*\n* Note that media-breakpoint-down begins at the next highest level!\n* So we should choose a media-breakpoint-down one *lower* than when we want to start\n* example: media-breakpoint-up(md) and media-breakpoint-down(sm) trigger at the same time\n* ref: https://github.com/twbs/bootstrap/issues/31214\n*/\n$breakpoint-sidebar-primary: lg; // When we collapse the primary sidebar\n$breakpoint-sidebar-secondary: xl; // When we collapse the secondary sidebar\n$breakpoint-page-width: 88rem; // taken from sphinx-basic-ng, which we are ultimately going to inherit\n\n/*******************************************************************************\n* Define the animation behaviour\n*/\n$animation-time: 200ms;\n\n/*******************************************************************************\n* UI shaping and padding\n*/\n$admonition-border-radius: 0.25rem;\n","html {\n  /*****************************************************************************\n  * Font features used in this theme\n  */\n\n  // base font size - applied at body/html level\n  --pst-font-size-base: 1rem;\n\n  // heading font sizes based on bootstrap sizing\n  --pst-font-size-h1: 2.5rem;\n  --pst-font-size-h2: 2rem;\n  --pst-font-size-h3: 1.75rem;\n  --pst-font-size-h4: 1.5rem;\n  --pst-font-size-h5: 1.25rem;\n  --pst-font-size-h6: 1.1rem;\n\n  // smaller than heading font sizes\n  --pst-font-size-milli: 0.9rem;\n\n  // Sidebar styles\n  --pst-sidebar-font-size: 0.9rem;\n  --pst-sidebar-font-size-mobile: 1.1rem;\n  --pst-sidebar-header-font-size: 1.2rem;\n  --pst-sidebar-header-font-weight: 600;\n\n  // Admonition styles\n  --pst-admonition-font-weight-heading: 600;\n\n  // Font weights\n  --pst-font-weight-caption: 300;\n  --pst-font-weight-heading: 400;\n\n  // Font family\n  // These are adapted from https://systemfontstack.com/ */\n  --pst-font-family-base-system: -apple-system, BlinkMacSystemFont, Segoe UI,\n    \"Helvetica Neue\", Arial, sans-serif, Apple Color Emoji, Segoe UI Emoji,\n    Segoe UI Symbol;\n  --pst-font-family-monospace-system: \"SFMono-Regular\", Menlo, Consolas, Monaco,\n    Liberation Mono, Lucida Console, monospace;\n\n  --pst-font-family-base: var(--pst-font-family-base-system);\n  --pst-font-family-heading: var(--pst-font-family-base-system);\n  --pst-font-family-monospace: var(--pst-font-family-monospace-system);\n}\n","html {\n  /*****************************************************************************\n  * Icon\n  */\n\n  // Font size across all icons\n  --pst-font-size-icon: 1.5rem;\n\n  // Font Awesome default icons\n  --pst-icon-check-circle: \"\\f058\"; // fa-solid fa-circle-check\n  --pst-icon-info-circle: \"\\f05a\"; // fa-solid fa-circle-info\n  --pst-icon-exclamation-triangle: \"\\f071\"; // fa-solid fa-triangle-exclamation\n  --pst-icon-exclamation-circle: \"\\f06a\"; // fa-solid fa-circle-exclamation\n  --pst-icon-times-circle: \"\\f057\"; // fa-solid fa-circle-xmark\n  --pst-icon-lightbulb: \"\\f0eb\"; // fa-solid fa-lightbulb\n  --pst-icon-download: \"\\f019\"; // fa-solid fa-download\n  --pst-icon-angle-left: \"\\f104\"; // fa-solid fa-angle-left\n  --pst-icon-angle-right: \"\\f105\"; // fa-solid fa-angle-right\n  --pst-icon-external-link: \"\\f35d\"; // fa-solid fa-up-right-from-square\n  --pst-icon-search-minus: \"\\f010\"; // fa-solid fa-magnifying-glass-minus\n  --pst-icon-github: \"\\f09b\"; // fa-brands fa-github\n  --pst-icon-gitlab: \"\\f296\"; // fa-brands fa-gitlab\n  --pst-icon-share: \"\\f064\"; // fa-solid fa-share\n  --pst-icon-bell: \"\\f0f3\"; // fa-solid fa-bell\n  --pst-icon-pencil: \"\\f303\"; // fa-solid fa-pencil\n\n  // Bootstrap icons\n  --pst-breadcrumb-divider: \"\\f105\";\n}\n","html {\n  /*****************************************************************************\n  * Admonitions\n  **/\n\n  --pst-icon-admonition-default: var(--pst-icon-bell);\n  --pst-icon-admonition-note: var(--pst-icon-info-circle);\n  --pst-icon-admonition-attention: var(--pst-icon-exclamation-circle);\n  --pst-icon-admonition-caution: var(--pst-icon-exclamation-triangle);\n  --pst-icon-admonition-warning: var(--pst-icon-exclamation-triangle);\n  --pst-icon-admonition-danger: var(--pst-icon-exclamation-triangle);\n  --pst-icon-admonition-error: var(--pst-icon-times-circle);\n  --pst-icon-admonition-hint: var(--pst-icon-lightbulb);\n  --pst-icon-admonition-tip: var(--pst-icon-lightbulb);\n  --pst-icon-admonition-important: var(--pst-icon-exclamation-circle);\n  --pst-icon-admonition-seealso: var(--pst-icon-share);\n  --pst-icon-admonition-todo: var(--pst-icon-pencil);\n}\n","html {\n  /*****************************************************************************\n  * versionmodified\n  **/\n\n  --pst-icon-versionmodified-default: var(--pst-icon-exclamation-circle);\n  --pst-icon-versionmodified-added: var(--pst-icon-exclamation-circle);\n  --pst-icon-versionmodified-changed: var(--pst-icon-exclamation-circle);\n  --pst-icon-versionmodified-deprecated: var(--pst-icon-exclamation-circle);\n}\n","html {\n  font-size: var(--pst-font-size-base);\n  scroll-padding-top: calc(var(--pst-header-height) + 1rem);\n}\n\nbody {\n  background-color: var(--pst-color-background);\n  font-family: var(--pst-font-family-base);\n  font-weight: 400;\n  line-height: 1.65;\n  color: var(--pst-color-text-base);\n  min-height: 100vh;\n  display: flex;\n  flex-direction: column;\n\n  // hack to avoid the black background on some browser including Safari\n  &::-webkit-scrollbar-track {\n    background: var(--pst-color-background);\n  }\n}\n\np {\n  margin-bottom: 1.15rem;\n  font-size: 1em;\n  color: var(--pst-color-text-base);\n\n  /* section header in docstring pages */\n  &.rubric {\n    border-bottom: 1px solid var(--pst-color-border);\n  }\n\n  &.centered {\n    text-align: center;\n  }\n}\n\na {\n  @include link-style-default;\n\n  // The # anchor that appears on hover over headings\n  &.headerlink {\n    color: var(--pst-color-secondary);\n    opacity: 0.7;\n    font-size: 0.8em;\n    padding: 0 4px 0 4px;\n    margin-left: 0.2em;\n    text-decoration: none;\n    transition: all 0.2s ease-out;\n    user-select: none;\n\n    &:hover {\n      opacity: 1;\n    }\n  }\n\n  // set up a icon next to the shorten links from github and gitlab\n  &.github,\n  &.gitlab {\n    &::before {\n      color: var(--pst-color-text-muted);\n      font: var(--fa-font-brands);\n      margin-right: 0.25rem;\n    }\n  }\n\n  &.github::before {\n    content: var(--pst-icon-github);\n  }\n\n  &.gitlab::before {\n    content: var(--pst-icon-gitlab);\n  }\n}\n\n.heading-style {\n  margin: 2.75rem 0 1.05rem;\n  font-family: var(--pst-font-family-heading);\n  font-weight: var(--pst-font-weight-heading);\n  line-height: 1.15;\n}\n\nh1 {\n  @extend .heading-style;\n  margin-top: 0;\n  font-size: var(--pst-font-size-h1);\n  color: var(--pst-heading-color);\n}\n\nh2 {\n  @extend .heading-style;\n  font-size: var(--pst-font-size-h2);\n  color: var(--pst-heading-color);\n}\n\nh3 {\n  @extend .heading-style;\n  font-size: var(--pst-font-size-h3);\n  color: var(--pst-heading-color);\n}\n\nh4 {\n  @extend .heading-style;\n  font-size: var(--pst-font-size-h4);\n  color: var(--pst-heading-color);\n}\n\nh5 {\n  @extend .heading-style;\n  font-size: var(--pst-font-size-h5);\n  color: var(--pst-color-text-base);\n}\n\nh6 {\n  @extend .heading-style;\n  font-size: var(--pst-font-size-h6);\n  color: var(--pst-color-text-base);\n}\n\nsmall,\n.text_small {\n  font-size: var(--pst-font-size-milli);\n}\n\nhr {\n  border: 0;\n  border-top: 1px solid var(--pst-color-border);\n}\n\npre,\ncode,\nkbd,\nsamp {\n  font-family: var(--pst-font-family-monospace);\n}\n\nkbd {\n  // use theme negative\n  background-color: var(--pst-color-on-background);\n  color: var(--pst-color-text-muted);\n\n  // Compound keyboard elements will have nested kbd in them, so this prevents double lines\n  &:not(.compound) {\n    border: 1px solid var(--pst-color-border);\n    margin: 0 0.1rem;\n    padding: 0.1rem 0.4rem;\n    box-shadow: 1px 1px 1px var(--pst-color-shadow);\n  }\n}\n\ncode {\n  color: var(--pst-color-inline-code);\n}\n\npre {\n  margin: 1.5em 0 1.5em 0;\n  padding: 1rem;\n  background-color: var(--pst-color-surface);\n  color: var(--pst-color-text-base);\n  line-height: 1.2em;\n  border: 1px solid var(--pst-color-border);\n  border-radius: $admonition-border-radius;\n\n  .linenos {\n    // minimum opacity to make the line numbers WCAG AA conformant\n    opacity: 0.8;\n    padding-right: 10px;\n  }\n}\n\n// the back to top btn\n#pst-back-to-top {\n  z-index: $zindex-tooltip;\n  position: fixed;\n  display: none;\n  top: 90vh;\n  left: 50vw;\n  transform: translate(-50%);\n  color: var(--pst-color-secondary-text);\n  background-color: var(--pst-color-secondary);\n  border: none;\n}\n","/**\n * Consistent styling for links\n **/\n\n// Define some useful variables for links styling consistency\n//\n//  Thickness of the underline for links\n// the default will be either:\n//  - 1px\n// - 0.0625rem if it's thicker than 1px because the user has changed the text\n//   size in their browser\n$link-underline-thickness: unquote(\"max(1px, .0625rem)\") !default;\n// Offset of link underlines from text baseline\n// The default is 3px expressed as ems, as calculated against the default body\n// font size (on desktop).\n$link-underline-offset: 0.1578em !default;\n// Thickness of link underlines in hover state\n// The default for each link will be the thickest of the following:\n//  - 3px\n//  - 0.1875rem, if it's thicker than 3px because the user has changed the text\n//    size in their browser\n//  - 0.12em (relative to the link's text size)\n$link-hover-decoration-thickness: unquote(\"max(3px, .1875rem, .12em)\") !default;\n\n//  Ensures links have an underline decoration by default - needed to meet\n// WCAG SC 1.4.1\n@mixin link-decoration {\n  text-decoration: underline;\n\n  @if $link-underline-thickness {\n    text-decoration-thickness: $link-underline-thickness;\n  }\n\n  @if $link-underline-offset {\n    text-underline-offset: $link-underline-offset;\n  }\n}\n\n// Ensures links have an underline decoration on hover - distinct from the\n// default behaviour\n@mixin link-decoration-hover {\n  @if $link-hover-decoration-thickness {\n    text-decoration-thickness: $link-hover-decoration-thickness;\n    // Disable ink skipping on underlines on hover. Browsers haven't\n    // standardised on this part of the spec yet, so set both properties\n    text-decoration-skip-ink: none; // Chromium, Firefox\n    text-decoration-skip: none; // Safari\n  }\n}\n\n// Simple hover style - can be used alone or in conjunction with other mixins\n// Add the text underline and change in thickness on hover\n@mixin link-style-hover {\n  &:hover {\n    @include link-decoration;\n    @include link-decoration-hover;\n    color: var(--pst-color-link-hover);\n  }\n}\n\n// Default link styles\n//\n// Defines: default unvisited, visited, hover, and active.\n// TODO: @trallard to improve focus styles in subsequent PR\n@mixin link-style-default {\n  // So that really long links don't spill out of their container\n  word-wrap: break-word;\n\n  color: var(--pst-color-link);\n  @include link-decoration;\n\n  &:hover {\n    color: var(--pst-color-link-hover);\n    @include link-decoration-hover;\n  }\n\n  // TODO: @trallard to add active styles in subsequent PR\n  &:active {\n    color: var(--pst-color-link);\n  }\n\n  // Visited should still be hoverable\n  &:visited {\n    color: var(--pst-color-link);\n    &:hover {\n      color: var(--pst-color-link-hover);\n    }\n  }\n  @include focus-indicator;\n}\n\n// Text link styles\n//\n// Makes links use the muted text colour and removes the underline.\n// Use this mixin for navigation bar links.\n@mixin link-style-text {\n  color: var(--pst-color-text-muted);\n  text-decoration: none;\n\n  &:hover {\n    color: var(--pst-color-link-hover);\n    @include link-decoration;\n    @include link-decoration-hover;\n  }\n  @include focus-indicator;\n}\n\n// Sidebar and TOC links\n//\n// Makes links use the muted text colour and removes the underline.\n// Use this mixin for navigation the primary sidebar and table of contents.\n// Active and hover should work together rather than one overriding the other.\n@mixin link-sidebar {\n  color: var(--pst-color-text-muted);\n  text-decoration: none;\n\n  &:hover {\n    text-decoration: underline;\n    background-color: transparent;\n    color: var(--pst-color-link-hover);\n    @include link-decoration-hover;\n  }\n\n  // TODO: @trallard to update active styles in subsequent PR\n  &:active {\n    color: var(--pst-color-link-hover);\n  }\n}\n\n// Sidebar current page link styles\n//\n// Adds a vertical line on the left hand side of the link to indicate that\n// it's the current page. Note this is distinct from an active state.\n// Used on the primary sidebar and the TOC.\n// We want the side box shadow to have the same thickness as the hover underline\n@mixin link-sidebar-current {\n  font-weight: 600;\n  color: var(--pst-color-primary);\n  @if $link-hover-decoration-thickness {\n    box-shadow: inset\n      $link-hover-decoration-thickness\n      0px\n      0px\n      var(--pst-color-primary);\n  }\n}\n\n// Navigation bar current page link styles\n//\n// Adds a bottom underline, this leaves enough space for the hover state without\n// cluttering the navbar.\n// We want the side box shadow to have the same thickness as the hover underline\n@mixin link-navbar-current {\n  font-weight: 600;\n  color: var(--pst-color-primary);\n  @if $link-hover-decoration-thickness {\n    border-bottom: $link-hover-decoration-thickness\n      solid\n      var(--pst-color-primary);\n  }\n}\n\n// Navigation bar icon links hover styles\n//\n// Adds a bottom box-shadow - since there is no text we cannot use text-decoration\n// We want the side box shadow to have the same thickness as the hover underline\n@mixin icon-navbar-hover {\n  &:hover {\n    color: var(--pst-color-link-hover);\n    @if $link-hover-decoration-thickness {\n      box-shadow: 0px\n        $link-hover-decoration-thickness\n        0px\n        var(--pst-color-link-hover);\n    }\n  }\n}\n\n// Focus indicator\n@mixin focus-indicator {\n  &:focus-visible {\n    outline: 2px solid var(--pst-color-accent);\n  }\n}\n","// Variables\n//\n// Variables should follow the `$component-state-property-size` formula for\n// consistent naming. Ex: $nav-link-disabled-color and $modal-content-box-shadow-xs.\n\n// Color system\n\n// scss-docs-start gray-color-variables\n$white:    #fff !default;\n$gray-100: #f8f9fa !default;\n$gray-200: #e9ecef !default;\n$gray-300: #dee2e6 !default;\n$gray-400: #ced4da !default;\n$gray-500: #adb5bd !default;\n$gray-600: #6c757d !default;\n$gray-700: #495057 !default;\n$gray-800: #343a40 !default;\n$gray-900: #212529 !default;\n$black:    #000 !default;\n// scss-docs-end gray-color-variables\n\n// fusv-disable\n// scss-docs-start gray-colors-map\n$grays: (\n  \"100\": $gray-100,\n  \"200\": $gray-200,\n  \"300\": $gray-300,\n  \"400\": $gray-400,\n  \"500\": $gray-500,\n  \"600\": $gray-600,\n  \"700\": $gray-700,\n  \"800\": $gray-800,\n  \"900\": $gray-900\n) !default;\n// scss-docs-end gray-colors-map\n// fusv-enable\n\n// scss-docs-start color-variables\n$blue:    #0d6efd !default;\n$indigo:  #6610f2 !default;\n$purple:  #6f42c1 !default;\n$pink:    #d63384 !default;\n$red:     #dc3545 !default;\n$orange:  #fd7e14 !default;\n$yellow:  #ffc107 !default;\n$green:   #198754 !default;\n$teal:    #20c997 !default;\n$cyan:    #0dcaf0 !default;\n// scss-docs-end color-variables\n\n// scss-docs-start colors-map\n$colors: (\n  \"blue\":       $blue,\n  \"indigo\":     $indigo,\n  \"purple\":     $purple,\n  \"pink\":       $pink,\n  \"red\":        $red,\n  \"orange\":     $orange,\n  \"yellow\":     $yellow,\n  \"green\":      $green,\n  \"teal\":       $teal,\n  \"cyan\":       $cyan,\n  \"black\":      $black,\n  \"white\":      $white,\n  \"gray\":       $gray-600,\n  \"gray-dark\":  $gray-800\n) !default;\n// scss-docs-end colors-map\n\n// The contrast ratio to reach against white, to determine if color changes from \"light\" to \"dark\". Acceptable values for WCAG 2.0 are 3, 4.5 and 7.\n// See https://www.w3.org/TR/WCAG20/#visual-audio-contrast-contrast\n$min-contrast-ratio:   4.5 !default;\n\n// Customize the light and dark text colors for use in our color contrast function.\n$color-contrast-dark:      $black !default;\n$color-contrast-light:     $white !default;\n\n// fusv-disable\n$blue-100: tint-color($blue, 80%) !default;\n$blue-200: tint-color($blue, 60%) !default;\n$blue-300: tint-color($blue, 40%) !default;\n$blue-400: tint-color($blue, 20%) !default;\n$blue-500: $blue !default;\n$blue-600: shade-color($blue, 20%) !default;\n$blue-700: shade-color($blue, 40%) !default;\n$blue-800: shade-color($blue, 60%) !default;\n$blue-900: shade-color($blue, 80%) !default;\n\n$indigo-100: tint-color($indigo, 80%) !default;\n$indigo-200: tint-color($indigo, 60%) !default;\n$indigo-300: tint-color($indigo, 40%) !default;\n$indigo-400: tint-color($indigo, 20%) !default;\n$indigo-500: $indigo !default;\n$indigo-600: shade-color($indigo, 20%) !default;\n$indigo-700: shade-color($indigo, 40%) !default;\n$indigo-800: shade-color($indigo, 60%) !default;\n$indigo-900: shade-color($indigo, 80%) !default;\n\n$purple-100: tint-color($purple, 80%) !default;\n$purple-200: tint-color($purple, 60%) !default;\n$purple-300: tint-color($purple, 40%) !default;\n$purple-400: tint-color($purple, 20%) !default;\n$purple-500: $purple !default;\n$purple-600: shade-color($purple, 20%) !default;\n$purple-700: shade-color($purple, 40%) !default;\n$purple-800: shade-color($purple, 60%) !default;\n$purple-900: shade-color($purple, 80%) !default;\n\n$pink-100: tint-color($pink, 80%) !default;\n$pink-200: tint-color($pink, 60%) !default;\n$pink-300: tint-color($pink, 40%) !default;\n$pink-400: tint-color($pink, 20%) !default;\n$pink-500: $pink !default;\n$pink-600: shade-color($pink, 20%) !default;\n$pink-700: shade-color($pink, 40%) !default;\n$pink-800: shade-color($pink, 60%) !default;\n$pink-900: shade-color($pink, 80%) !default;\n\n$red-100: tint-color($red, 80%) !default;\n$red-200: tint-color($red, 60%) !default;\n$red-300: tint-color($red, 40%) !default;\n$red-400: tint-color($red, 20%) !default;\n$red-500: $red !default;\n$red-600: shade-color($red, 20%) !default;\n$red-700: shade-color($red, 40%) !default;\n$red-800: shade-color($red, 60%) !default;\n$red-900: shade-color($red, 80%) !default;\n\n$orange-100: tint-color($orange, 80%) !default;\n$orange-200: tint-color($orange, 60%) !default;\n$orange-300: tint-color($orange, 40%) !default;\n$orange-400: tint-color($orange, 20%) !default;\n$orange-500: $orange !default;\n$orange-600: shade-color($orange, 20%) !default;\n$orange-700: shade-color($orange, 40%) !default;\n$orange-800: shade-color($orange, 60%) !default;\n$orange-900: shade-color($orange, 80%) !default;\n\n$yellow-100: tint-color($yellow, 80%) !default;\n$yellow-200: tint-color($yellow, 60%) !default;\n$yellow-300: tint-color($yellow, 40%) !default;\n$yellow-400: tint-color($yellow, 20%) !default;\n$yellow-500: $yellow !default;\n$yellow-600: shade-color($yellow, 20%) !default;\n$yellow-700: shade-color($yellow, 40%) !default;\n$yellow-800: shade-color($yellow, 60%) !default;\n$yellow-900: shade-color($yellow, 80%) !default;\n\n$green-100: tint-color($green, 80%) !default;\n$green-200: tint-color($green, 60%) !default;\n$green-300: tint-color($green, 40%) !default;\n$green-400: tint-color($green, 20%) !default;\n$green-500: $green !default;\n$green-600: shade-color($green, 20%) !default;\n$green-700: shade-color($green, 40%) !default;\n$green-800: shade-color($green, 60%) !default;\n$green-900: shade-color($green, 80%) !default;\n\n$teal-100: tint-color($teal, 80%) !default;\n$teal-200: tint-color($teal, 60%) !default;\n$teal-300: tint-color($teal, 40%) !default;\n$teal-400: tint-color($teal, 20%) !default;\n$teal-500: $teal !default;\n$teal-600: shade-color($teal, 20%) !default;\n$teal-700: shade-color($teal, 40%) !default;\n$teal-800: shade-color($teal, 60%) !default;\n$teal-900: shade-color($teal, 80%) !default;\n\n$cyan-100: tint-color($cyan, 80%) !default;\n$cyan-200: tint-color($cyan, 60%) !default;\n$cyan-300: tint-color($cyan, 40%) !default;\n$cyan-400: tint-color($cyan, 20%) !default;\n$cyan-500: $cyan !default;\n$cyan-600: shade-color($cyan, 20%) !default;\n$cyan-700: shade-color($cyan, 40%) !default;\n$cyan-800: shade-color($cyan, 60%) !default;\n$cyan-900: shade-color($cyan, 80%) !default;\n\n$blues: (\n  \"blue-100\": $blue-100,\n  \"blue-200\": $blue-200,\n  \"blue-300\": $blue-300,\n  \"blue-400\": $blue-400,\n  \"blue-500\": $blue-500,\n  \"blue-600\": $blue-600,\n  \"blue-700\": $blue-700,\n  \"blue-800\": $blue-800,\n  \"blue-900\": $blue-900\n) !default;\n\n$indigos: (\n  \"indigo-100\": $indigo-100,\n  \"indigo-200\": $indigo-200,\n  \"indigo-300\": $indigo-300,\n  \"indigo-400\": $indigo-400,\n  \"indigo-500\": $indigo-500,\n  \"indigo-600\": $indigo-600,\n  \"indigo-700\": $indigo-700,\n  \"indigo-800\": $indigo-800,\n  \"indigo-900\": $indigo-900\n) !default;\n\n$purples: (\n  \"purple-100\": $purple-100,\n  \"purple-200\": $purple-200,\n  \"purple-300\": $purple-300,\n  \"purple-400\": $purple-400,\n  \"purple-500\": $purple-500,\n  \"purple-600\": $purple-600,\n  \"purple-700\": $purple-700,\n  \"purple-800\": $purple-800,\n  \"purple-900\": $purple-900\n) !default;\n\n$pinks: (\n  \"pink-100\": $pink-100,\n  \"pink-200\": $pink-200,\n  \"pink-300\": $pink-300,\n  \"pink-400\": $pink-400,\n  \"pink-500\": $pink-500,\n  \"pink-600\": $pink-600,\n  \"pink-700\": $pink-700,\n  \"pink-800\": $pink-800,\n  \"pink-900\": $pink-900\n) !default;\n\n$reds: (\n  \"red-100\": $red-100,\n  \"red-200\": $red-200,\n  \"red-300\": $red-300,\n  \"red-400\": $red-400,\n  \"red-500\": $red-500,\n  \"red-600\": $red-600,\n  \"red-700\": $red-700,\n  \"red-800\": $red-800,\n  \"red-900\": $red-900\n) !default;\n\n$oranges: (\n  \"orange-100\": $orange-100,\n  \"orange-200\": $orange-200,\n  \"orange-300\": $orange-300,\n  \"orange-400\": $orange-400,\n  \"orange-500\": $orange-500,\n  \"orange-600\": $orange-600,\n  \"orange-700\": $orange-700,\n  \"orange-800\": $orange-800,\n  \"orange-900\": $orange-900\n) !default;\n\n$yellows: (\n  \"yellow-100\": $yellow-100,\n  \"yellow-200\": $yellow-200,\n  \"yellow-300\": $yellow-300,\n  \"yellow-400\": $yellow-400,\n  \"yellow-500\": $yellow-500,\n  \"yellow-600\": $yellow-600,\n  \"yellow-700\": $yellow-700,\n  \"yellow-800\": $yellow-800,\n  \"yellow-900\": $yellow-900\n) !default;\n\n$greens: (\n  \"green-100\": $green-100,\n  \"green-200\": $green-200,\n  \"green-300\": $green-300,\n  \"green-400\": $green-400,\n  \"green-500\": $green-500,\n  \"green-600\": $green-600,\n  \"green-700\": $green-700,\n  \"green-800\": $green-800,\n  \"green-900\": $green-900\n) !default;\n\n$teals: (\n  \"teal-100\": $teal-100,\n  \"teal-200\": $teal-200,\n  \"teal-300\": $teal-300,\n  \"teal-400\": $teal-400,\n  \"teal-500\": $teal-500,\n  \"teal-600\": $teal-600,\n  \"teal-700\": $teal-700,\n  \"teal-800\": $teal-800,\n  \"teal-900\": $teal-900\n) !default;\n\n$cyans: (\n  \"cyan-100\": $cyan-100,\n  \"cyan-200\": $cyan-200,\n  \"cyan-300\": $cyan-300,\n  \"cyan-400\": $cyan-400,\n  \"cyan-500\": $cyan-500,\n  \"cyan-600\": $cyan-600,\n  \"cyan-700\": $cyan-700,\n  \"cyan-800\": $cyan-800,\n  \"cyan-900\": $cyan-900\n) !default;\n// fusv-enable\n\n// scss-docs-start theme-color-variables\n$primary:       $blue !default;\n$secondary:     $gray-600 !default;\n$success:       $green !default;\n$info:          $cyan !default;\n$warning:       $yellow !default;\n$danger:        $red !default;\n$light:         $gray-100 !default;\n$dark:          $gray-900 !default;\n// scss-docs-end theme-color-variables\n\n// scss-docs-start theme-colors-map\n$theme-colors: (\n  \"primary\":    $primary,\n  \"secondary\":  $secondary,\n  \"success\":    $success,\n  \"info\":       $info,\n  \"warning\":    $warning,\n  \"danger\":     $danger,\n  \"light\":      $light,\n  \"dark\":       $dark\n) !default;\n// scss-docs-end theme-colors-map\n\n// scss-docs-start theme-text-variables\n$primary-text-emphasis:   shade-color($primary, 60%) !default;\n$secondary-text-emphasis: shade-color($secondary, 60%) !default;\n$success-text-emphasis:   shade-color($success, 60%) !default;\n$info-text-emphasis:      shade-color($info, 60%) !default;\n$warning-text-emphasis:   shade-color($warning, 60%) !default;\n$danger-text-emphasis:    shade-color($danger, 60%) !default;\n$light-text-emphasis:     $gray-700 !default;\n$dark-text-emphasis:      $gray-700 !default;\n// scss-docs-end theme-text-variables\n\n// scss-docs-start theme-bg-subtle-variables\n$primary-bg-subtle:       tint-color($primary, 80%) !default;\n$secondary-bg-subtle:     tint-color($secondary, 80%) !default;\n$success-bg-subtle:       tint-color($success, 80%) !default;\n$info-bg-subtle:          tint-color($info, 80%) !default;\n$warning-bg-subtle:       tint-color($warning, 80%) !default;\n$danger-bg-subtle:        tint-color($danger, 80%) !default;\n$light-bg-subtle:         mix($gray-100, $white) !default;\n$dark-bg-subtle:          $gray-400 !default;\n// scss-docs-end theme-bg-subtle-variables\n\n// scss-docs-start theme-border-subtle-variables\n$primary-border-subtle:   tint-color($primary, 60%) !default;\n$secondary-border-subtle: tint-color($secondary, 60%) !default;\n$success-border-subtle:   tint-color($success, 60%) !default;\n$info-border-subtle:      tint-color($info, 60%) !default;\n$warning-border-subtle:   tint-color($warning, 60%) !default;\n$danger-border-subtle:    tint-color($danger, 60%) !default;\n$light-border-subtle:     $gray-200 !default;\n$dark-border-subtle:      $gray-500 !default;\n// scss-docs-end theme-border-subtle-variables\n\n// Characters which are escaped by the escape-svg function\n$escaped-characters: (\n  (\"<\", \"%3c\"),\n  (\">\", \"%3e\"),\n  (\"#\", \"%23\"),\n  (\"(\", \"%28\"),\n  (\")\", \"%29\"),\n) !default;\n\n// Options\n//\n// Quickly modify global styling by enabling or disabling optional features.\n\n$enable-caret:                true !default;\n$enable-rounded:              true !default;\n$enable-shadows:              false !default;\n$enable-gradients:            false !default;\n$enable-transitions:          true !default;\n$enable-reduced-motion:       true !default;\n$enable-smooth-scroll:        true !default;\n$enable-grid-classes:         true !default;\n$enable-container-classes:    true !default;\n$enable-cssgrid:              false !default;\n$enable-button-pointers:      true !default;\n$enable-rfs:                  true !default;\n$enable-validation-icons:     true !default;\n$enable-negative-margins:     false !default;\n$enable-deprecation-messages: true !default;\n$enable-important-utilities:  true !default;\n\n$enable-dark-mode:            true !default;\n$color-mode-type:             data !default; // `data` or `media-query`\n\n// Prefix for :root CSS variables\n\n$variable-prefix:             bs- !default; // Deprecated in v5.2.0 for the shorter `$prefix`\n$prefix:                      $variable-prefix !default;\n\n// Gradient\n//\n// The gradient which is added to components if `$enable-gradients` is `true`\n// This gradient is also added to elements with `.bg-gradient`\n// scss-docs-start variable-gradient\n$gradient: linear-gradient(180deg, rgba($white, .15), rgba($white, 0)) !default;\n// scss-docs-end variable-gradient\n\n// Spacing\n//\n// Control the default styling of most Bootstrap elements by modifying these\n// variables. Mostly focused on spacing.\n// You can add more entries to the $spacers map, should you need more variation.\n\n// scss-docs-start spacer-variables-maps\n$spacer: 1rem !default;\n$spacers: (\n  0: 0,\n  1: $spacer * .25,\n  2: $spacer * .5,\n  3: $spacer,\n  4: $spacer * 1.5,\n  5: $spacer * 3,\n) !default;\n// scss-docs-end spacer-variables-maps\n\n// Position\n//\n// Define the edge positioning anchors of the position utilities.\n\n// scss-docs-start position-map\n$position-values: (\n  0: 0,\n  50: 50%,\n  100: 100%\n) !default;\n// scss-docs-end position-map\n\n// Body\n//\n// Settings for the `<body>` element.\n\n$body-text-align:           null !default;\n$body-color:                $gray-900 !default;\n$body-bg:                   $white !default;\n\n$body-secondary-color:      rgba($body-color, .75) !default;\n$body-secondary-bg:         $gray-200 !default;\n\n$body-tertiary-color:       rgba($body-color, .5) !default;\n$body-tertiary-bg:          $gray-100 !default;\n\n$body-emphasis-color:       $black !default;\n\n// Links\n//\n// Style anchor elements.\n\n$link-color:                              $primary !default;\n$link-decoration:                         underline !default;\n$link-shade-percentage:                   20% !default;\n$link-hover-color:                        shift-color($link-color, $link-shade-percentage) !default;\n$link-hover-decoration:                   null !default;\n\n$stretched-link-pseudo-element:           after !default;\n$stretched-link-z-index:                  1 !default;\n\n// Icon links\n// scss-docs-start icon-link-variables\n$icon-link-gap:               .375rem !default;\n$icon-link-underline-offset:  .25em !default;\n$icon-link-icon-size:         1em !default;\n$icon-link-icon-transition:   .2s ease-in-out transform !default;\n$icon-link-icon-transform:    translate3d(.25em, 0, 0) !default;\n// scss-docs-end icon-link-variables\n\n// Paragraphs\n//\n// Style p element.\n\n$paragraph-margin-bottom:   1rem !default;\n\n\n// Grid breakpoints\n//\n// Define the minimum dimensions at which your layout will change,\n// adapting to different screen sizes, for use in media queries.\n\n// scss-docs-start grid-breakpoints\n$grid-breakpoints: (\n  xs: 0,\n  sm: 576px,\n  md: 768px,\n  lg: 992px,\n  xl: 1200px,\n  xxl: 1400px\n) !default;\n// scss-docs-end grid-breakpoints\n\n@include _assert-ascending($grid-breakpoints, \"$grid-breakpoints\");\n@include _assert-starts-at-zero($grid-breakpoints, \"$grid-breakpoints\");\n\n\n// Grid containers\n//\n// Define the maximum width of `.container` for different screen sizes.\n\n// scss-docs-start container-max-widths\n$container-max-widths: (\n  sm: 540px,\n  md: 720px,\n  lg: 960px,\n  xl: 1140px,\n  xxl: 1320px\n) !default;\n// scss-docs-end container-max-widths\n\n@include _assert-ascending($container-max-widths, \"$container-max-widths\");\n\n\n// Grid columns\n//\n// Set the number of columns and specify the width of the gutters.\n\n$grid-columns:                12 !default;\n$grid-gutter-width:           1.5rem !default;\n$grid-row-columns:            6 !default;\n\n// Container padding\n\n$container-padding-x: $grid-gutter-width !default;\n\n\n// Components\n//\n// Define common padding and border radius sizes and more.\n\n// scss-docs-start border-variables\n$border-width:                1px !default;\n$border-widths: (\n  1: 1px,\n  2: 2px,\n  3: 3px,\n  4: 4px,\n  5: 5px\n) !default;\n$border-style:                solid !default;\n$border-color:                $gray-300 !default;\n$border-color-translucent:    rgba($black, .175) !default;\n// scss-docs-end border-variables\n\n// scss-docs-start border-radius-variables\n$border-radius:               .375rem !default;\n$border-radius-sm:            .25rem !default;\n$border-radius-lg:            .5rem !default;\n$border-radius-xl:            1rem !default;\n$border-radius-xxl:           2rem !default;\n$border-radius-pill:          50rem !default;\n// scss-docs-end border-radius-variables\n// fusv-disable\n$border-radius-2xl:           $border-radius-xxl !default; // Deprecated in v5.3.0\n// fusv-enable\n\n// scss-docs-start box-shadow-variables\n$box-shadow:                  0 .5rem 1rem rgba($black, .15) !default;\n$box-shadow-sm:               0 .125rem .25rem rgba($black, .075) !default;\n$box-shadow-lg:               0 1rem 3rem rgba($black, .175) !default;\n$box-shadow-inset:            inset 0 1px 2px rgba($black, .075) !default;\n// scss-docs-end box-shadow-variables\n\n$component-active-color:      $white !default;\n$component-active-bg:         $primary !default;\n\n// scss-docs-start focus-ring-variables\n$focus-ring-width:      .25rem !default;\n$focus-ring-opacity:    .25 !default;\n$focus-ring-color:      rgba($primary, $focus-ring-opacity) !default;\n$focus-ring-blur:       0 !default;\n$focus-ring-box-shadow: 0 0 $focus-ring-blur $focus-ring-width $focus-ring-color !default;\n// scss-docs-end focus-ring-variables\n\n// scss-docs-start caret-variables\n$caret-width:                 .3em !default;\n$caret-vertical-align:        $caret-width * .85 !default;\n$caret-spacing:               $caret-width * .85 !default;\n// scss-docs-end caret-variables\n\n$transition-base:             all .2s ease-in-out !default;\n$transition-fade:             opacity .15s linear !default;\n// scss-docs-start collapse-transition\n$transition-collapse:         height .35s ease !default;\n$transition-collapse-width:   width .35s ease !default;\n// scss-docs-end collapse-transition\n\n// stylelint-disable function-disallowed-list\n// scss-docs-start aspect-ratios\n$aspect-ratios: (\n  \"1x1\": 100%,\n  \"4x3\": calc(3 / 4 * 100%),\n  \"16x9\": calc(9 / 16 * 100%),\n  \"21x9\": calc(9 / 21 * 100%)\n) !default;\n// scss-docs-end aspect-ratios\n// stylelint-enable function-disallowed-list\n\n// Typography\n//\n// Font, line-height, and color for body text, headings, and more.\n\n// scss-docs-start font-variables\n// stylelint-disable value-keyword-case\n$font-family-sans-serif:      system-ui, -apple-system, \"Segoe UI\", Roboto, \"Helvetica Neue\", \"Noto Sans\", \"Liberation Sans\", Arial, sans-serif, \"Apple Color Emoji\", \"Segoe UI Emoji\", \"Segoe UI Symbol\", \"Noto Color Emoji\" !default;\n$font-family-monospace:       SFMono-Regular, Menlo, Monaco, Consolas, \"Liberation Mono\", \"Courier New\", monospace !default;\n// stylelint-enable value-keyword-case\n$font-family-base:            var(--#{$prefix}font-sans-serif) !default;\n$font-family-code:            var(--#{$prefix}font-monospace) !default;\n\n// $font-size-root affects the value of `rem`, which is used for as well font sizes, paddings, and margins\n// $font-size-base affects the font size of the body text\n$font-size-root:              null !default;\n$font-size-base:              1rem !default; // Assumes the browser default, typically `16px`\n$font-size-sm:                $font-size-base * .875 !default;\n$font-size-lg:                $font-size-base * 1.25 !default;\n\n$font-weight-lighter:         lighter !default;\n$font-weight-light:           300 !default;\n$font-weight-normal:          400 !default;\n$font-weight-medium:          500 !default;\n$font-weight-semibold:        600 !default;\n$font-weight-bold:            700 !default;\n$font-weight-bolder:          bolder !default;\n\n$font-weight-base:            $font-weight-normal !default;\n\n$line-height-base:            1.5 !default;\n$line-height-sm:              1.25 !default;\n$line-height-lg:              2 !default;\n\n$h1-font-size:                $font-size-base * 2.5 !default;\n$h2-font-size:                $font-size-base * 2 !default;\n$h3-font-size:                $font-size-base * 1.75 !default;\n$h4-font-size:                $font-size-base * 1.5 !default;\n$h5-font-size:                $font-size-base * 1.25 !default;\n$h6-font-size:                $font-size-base !default;\n// scss-docs-end font-variables\n\n// scss-docs-start font-sizes\n$font-sizes: (\n  1: $h1-font-size,\n  2: $h2-font-size,\n  3: $h3-font-size,\n  4: $h4-font-size,\n  5: $h5-font-size,\n  6: $h6-font-size\n) !default;\n// scss-docs-end font-sizes\n\n// scss-docs-start headings-variables\n$headings-margin-bottom:      $spacer * .5 !default;\n$headings-font-family:        null !default;\n$headings-font-style:         null !default;\n$headings-font-weight:        500 !default;\n$headings-line-height:        1.2 !default;\n$headings-color:              inherit !default;\n// scss-docs-end headings-variables\n\n// scss-docs-start display-headings\n$display-font-sizes: (\n  1: 5rem,\n  2: 4.5rem,\n  3: 4rem,\n  4: 3.5rem,\n  5: 3rem,\n  6: 2.5rem\n) !default;\n\n$display-font-family: null !default;\n$display-font-style:  null !default;\n$display-font-weight: 300 !default;\n$display-line-height: $headings-line-height !default;\n// scss-docs-end display-headings\n\n// scss-docs-start type-variables\n$lead-font-size:              $font-size-base * 1.25 !default;\n$lead-font-weight:            300 !default;\n\n$small-font-size:             .875em !default;\n\n$sub-sup-font-size:           .75em !default;\n\n// fusv-disable\n$text-muted:                  var(--#{$prefix}secondary-color) !default; // Deprecated in 5.3.0\n// fusv-enable\n\n$initialism-font-size:        $small-font-size !default;\n\n$blockquote-margin-y:         $spacer !default;\n$blockquote-font-size:        $font-size-base * 1.25 !default;\n$blockquote-footer-color:     $gray-600 !default;\n$blockquote-footer-font-size: $small-font-size !default;\n\n$hr-margin-y:                 $spacer !default;\n$hr-color:                    inherit !default;\n\n// fusv-disable\n$hr-bg-color:                 null !default; // Deprecated in v5.2.0\n$hr-height:                   null !default; // Deprecated in v5.2.0\n// fusv-enable\n\n$hr-border-color:             null !default; // Allows for inherited colors\n$hr-border-width:             var(--#{$prefix}border-width) !default;\n$hr-opacity:                  .25 !default;\n\n// scss-docs-start vr-variables\n$vr-border-width:             var(--#{$prefix}border-width) !default;\n// scss-docs-end vr-variables\n\n$legend-margin-bottom:        .5rem !default;\n$legend-font-size:            1.5rem !default;\n$legend-font-weight:          null !default;\n\n$dt-font-weight:              $font-weight-bold !default;\n\n$list-inline-padding:         .5rem !default;\n\n$mark-padding:                .1875em !default;\n$mark-color:                  $body-color !default;\n$mark-bg:                     $yellow-100 !default;\n// scss-docs-end type-variables\n\n\n// Tables\n//\n// Customizes the `.table` component with basic values, each used across all table variations.\n\n// scss-docs-start table-variables\n$table-cell-padding-y:        .5rem !default;\n$table-cell-padding-x:        .5rem !default;\n$table-cell-padding-y-sm:     .25rem !default;\n$table-cell-padding-x-sm:     .25rem !default;\n\n$table-cell-vertical-align:   top !default;\n\n$table-color:                 var(--#{$prefix}emphasis-color) !default;\n$table-bg:                    var(--#{$prefix}body-bg) !default;\n$table-accent-bg:             transparent !default;\n\n$table-th-font-weight:        null !default;\n\n$table-striped-color:         $table-color !default;\n$table-striped-bg-factor:     .05 !default;\n$table-striped-bg:            rgba(var(--#{$prefix}emphasis-color-rgb), $table-striped-bg-factor) !default;\n\n$table-active-color:          $table-color !default;\n$table-active-bg-factor:      .1 !default;\n$table-active-bg:             rgba(var(--#{$prefix}emphasis-color-rgb), $table-active-bg-factor) !default;\n\n$table-hover-color:           $table-color !default;\n$table-hover-bg-factor:       .075 !default;\n$table-hover-bg:              rgba(var(--#{$prefix}emphasis-color-rgb), $table-hover-bg-factor) !default;\n\n$table-border-factor:         .2 !default;\n$table-border-width:          var(--#{$prefix}border-width) !default;\n$table-border-color:          var(--#{$prefix}border-color) !default;\n\n$table-striped-order:         odd !default;\n$table-striped-columns-order: even !default;\n\n$table-group-separator-color: currentcolor !default;\n\n$table-caption-color:         var(--#{$prefix}secondary-color) !default;\n\n$table-bg-scale:              -80% !default;\n// scss-docs-end table-variables\n\n// scss-docs-start table-loop\n$table-variants: (\n  \"primary\":    shift-color($primary, $table-bg-scale),\n  \"secondary\":  shift-color($secondary, $table-bg-scale),\n  \"success\":    shift-color($success, $table-bg-scale),\n  \"info\":       shift-color($info, $table-bg-scale),\n  \"warning\":    shift-color($warning, $table-bg-scale),\n  \"danger\":     shift-color($danger, $table-bg-scale),\n  \"light\":      $light,\n  \"dark\":       $dark,\n) !default;\n// scss-docs-end table-loop\n\n\n// Buttons + Forms\n//\n// Shared variables that are reassigned to `$input-` and `$btn-` specific variables.\n\n// scss-docs-start input-btn-variables\n$input-btn-padding-y:         .375rem !default;\n$input-btn-padding-x:         .75rem !default;\n$input-btn-font-family:       null !default;\n$input-btn-font-size:         $font-size-base !default;\n$input-btn-line-height:       $line-height-base !default;\n\n$input-btn-focus-width:         $focus-ring-width !default;\n$input-btn-focus-color-opacity: $focus-ring-opacity !default;\n$input-btn-focus-color:         $focus-ring-color !default;\n$input-btn-focus-blur:          $focus-ring-blur !default;\n$input-btn-focus-box-shadow:    $focus-ring-box-shadow !default;\n\n$input-btn-padding-y-sm:      .25rem !default;\n$input-btn-padding-x-sm:      .5rem !default;\n$input-btn-font-size-sm:      $font-size-sm !default;\n\n$input-btn-padding-y-lg:      .5rem !default;\n$input-btn-padding-x-lg:      1rem !default;\n$input-btn-font-size-lg:      $font-size-lg !default;\n\n$input-btn-border-width:      var(--#{$prefix}border-width) !default;\n// scss-docs-end input-btn-variables\n\n\n// Buttons\n//\n// For each of Bootstrap's buttons, define text, background, and border color.\n\n// scss-docs-start btn-variables\n$btn-color:                   var(--#{$prefix}body-color) !default;\n$btn-padding-y:               $input-btn-padding-y !default;\n$btn-padding-x:               $input-btn-padding-x !default;\n$btn-font-family:             $input-btn-font-family !default;\n$btn-font-size:               $input-btn-font-size !default;\n$btn-line-height:             $input-btn-line-height !default;\n$btn-white-space:             null !default; // Set to `nowrap` to prevent text wrapping\n\n$btn-padding-y-sm:            $input-btn-padding-y-sm !default;\n$btn-padding-x-sm:            $input-btn-padding-x-sm !default;\n$btn-font-size-sm:            $input-btn-font-size-sm !default;\n\n$btn-padding-y-lg:            $input-btn-padding-y-lg !default;\n$btn-padding-x-lg:            $input-btn-padding-x-lg !default;\n$btn-font-size-lg:            $input-btn-font-size-lg !default;\n\n$btn-border-width:            $input-btn-border-width !default;\n\n$btn-font-weight:             $font-weight-normal !default;\n$btn-box-shadow:              inset 0 1px 0 rgba($white, .15), 0 1px 1px rgba($black, .075) !default;\n$btn-focus-width:             $input-btn-focus-width !default;\n$btn-focus-box-shadow:        $input-btn-focus-box-shadow !default;\n$btn-disabled-opacity:        .65 !default;\n$btn-active-box-shadow:       inset 0 3px 5px rgba($black, .125) !default;\n\n$btn-link-color:              var(--#{$prefix}link-color) !default;\n$btn-link-hover-color:        var(--#{$prefix}link-hover-color) !default;\n$btn-link-disabled-color:     $gray-600 !default;\n$btn-link-focus-shadow-rgb:   to-rgb(mix(color-contrast($link-color), $link-color, 15%)) !default;\n\n// Allows for customizing button radius independently from global border radius\n$btn-border-radius:           var(--#{$prefix}border-radius) !default;\n$btn-border-radius-sm:        var(--#{$prefix}border-radius-sm) !default;\n$btn-border-radius-lg:        var(--#{$prefix}border-radius-lg) !default;\n\n$btn-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$btn-hover-bg-shade-amount:       15% !default;\n$btn-hover-bg-tint-amount:        15% !default;\n$btn-hover-border-shade-amount:   20% !default;\n$btn-hover-border-tint-amount:    10% !default;\n$btn-active-bg-shade-amount:      20% !default;\n$btn-active-bg-tint-amount:       20% !default;\n$btn-active-border-shade-amount:  25% !default;\n$btn-active-border-tint-amount:   10% !default;\n// scss-docs-end btn-variables\n\n\n// Forms\n\n// scss-docs-start form-text-variables\n$form-text-margin-top:                  .25rem !default;\n$form-text-font-size:                   $small-font-size !default;\n$form-text-font-style:                  null !default;\n$form-text-font-weight:                 null !default;\n$form-text-color:                       var(--#{$prefix}secondary-color) !default;\n// scss-docs-end form-text-variables\n\n// scss-docs-start form-label-variables\n$form-label-margin-bottom:              .5rem !default;\n$form-label-font-size:                  null !default;\n$form-label-font-style:                 null !default;\n$form-label-font-weight:                null !default;\n$form-label-color:                      null !default;\n// scss-docs-end form-label-variables\n\n// scss-docs-start form-input-variables\n$input-padding-y:                       $input-btn-padding-y !default;\n$input-padding-x:                       $input-btn-padding-x !default;\n$input-font-family:                     $input-btn-font-family !default;\n$input-font-size:                       $input-btn-font-size !default;\n$input-font-weight:                     $font-weight-base !default;\n$input-line-height:                     $input-btn-line-height !default;\n\n$input-padding-y-sm:                    $input-btn-padding-y-sm !default;\n$input-padding-x-sm:                    $input-btn-padding-x-sm !default;\n$input-font-size-sm:                    $input-btn-font-size-sm !default;\n\n$input-padding-y-lg:                    $input-btn-padding-y-lg !default;\n$input-padding-x-lg:                    $input-btn-padding-x-lg !default;\n$input-font-size-lg:                    $input-btn-font-size-lg !default;\n\n$input-bg:                              var(--#{$prefix}body-bg) !default;\n$input-disabled-color:                  null !default;\n$input-disabled-bg:                     var(--#{$prefix}secondary-bg) !default;\n$input-disabled-border-color:           null !default;\n\n$input-color:                           var(--#{$prefix}body-color) !default;\n$input-border-color:                    var(--#{$prefix}border-color) !default;\n$input-border-width:                    $input-btn-border-width !default;\n$input-box-shadow:                      var(--#{$prefix}box-shadow-inset) !default;\n\n$input-border-radius:                   var(--#{$prefix}border-radius) !default;\n$input-border-radius-sm:                var(--#{$prefix}border-radius-sm) !default;\n$input-border-radius-lg:                var(--#{$prefix}border-radius-lg) !default;\n\n$input-focus-bg:                        $input-bg !default;\n$input-focus-border-color:              tint-color($component-active-bg, 50%) !default;\n$input-focus-color:                     $input-color !default;\n$input-focus-width:                     $input-btn-focus-width !default;\n$input-focus-box-shadow:                $input-btn-focus-box-shadow !default;\n\n$input-placeholder-color:               var(--#{$prefix}secondary-color) !default;\n$input-plaintext-color:                 var(--#{$prefix}body-color) !default;\n\n$input-height-border:                   calc(#{$input-border-width} * 2) !default; // stylelint-disable-line function-disallowed-list\n\n$input-height-inner:                    add($input-line-height * 1em, $input-padding-y * 2) !default;\n$input-height-inner-half:               add($input-line-height * .5em, $input-padding-y) !default;\n$input-height-inner-quarter:            add($input-line-height * .25em, $input-padding-y * .5) !default;\n\n$input-height:                          add($input-line-height * 1em, add($input-padding-y * 2, $input-height-border, false)) !default;\n$input-height-sm:                       add($input-line-height * 1em, add($input-padding-y-sm * 2, $input-height-border, false)) !default;\n$input-height-lg:                       add($input-line-height * 1em, add($input-padding-y-lg * 2, $input-height-border, false)) !default;\n\n$input-transition:                      border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$form-color-width:                      3rem !default;\n// scss-docs-end form-input-variables\n\n// scss-docs-start form-check-variables\n$form-check-input-width:                  1em !default;\n$form-check-min-height:                   $font-size-base * $line-height-base !default;\n$form-check-padding-start:                $form-check-input-width + .5em !default;\n$form-check-margin-bottom:                .125rem !default;\n$form-check-label-color:                  null !default;\n$form-check-label-cursor:                 null !default;\n$form-check-transition:                   null !default;\n\n$form-check-input-active-filter:          brightness(90%) !default;\n\n$form-check-input-bg:                     $input-bg !default;\n$form-check-input-border:                 var(--#{$prefix}border-width) solid var(--#{$prefix}border-color) !default;\n$form-check-input-border-radius:          .25em !default;\n$form-check-radio-border-radius:          50% !default;\n$form-check-input-focus-border:           $input-focus-border-color !default;\n$form-check-input-focus-box-shadow:       $focus-ring-box-shadow !default;\n\n$form-check-input-checked-color:          $component-active-color !default;\n$form-check-input-checked-bg-color:       $component-active-bg !default;\n$form-check-input-checked-border-color:   $form-check-input-checked-bg-color !default;\n$form-check-input-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-checked-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='m6 10 3 3 6-6'/></svg>\") !default;\n$form-check-radio-checked-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='2' fill='#{$form-check-input-checked-color}'/></svg>\") !default;\n\n$form-check-input-indeterminate-color:          $component-active-color !default;\n$form-check-input-indeterminate-bg-color:       $component-active-bg !default;\n$form-check-input-indeterminate-border-color:   $form-check-input-indeterminate-bg-color !default;\n$form-check-input-indeterminate-bg-image:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 20 20'><path fill='none' stroke='#{$form-check-input-indeterminate-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='3' d='M6 10h8'/></svg>\") !default;\n\n$form-check-input-disabled-opacity:        .5 !default;\n$form-check-label-disabled-opacity:        $form-check-input-disabled-opacity !default;\n$form-check-btn-check-disabled-opacity:    $btn-disabled-opacity !default;\n\n$form-check-inline-margin-end:    1rem !default;\n// scss-docs-end form-check-variables\n\n// scss-docs-start form-switch-variables\n$form-switch-color:               rgba($black, .25) !default;\n$form-switch-width:               2em !default;\n$form-switch-padding-start:       $form-switch-width + .5em !default;\n$form-switch-bg-image:            url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-color}'/></svg>\") !default;\n$form-switch-border-radius:       $form-switch-width !default;\n$form-switch-transition:          background-position .15s ease-in-out !default;\n\n$form-switch-focus-color:         $input-focus-border-color !default;\n$form-switch-focus-bg-image:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-focus-color}'/></svg>\") !default;\n\n$form-switch-checked-color:       $component-active-color !default;\n$form-switch-checked-bg-image:    url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='-4 -4 8 8'><circle r='3' fill='#{$form-switch-checked-color}'/></svg>\") !default;\n$form-switch-checked-bg-position: right center !default;\n// scss-docs-end form-switch-variables\n\n// scss-docs-start input-group-variables\n$input-group-addon-padding-y:           $input-padding-y !default;\n$input-group-addon-padding-x:           $input-padding-x !default;\n$input-group-addon-font-weight:         $input-font-weight !default;\n$input-group-addon-color:               $input-color !default;\n$input-group-addon-bg:                  var(--#{$prefix}tertiary-bg) !default;\n$input-group-addon-border-color:        $input-border-color !default;\n// scss-docs-end input-group-variables\n\n// scss-docs-start form-select-variables\n$form-select-padding-y:             $input-padding-y !default;\n$form-select-padding-x:             $input-padding-x !default;\n$form-select-font-family:           $input-font-family !default;\n$form-select-font-size:             $input-font-size !default;\n$form-select-indicator-padding:     $form-select-padding-x * 3 !default; // Extra padding for background-image\n$form-select-font-weight:           $input-font-weight !default;\n$form-select-line-height:           $input-line-height !default;\n$form-select-color:                 $input-color !default;\n$form-select-bg:                    $input-bg !default;\n$form-select-disabled-color:        null !default;\n$form-select-disabled-bg:           $input-disabled-bg !default;\n$form-select-disabled-border-color: $input-disabled-border-color !default;\n$form-select-bg-position:           right $form-select-padding-x center !default;\n$form-select-bg-size:               16px 12px !default; // In pixels because image dimensions\n$form-select-indicator-color:       $gray-800 !default;\n$form-select-indicator:             url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16'><path fill='none' stroke='#{$form-select-indicator-color}' stroke-linecap='round' stroke-linejoin='round' stroke-width='2' d='m2 5 6 6 6-6'/></svg>\") !default;\n\n$form-select-feedback-icon-padding-end: $form-select-padding-x * 2.5 + $form-select-indicator-padding !default;\n$form-select-feedback-icon-position:    center right $form-select-indicator-padding !default;\n$form-select-feedback-icon-size:        $input-height-inner-half $input-height-inner-half !default;\n\n$form-select-border-width:        $input-border-width !default;\n$form-select-border-color:        $input-border-color !default;\n$form-select-border-radius:       $input-border-radius !default;\n$form-select-box-shadow:          var(--#{$prefix}box-shadow-inset) !default;\n\n$form-select-focus-border-color:  $input-focus-border-color !default;\n$form-select-focus-width:         $input-focus-width !default;\n$form-select-focus-box-shadow:    0 0 0 $form-select-focus-width $input-btn-focus-color !default;\n\n$form-select-padding-y-sm:        $input-padding-y-sm !default;\n$form-select-padding-x-sm:        $input-padding-x-sm !default;\n$form-select-font-size-sm:        $input-font-size-sm !default;\n$form-select-border-radius-sm:    $input-border-radius-sm !default;\n\n$form-select-padding-y-lg:        $input-padding-y-lg !default;\n$form-select-padding-x-lg:        $input-padding-x-lg !default;\n$form-select-font-size-lg:        $input-font-size-lg !default;\n$form-select-border-radius-lg:    $input-border-radius-lg !default;\n\n$form-select-transition:          $input-transition !default;\n// scss-docs-end form-select-variables\n\n// scss-docs-start form-range-variables\n$form-range-track-width:          100% !default;\n$form-range-track-height:         .5rem !default;\n$form-range-track-cursor:         pointer !default;\n$form-range-track-bg:             var(--#{$prefix}secondary-bg) !default;\n$form-range-track-border-radius:  1rem !default;\n$form-range-track-box-shadow:     var(--#{$prefix}box-shadow-inset) !default;\n\n$form-range-thumb-width:                   1rem !default;\n$form-range-thumb-height:                  $form-range-thumb-width !default;\n$form-range-thumb-bg:                      $component-active-bg !default;\n$form-range-thumb-border:                  0 !default;\n$form-range-thumb-border-radius:           1rem !default;\n$form-range-thumb-box-shadow:              0 .1rem .25rem rgba($black, .1) !default;\n$form-range-thumb-focus-box-shadow:        0 0 0 1px $body-bg, $input-focus-box-shadow !default;\n$form-range-thumb-focus-box-shadow-width:  $input-focus-width !default; // For focus box shadow issue in Edge\n$form-range-thumb-active-bg:               tint-color($component-active-bg, 70%) !default;\n$form-range-thumb-disabled-bg:             var(--#{$prefix}secondary-color) !default;\n$form-range-thumb-transition:              background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n// scss-docs-end form-range-variables\n\n// scss-docs-start form-file-variables\n$form-file-button-color:          $input-color !default;\n$form-file-button-bg:             var(--#{$prefix}tertiary-bg) !default;\n$form-file-button-hover-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end form-file-variables\n\n// scss-docs-start form-floating-variables\n$form-floating-height:                  add(3.5rem, $input-height-border) !default;\n$form-floating-line-height:             1.25 !default;\n$form-floating-padding-x:               $input-padding-x !default;\n$form-floating-padding-y:               1rem !default;\n$form-floating-input-padding-t:         1.625rem !default;\n$form-floating-input-padding-b:         .625rem !default;\n$form-floating-label-height:            1.5em !default;\n$form-floating-label-opacity:           .65 !default;\n$form-floating-label-transform:         scale(.85) translateY(-.5rem) translateX(.15rem) !default;\n$form-floating-label-disabled-color:    $gray-600 !default;\n$form-floating-transition:              opacity .1s ease-in-out, transform .1s ease-in-out !default;\n// scss-docs-end form-floating-variables\n\n// Form validation\n\n// scss-docs-start form-feedback-variables\n$form-feedback-margin-top:          $form-text-margin-top !default;\n$form-feedback-font-size:           $form-text-font-size !default;\n$form-feedback-font-style:          $form-text-font-style !default;\n$form-feedback-valid-color:         $success !default;\n$form-feedback-invalid-color:       $danger !default;\n\n$form-feedback-icon-valid-color:    $form-feedback-valid-color !default;\n$form-feedback-icon-valid:          url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 8 8'><path fill='#{$form-feedback-icon-valid-color}' d='M2.3 6.73.6 4.53c-.4-1.04.46-1.4 1.1-.8l1.1 1.4 3.4-3.8c.6-.63 1.6-.27 1.2.7l-4 4.6c-.43.5-.8.4-1.1.1z'/></svg>\") !default;\n$form-feedback-icon-invalid-color:  $form-feedback-invalid-color !default;\n$form-feedback-icon-invalid:        url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 12 12' width='12' height='12' fill='none' stroke='#{$form-feedback-icon-invalid-color}'><circle cx='6' cy='6' r='4.5'/><path stroke-linejoin='round' d='M5.8 3.6h.4L6 6.5z'/><circle cx='6' cy='8.2' r='.6' fill='#{$form-feedback-icon-invalid-color}' stroke='none'/></svg>\") !default;\n// scss-docs-end form-feedback-variables\n\n// scss-docs-start form-validation-colors\n$form-valid-color:                  $form-feedback-valid-color !default;\n$form-valid-border-color:           $form-feedback-valid-color !default;\n$form-invalid-color:                $form-feedback-invalid-color !default;\n$form-invalid-border-color:         $form-feedback-invalid-color !default;\n// scss-docs-end form-validation-colors\n\n// scss-docs-start form-validation-states\n$form-validation-states: (\n  \"valid\": (\n    \"color\": var(--#{$prefix}form-valid-color),\n    \"icon\": $form-feedback-icon-valid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}success),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}success-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-valid-border-color),\n  ),\n  \"invalid\": (\n    \"color\": var(--#{$prefix}form-invalid-color),\n    \"icon\": $form-feedback-icon-invalid,\n    \"tooltip-color\": #fff,\n    \"tooltip-bg-color\": var(--#{$prefix}danger),\n    \"focus-box-shadow\": 0 0 $input-btn-focus-blur $input-focus-width rgba(var(--#{$prefix}danger-rgb), $input-btn-focus-color-opacity),\n    \"border-color\": var(--#{$prefix}form-invalid-border-color),\n  )\n) !default;\n// scss-docs-end form-validation-states\n\n// Z-index master list\n//\n// Warning: Avoid customizing these values. They're used for a bird's eye view\n// of components dependent on the z-axis and are designed to all work together.\n\n// scss-docs-start zindex-stack\n$zindex-dropdown:                   1000 !default;\n$zindex-sticky:                     1020 !default;\n$zindex-fixed:                      1030 !default;\n$zindex-offcanvas-backdrop:         1040 !default;\n$zindex-offcanvas:                  1045 !default;\n$zindex-modal-backdrop:             1050 !default;\n$zindex-modal:                      1055 !default;\n$zindex-popover:                    1070 !default;\n$zindex-tooltip:                    1080 !default;\n$zindex-toast:                      1090 !default;\n// scss-docs-end zindex-stack\n\n// scss-docs-start zindex-levels-map\n$zindex-levels: (\n  n1: -1,\n  0: 0,\n  1: 1,\n  2: 2,\n  3: 3\n) !default;\n// scss-docs-end zindex-levels-map\n\n\n// Navs\n\n// scss-docs-start nav-variables\n$nav-link-padding-y:                .5rem !default;\n$nav-link-padding-x:                1rem !default;\n$nav-link-font-size:                null !default;\n$nav-link-font-weight:              null !default;\n$nav-link-color:                    var(--#{$prefix}link-color) !default;\n$nav-link-hover-color:              var(--#{$prefix}link-hover-color) !default;\n$nav-link-transition:               color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out !default;\n$nav-link-disabled-color:           var(--#{$prefix}secondary-color) !default;\n$nav-link-focus-box-shadow:         $focus-ring-box-shadow !default;\n\n$nav-tabs-border-color:             var(--#{$prefix}border-color) !default;\n$nav-tabs-border-width:             var(--#{$prefix}border-width) !default;\n$nav-tabs-border-radius:            var(--#{$prefix}border-radius) !default;\n$nav-tabs-link-hover-border-color:  var(--#{$prefix}secondary-bg) var(--#{$prefix}secondary-bg) $nav-tabs-border-color !default;\n$nav-tabs-link-active-color:        var(--#{$prefix}emphasis-color) !default;\n$nav-tabs-link-active-bg:           var(--#{$prefix}body-bg) !default;\n$nav-tabs-link-active-border-color: var(--#{$prefix}border-color) var(--#{$prefix}border-color) $nav-tabs-link-active-bg !default;\n\n$nav-pills-border-radius:           var(--#{$prefix}border-radius) !default;\n$nav-pills-link-active-color:       $component-active-color !default;\n$nav-pills-link-active-bg:          $component-active-bg !default;\n\n$nav-underline-gap:                 1rem !default;\n$nav-underline-border-width:        .125rem !default;\n$nav-underline-link-active-color:   var(--#{$prefix}emphasis-color) !default;\n// scss-docs-end nav-variables\n\n\n// Navbar\n\n// scss-docs-start navbar-variables\n$navbar-padding-y:                  $spacer * .5 !default;\n$navbar-padding-x:                  null !default;\n\n$navbar-nav-link-padding-x:         .5rem !default;\n\n$navbar-brand-font-size:            $font-size-lg !default;\n// Compute the navbar-brand padding-y so the navbar-brand will have the same height as navbar-text and nav-link\n$nav-link-height:                   $font-size-base * $line-height-base + $nav-link-padding-y * 2 !default;\n$navbar-brand-height:               $navbar-brand-font-size * $line-height-base !default;\n$navbar-brand-padding-y:            ($nav-link-height - $navbar-brand-height) * .5 !default;\n$navbar-brand-margin-end:           1rem !default;\n\n$navbar-toggler-padding-y:          .25rem !default;\n$navbar-toggler-padding-x:          .75rem !default;\n$navbar-toggler-font-size:          $font-size-lg !default;\n$navbar-toggler-border-radius:      $btn-border-radius !default;\n$navbar-toggler-focus-width:        $btn-focus-width !default;\n$navbar-toggler-transition:         box-shadow .15s ease-in-out !default;\n\n$navbar-light-color:                rgba(var(--#{$prefix}emphasis-color-rgb), .65) !default;\n$navbar-light-hover-color:          rgba(var(--#{$prefix}emphasis-color-rgb), .8) !default;\n$navbar-light-active-color:         rgba(var(--#{$prefix}emphasis-color-rgb), 1) !default;\n$navbar-light-disabled-color:       rgba(var(--#{$prefix}emphasis-color-rgb), .3) !default;\n$navbar-light-icon-color:           rgba($body-color, .75) !default;\n$navbar-light-toggler-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-light-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-light-toggler-border-color: rgba(var(--#{$prefix}emphasis-color-rgb), .15) !default;\n$navbar-light-brand-color:          $navbar-light-active-color !default;\n$navbar-light-brand-hover-color:    $navbar-light-active-color !default;\n// scss-docs-end navbar-variables\n\n// scss-docs-start navbar-dark-variables\n$navbar-dark-color:                 rgba($white, .55) !default;\n$navbar-dark-hover-color:           rgba($white, .75) !default;\n$navbar-dark-active-color:          $white !default;\n$navbar-dark-disabled-color:        rgba($white, .25) !default;\n$navbar-dark-icon-color:            $navbar-dark-color !default;\n$navbar-dark-toggler-icon-bg:       url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'><path stroke='#{$navbar-dark-icon-color}' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/></svg>\") !default;\n$navbar-dark-toggler-border-color:  rgba($white, .1) !default;\n$navbar-dark-brand-color:           $navbar-dark-active-color !default;\n$navbar-dark-brand-hover-color:     $navbar-dark-active-color !default;\n// scss-docs-end navbar-dark-variables\n\n\n// Dropdowns\n//\n// Dropdown menu container and contents.\n\n// scss-docs-start dropdown-variables\n$dropdown-min-width:                10rem !default;\n$dropdown-padding-x:                0 !default;\n$dropdown-padding-y:                .5rem !default;\n$dropdown-spacer:                   .125rem !default;\n$dropdown-font-size:                $font-size-base !default;\n$dropdown-color:                    var(--#{$prefix}body-color) !default;\n$dropdown-bg:                       var(--#{$prefix}body-bg) !default;\n$dropdown-border-color:             var(--#{$prefix}border-color-translucent) !default;\n$dropdown-border-radius:            var(--#{$prefix}border-radius) !default;\n$dropdown-border-width:             var(--#{$prefix}border-width) !default;\n$dropdown-inner-border-radius:      calc(#{$dropdown-border-radius} - #{$dropdown-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$dropdown-divider-bg:               $dropdown-border-color !default;\n$dropdown-divider-margin-y:         $spacer * .5 !default;\n$dropdown-box-shadow:               var(--#{$prefix}box-shadow) !default;\n\n$dropdown-link-color:               var(--#{$prefix}body-color) !default;\n$dropdown-link-hover-color:         $dropdown-link-color !default;\n$dropdown-link-hover-bg:            var(--#{$prefix}tertiary-bg) !default;\n\n$dropdown-link-active-color:        $component-active-color !default;\n$dropdown-link-active-bg:           $component-active-bg !default;\n\n$dropdown-link-disabled-color:      var(--#{$prefix}tertiary-color) !default;\n\n$dropdown-item-padding-y:           $spacer * .25 !default;\n$dropdown-item-padding-x:           $spacer !default;\n\n$dropdown-header-color:             $gray-600 !default;\n$dropdown-header-padding-x:         $dropdown-item-padding-x !default;\n$dropdown-header-padding-y:         $dropdown-padding-y !default;\n// fusv-disable\n$dropdown-header-padding:           $dropdown-header-padding-y $dropdown-header-padding-x !default; // Deprecated in v5.2.0\n// fusv-enable\n// scss-docs-end dropdown-variables\n\n// scss-docs-start dropdown-dark-variables\n$dropdown-dark-color:               $gray-300 !default;\n$dropdown-dark-bg:                  $gray-800 !default;\n$dropdown-dark-border-color:        $dropdown-border-color !default;\n$dropdown-dark-divider-bg:          $dropdown-divider-bg !default;\n$dropdown-dark-box-shadow:          null !default;\n$dropdown-dark-link-color:          $dropdown-dark-color !default;\n$dropdown-dark-link-hover-color:    $white !default;\n$dropdown-dark-link-hover-bg:       rgba($white, .15) !default;\n$dropdown-dark-link-active-color:   $dropdown-link-active-color !default;\n$dropdown-dark-link-active-bg:      $dropdown-link-active-bg !default;\n$dropdown-dark-link-disabled-color: $gray-500 !default;\n$dropdown-dark-header-color:        $gray-500 !default;\n// scss-docs-end dropdown-dark-variables\n\n\n// Pagination\n\n// scss-docs-start pagination-variables\n$pagination-padding-y:              .375rem !default;\n$pagination-padding-x:              .75rem !default;\n$pagination-padding-y-sm:           .25rem !default;\n$pagination-padding-x-sm:           .5rem !default;\n$pagination-padding-y-lg:           .75rem !default;\n$pagination-padding-x-lg:           1.5rem !default;\n\n$pagination-font-size:              $font-size-base !default;\n\n$pagination-color:                  var(--#{$prefix}link-color) !default;\n$pagination-bg:                     var(--#{$prefix}body-bg) !default;\n$pagination-border-radius:          var(--#{$prefix}border-radius) !default;\n$pagination-border-width:           var(--#{$prefix}border-width) !default;\n$pagination-margin-start:           calc(#{$pagination-border-width} * -1) !default; // stylelint-disable-line function-disallowed-list\n$pagination-border-color:           var(--#{$prefix}border-color) !default;\n\n$pagination-focus-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-focus-bg:               var(--#{$prefix}secondary-bg) !default;\n$pagination-focus-box-shadow:       $focus-ring-box-shadow !default;\n$pagination-focus-outline:          0 !default;\n\n$pagination-hover-color:            var(--#{$prefix}link-hover-color) !default;\n$pagination-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$pagination-hover-border-color:     var(--#{$prefix}border-color) !default; // Todo in v6: remove this?\n\n$pagination-active-color:           $component-active-color !default;\n$pagination-active-bg:              $component-active-bg !default;\n$pagination-active-border-color:    $component-active-bg !default;\n\n$pagination-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$pagination-disabled-bg:            var(--#{$prefix}secondary-bg) !default;\n$pagination-disabled-border-color:  var(--#{$prefix}border-color) !default;\n\n$pagination-transition:              color .15s ease-in-out, background-color .15s ease-in-out, border-color .15s ease-in-out, box-shadow .15s ease-in-out !default;\n\n$pagination-border-radius-sm:       var(--#{$prefix}border-radius-sm) !default;\n$pagination-border-radius-lg:       var(--#{$prefix}border-radius-lg) !default;\n// scss-docs-end pagination-variables\n\n\n// Placeholders\n\n// scss-docs-start placeholders\n$placeholder-opacity-max:           .5 !default;\n$placeholder-opacity-min:           .2 !default;\n// scss-docs-end placeholders\n\n// Cards\n\n// scss-docs-start card-variables\n$card-spacer-y:                     $spacer !default;\n$card-spacer-x:                     $spacer !default;\n$card-title-spacer-y:               $spacer * .5 !default;\n$card-title-color:                  null !default;\n$card-subtitle-color:               null !default;\n$card-border-width:                 var(--#{$prefix}border-width) !default;\n$card-border-color:                 var(--#{$prefix}border-color-translucent) !default;\n$card-border-radius:                var(--#{$prefix}border-radius) !default;\n$card-box-shadow:                   null !default;\n$card-inner-border-radius:          subtract($card-border-radius, $card-border-width) !default;\n$card-cap-padding-y:                $card-spacer-y * .5 !default;\n$card-cap-padding-x:                $card-spacer-x !default;\n$card-cap-bg:                       rgba(var(--#{$prefix}body-color-rgb), .03) !default;\n$card-cap-color:                    null !default;\n$card-height:                       null !default;\n$card-color:                        null !default;\n$card-bg:                           var(--#{$prefix}body-bg) !default;\n$card-img-overlay-padding:          $spacer !default;\n$card-group-margin:                 $grid-gutter-width * .5 !default;\n// scss-docs-end card-variables\n\n// Accordion\n\n// scss-docs-start accordion-variables\n$accordion-padding-y:                     1rem !default;\n$accordion-padding-x:                     1.25rem !default;\n$accordion-color:                         var(--#{$prefix}body-color) !default;\n$accordion-bg:                            var(--#{$prefix}body-bg) !default;\n$accordion-border-width:                  var(--#{$prefix}border-width) !default;\n$accordion-border-color:                  var(--#{$prefix}border-color) !default;\n$accordion-border-radius:                 var(--#{$prefix}border-radius) !default;\n$accordion-inner-border-radius:           subtract($accordion-border-radius, $accordion-border-width) !default;\n\n$accordion-body-padding-y:                $accordion-padding-y !default;\n$accordion-body-padding-x:                $accordion-padding-x !default;\n\n$accordion-button-padding-y:              $accordion-padding-y !default;\n$accordion-button-padding-x:              $accordion-padding-x !default;\n$accordion-button-color:                  var(--#{$prefix}body-color) !default;\n$accordion-button-bg:                     var(--#{$prefix}accordion-bg) !default;\n$accordion-transition:                    $btn-transition, border-radius .15s ease !default;\n$accordion-button-active-bg:              var(--#{$prefix}primary-bg-subtle) !default;\n$accordion-button-active-color:           var(--#{$prefix}primary-text-emphasis) !default;\n\n$accordion-button-focus-border-color:     $input-focus-border-color !default;\n$accordion-button-focus-box-shadow:       $btn-focus-box-shadow !default;\n\n$accordion-icon-width:                    1.25rem !default;\n$accordion-icon-color:                    $body-color !default;\n$accordion-icon-active-color:             $primary-text-emphasis !default;\n$accordion-icon-transition:               transform .2s ease-in-out !default;\n$accordion-icon-transform:                rotate(-180deg) !default;\n\n$accordion-button-icon:         url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$accordion-icon-color}'><path fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/></svg>\") !default;\n$accordion-button-active-icon:  url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$accordion-icon-active-color}'><path fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/></svg>\") !default;\n// scss-docs-end accordion-variables\n\n// Tooltips\n\n// scss-docs-start tooltip-variables\n$tooltip-font-size:                 $font-size-sm !default;\n$tooltip-max-width:                 200px !default;\n$tooltip-color:                     var(--#{$prefix}body-bg) !default;\n$tooltip-bg:                        var(--#{$prefix}emphasis-color) !default;\n$tooltip-border-radius:             var(--#{$prefix}border-radius) !default;\n$tooltip-opacity:                   .9 !default;\n$tooltip-padding-y:                 $spacer * .25 !default;\n$tooltip-padding-x:                 $spacer * .5 !default;\n$tooltip-margin:                    null !default; // TODO: remove this in v6\n\n$tooltip-arrow-width:               .8rem !default;\n$tooltip-arrow-height:              .4rem !default;\n// fusv-disable\n$tooltip-arrow-color:               null !default; // Deprecated in Bootstrap 5.2.0 for CSS variables\n// fusv-enable\n// scss-docs-end tooltip-variables\n\n// Form tooltips must come after regular tooltips\n// scss-docs-start tooltip-feedback-variables\n$form-feedback-tooltip-padding-y:     $tooltip-padding-y !default;\n$form-feedback-tooltip-padding-x:     $tooltip-padding-x !default;\n$form-feedback-tooltip-font-size:     $tooltip-font-size !default;\n$form-feedback-tooltip-line-height:   null !default;\n$form-feedback-tooltip-opacity:       $tooltip-opacity !default;\n$form-feedback-tooltip-border-radius: $tooltip-border-radius !default;\n// scss-docs-end tooltip-feedback-variables\n\n\n// Popovers\n\n// scss-docs-start popover-variables\n$popover-font-size:                 $font-size-sm !default;\n$popover-bg:                        var(--#{$prefix}body-bg) !default;\n$popover-max-width:                 276px !default;\n$popover-border-width:              var(--#{$prefix}border-width) !default;\n$popover-border-color:              var(--#{$prefix}border-color-translucent) !default;\n$popover-border-radius:             var(--#{$prefix}border-radius-lg) !default;\n$popover-inner-border-radius:       calc(#{$popover-border-radius} - #{$popover-border-width}) !default; // stylelint-disable-line function-disallowed-list\n$popover-box-shadow:                var(--#{$prefix}box-shadow) !default;\n\n$popover-header-font-size:          $font-size-base !default;\n$popover-header-bg:                 var(--#{$prefix}secondary-bg) !default;\n$popover-header-color:              $headings-color !default;\n$popover-header-padding-y:          .5rem !default;\n$popover-header-padding-x:          $spacer !default;\n\n$popover-body-color:                var(--#{$prefix}body-color) !default;\n$popover-body-padding-y:            $spacer !default;\n$popover-body-padding-x:            $spacer !default;\n\n$popover-arrow-width:               1rem !default;\n$popover-arrow-height:              .5rem !default;\n// scss-docs-end popover-variables\n\n// fusv-disable\n// Deprecated in Bootstrap 5.2.0 for CSS variables\n$popover-arrow-color:               $popover-bg !default;\n$popover-arrow-outer-color:         var(--#{$prefix}border-color-translucent) !default;\n// fusv-enable\n\n\n// Toasts\n\n// scss-docs-start toast-variables\n$toast-max-width:                   350px !default;\n$toast-padding-x:                   .75rem !default;\n$toast-padding-y:                   .5rem !default;\n$toast-font-size:                   .875rem !default;\n$toast-color:                       null !default;\n$toast-background-color:            rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-border-width:                var(--#{$prefix}border-width) !default;\n$toast-border-color:                var(--#{$prefix}border-color-translucent) !default;\n$toast-border-radius:               var(--#{$prefix}border-radius) !default;\n$toast-box-shadow:                  var(--#{$prefix}box-shadow) !default;\n$toast-spacing:                     $container-padding-x !default;\n\n$toast-header-color:                var(--#{$prefix}secondary-color) !default;\n$toast-header-background-color:     rgba(var(--#{$prefix}body-bg-rgb), .85) !default;\n$toast-header-border-color:         $toast-border-color !default;\n// scss-docs-end toast-variables\n\n\n// Badges\n\n// scss-docs-start badge-variables\n$badge-font-size:                   .75em !default;\n$badge-font-weight:                 $font-weight-bold !default;\n$badge-color:                       $white !default;\n$badge-padding-y:                   .35em !default;\n$badge-padding-x:                   .65em !default;\n$badge-border-radius:               var(--#{$prefix}border-radius) !default;\n// scss-docs-end badge-variables\n\n\n// Modals\n\n// scss-docs-start modal-variables\n$modal-inner-padding:               $spacer !default;\n\n$modal-footer-margin-between:       .5rem !default;\n\n$modal-dialog-margin:               .5rem !default;\n$modal-dialog-margin-y-sm-up:       1.75rem !default;\n\n$modal-title-line-height:           $line-height-base !default;\n\n$modal-content-color:               null !default;\n$modal-content-bg:                  var(--#{$prefix}body-bg) !default;\n$modal-content-border-color:        var(--#{$prefix}border-color-translucent) !default;\n$modal-content-border-width:        var(--#{$prefix}border-width) !default;\n$modal-content-border-radius:       var(--#{$prefix}border-radius-lg) !default;\n$modal-content-inner-border-radius: subtract($modal-content-border-radius, $modal-content-border-width) !default;\n$modal-content-box-shadow-xs:       var(--#{$prefix}box-shadow-sm) !default;\n$modal-content-box-shadow-sm-up:    var(--#{$prefix}box-shadow) !default;\n\n$modal-backdrop-bg:                 $black !default;\n$modal-backdrop-opacity:            .5 !default;\n\n$modal-header-border-color:         var(--#{$prefix}border-color) !default;\n$modal-header-border-width:         $modal-content-border-width !default;\n$modal-header-padding-y:            $modal-inner-padding !default;\n$modal-header-padding-x:            $modal-inner-padding !default;\n$modal-header-padding:              $modal-header-padding-y $modal-header-padding-x !default; // Keep this for backwards compatibility\n\n$modal-footer-bg:                   null !default;\n$modal-footer-border-color:         $modal-header-border-color !default;\n$modal-footer-border-width:         $modal-header-border-width !default;\n\n$modal-sm:                          300px !default;\n$modal-md:                          500px !default;\n$modal-lg:                          800px !default;\n$modal-xl:                          1140px !default;\n\n$modal-fade-transform:              translate(0, -50px) !default;\n$modal-show-transform:              none !default;\n$modal-transition:                  transform .3s ease-out !default;\n$modal-scale-transform:             scale(1.02) !default;\n// scss-docs-end modal-variables\n\n\n// Alerts\n//\n// Define alert colors, border radius, and padding.\n\n// scss-docs-start alert-variables\n$alert-padding-y:               $spacer !default;\n$alert-padding-x:               $spacer !default;\n$alert-margin-bottom:           1rem !default;\n$alert-border-radius:           var(--#{$prefix}border-radius) !default;\n$alert-link-font-weight:        $font-weight-bold !default;\n$alert-border-width:            var(--#{$prefix}border-width) !default;\n$alert-dismissible-padding-r:   $alert-padding-x * 3 !default; // 3x covers width of x plus default padding on either side\n// scss-docs-end alert-variables\n\n// fusv-disable\n$alert-bg-scale:                -80% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-border-scale:            -70% !default; // Deprecated in v5.2.0, to be removed in v6\n$alert-color-scale:             40% !default; // Deprecated in v5.2.0, to be removed in v6\n// fusv-enable\n\n// Progress bars\n\n// scss-docs-start progress-variables\n$progress-height:                   1rem !default;\n$progress-font-size:                $font-size-base * .75 !default;\n$progress-bg:                       var(--#{$prefix}secondary-bg) !default;\n$progress-border-radius:            var(--#{$prefix}border-radius) !default;\n$progress-box-shadow:               var(--#{$prefix}box-shadow-inset) !default;\n$progress-bar-color:                $white !default;\n$progress-bar-bg:                   $primary !default;\n$progress-bar-animation-timing:     1s linear infinite !default;\n$progress-bar-transition:           width .6s ease !default;\n// scss-docs-end progress-variables\n\n\n// List group\n\n// scss-docs-start list-group-variables\n$list-group-color:                  var(--#{$prefix}body-color) !default;\n$list-group-bg:                     var(--#{$prefix}body-bg) !default;\n$list-group-border-color:           var(--#{$prefix}border-color) !default;\n$list-group-border-width:           var(--#{$prefix}border-width) !default;\n$list-group-border-radius:          var(--#{$prefix}border-radius) !default;\n\n$list-group-item-padding-y:         $spacer * .5 !default;\n$list-group-item-padding-x:         $spacer !default;\n// fusv-disable\n$list-group-item-bg-scale:          -80% !default; // Deprecated in v5.3.0\n$list-group-item-color-scale:       40% !default; // Deprecated in v5.3.0\n// fusv-enable\n\n$list-group-hover-bg:               var(--#{$prefix}tertiary-bg) !default;\n$list-group-active-color:           $component-active-color !default;\n$list-group-active-bg:              $component-active-bg !default;\n$list-group-active-border-color:    $list-group-active-bg !default;\n\n$list-group-disabled-color:         var(--#{$prefix}secondary-color) !default;\n$list-group-disabled-bg:            $list-group-bg !default;\n\n$list-group-action-color:           var(--#{$prefix}secondary-color) !default;\n$list-group-action-hover-color:     var(--#{$prefix}emphasis-color) !default;\n\n$list-group-action-active-color:    var(--#{$prefix}body-color) !default;\n$list-group-action-active-bg:       var(--#{$prefix}secondary-bg) !default;\n// scss-docs-end list-group-variables\n\n\n// Image thumbnails\n\n// scss-docs-start thumbnail-variables\n$thumbnail-padding:                 .25rem !default;\n$thumbnail-bg:                      var(--#{$prefix}body-bg) !default;\n$thumbnail-border-width:            var(--#{$prefix}border-width) !default;\n$thumbnail-border-color:            var(--#{$prefix}border-color) !default;\n$thumbnail-border-radius:           var(--#{$prefix}border-radius) !default;\n$thumbnail-box-shadow:              var(--#{$prefix}box-shadow-sm) !default;\n// scss-docs-end thumbnail-variables\n\n\n// Figures\n\n// scss-docs-start figure-variables\n$figure-caption-font-size:          $small-font-size !default;\n$figure-caption-color:              var(--#{$prefix}secondary-color) !default;\n// scss-docs-end figure-variables\n\n\n// Breadcrumbs\n\n// scss-docs-start breadcrumb-variables\n$breadcrumb-font-size:              null !default;\n$breadcrumb-padding-y:              0 !default;\n$breadcrumb-padding-x:              0 !default;\n$breadcrumb-item-padding-x:         .5rem !default;\n$breadcrumb-margin-bottom:          1rem !default;\n$breadcrumb-bg:                     null !default;\n$breadcrumb-divider-color:          var(--#{$prefix}secondary-color) !default;\n$breadcrumb-active-color:           var(--#{$prefix}secondary-color) !default;\n$breadcrumb-divider:                quote(\"/\") !default;\n$breadcrumb-divider-flipped:        $breadcrumb-divider !default;\n$breadcrumb-border-radius:          null !default;\n// scss-docs-end breadcrumb-variables\n\n// Carousel\n\n// scss-docs-start carousel-variables\n$carousel-control-color:             $white !default;\n$carousel-control-width:             15% !default;\n$carousel-control-opacity:           .5 !default;\n$carousel-control-hover-opacity:     .9 !default;\n$carousel-control-transition:        opacity .15s ease !default;\n\n$carousel-indicator-width:           30px !default;\n$carousel-indicator-height:          3px !default;\n$carousel-indicator-hit-area-height: 10px !default;\n$carousel-indicator-spacer:          3px !default;\n$carousel-indicator-opacity:         .5 !default;\n$carousel-indicator-active-bg:       $white !default;\n$carousel-indicator-active-opacity:  1 !default;\n$carousel-indicator-transition:      opacity .6s ease !default;\n\n$carousel-caption-width:             70% !default;\n$carousel-caption-color:             $white !default;\n$carousel-caption-padding-y:         1.25rem !default;\n$carousel-caption-spacer:            1.25rem !default;\n\n$carousel-control-icon-width:        2rem !default;\n\n$carousel-control-prev-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M11.354 1.646a.5.5 0 0 1 0 .708L5.707 8l5.647 5.646a.5.5 0 0 1-.708.708l-6-6a.5.5 0 0 1 0-.708l6-6a.5.5 0 0 1 .708 0z'/></svg>\") !default;\n$carousel-control-next-icon-bg:      url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$carousel-control-color}'><path d='M4.646 1.646a.5.5 0 0 1 .708 0l6 6a.5.5 0 0 1 0 .708l-6 6a.5.5 0 0 1-.708-.708L10.293 8 4.646 2.354a.5.5 0 0 1 0-.708z'/></svg>\") !default;\n\n$carousel-transition-duration:       .6s !default;\n$carousel-transition:                transform $carousel-transition-duration ease-in-out !default; // Define transform transition first if using multiple transitions (e.g., `transform 2s ease, opacity .5s ease-out`)\n// scss-docs-end carousel-variables\n\n// scss-docs-start carousel-dark-variables\n$carousel-dark-indicator-active-bg:  $black !default;\n$carousel-dark-caption-color:        $black !default;\n$carousel-dark-control-icon-filter:  invert(1) grayscale(100) !default;\n// scss-docs-end carousel-dark-variables\n\n\n// Spinners\n\n// scss-docs-start spinner-variables\n$spinner-width:           2rem !default;\n$spinner-height:          $spinner-width !default;\n$spinner-vertical-align:  -.125em !default;\n$spinner-border-width:    .25em !default;\n$spinner-animation-speed: .75s !default;\n\n$spinner-width-sm:        1rem !default;\n$spinner-height-sm:       $spinner-width-sm !default;\n$spinner-border-width-sm: .2em !default;\n// scss-docs-end spinner-variables\n\n\n// Close\n\n// scss-docs-start close-variables\n$btn-close-width:            1em !default;\n$btn-close-height:           $btn-close-width !default;\n$btn-close-padding-x:        .25em !default;\n$btn-close-padding-y:        $btn-close-padding-x !default;\n$btn-close-color:            $black !default;\n$btn-close-bg:               url(\"data:image/svg+xml,<svg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='#{$btn-close-color}'><path d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/></svg>\") !default;\n$btn-close-focus-shadow:     $focus-ring-box-shadow !default;\n$btn-close-opacity:          .5 !default;\n$btn-close-hover-opacity:    .75 !default;\n$btn-close-focus-opacity:    1 !default;\n$btn-close-disabled-opacity: .25 !default;\n$btn-close-white-filter:     invert(1) grayscale(100%) brightness(200%) !default;\n// scss-docs-end close-variables\n\n\n// Offcanvas\n\n// scss-docs-start offcanvas-variables\n$offcanvas-padding-y:               $modal-inner-padding !default;\n$offcanvas-padding-x:               $modal-inner-padding !default;\n$offcanvas-horizontal-width:        400px !default;\n$offcanvas-vertical-height:         30vh !default;\n$offcanvas-transition-duration:     .3s !default;\n$offcanvas-border-color:            $modal-content-border-color !default;\n$offcanvas-border-width:            $modal-content-border-width !default;\n$offcanvas-title-line-height:       $modal-title-line-height !default;\n$offcanvas-bg-color:                var(--#{$prefix}body-bg) !default;\n$offcanvas-color:                   var(--#{$prefix}body-color) !default;\n$offcanvas-box-shadow:              $modal-content-box-shadow-xs !default;\n$offcanvas-backdrop-bg:             $modal-backdrop-bg !default;\n$offcanvas-backdrop-opacity:        $modal-backdrop-opacity !default;\n// scss-docs-end offcanvas-variables\n\n// Code\n\n$code-font-size:                    $small-font-size !default;\n$code-color:                        $pink !default;\n\n$kbd-padding-y:                     .1875rem !default;\n$kbd-padding-x:                     .375rem !default;\n$kbd-font-size:                     $code-font-size !default;\n$kbd-color:                         var(--#{$prefix}body-bg) !default;\n$kbd-bg:                            var(--#{$prefix}body-color) !default;\n$nested-kbd-font-weight:            null !default; // Deprecated in v5.2.0, removing in v6\n\n$pre-color:                         null !default;\n","/***\n * Rules for the UX/UI of skip navigation link btn. \n *It's only visible to people\n * navigating with keyboard for accessibility purposes\n * ref: https://www.youtube.com/watch?v=VUR0I5mqq7I\n ***/\n\n.skip-link {\n  position: fixed;\n  top: 0;\n  left: 0;\n  right: 0;\n  text-align: center;\n  background-color: var(--pst-color-warning);\n  // Ensure we are using a WCAG conformant colour\n  color: var(--pst-color-warning-text) !important;\n  padding: 0.5rem;\n  z-index: $zindex-modal;\n  border-bottom: 1px solid var(--pst-color-border);\n\n  // This shows / hides the button\n  transform: translateY(-100%);\n  transition: transform 150ms ease-in-out;\n  &:focus {\n    transform: translateY(0%);\n    // ensure this is visible\n    outline: 3px solid $foundation-black;\n  }\n}\n",".bd-container {\n  flex-grow: 1;\n  display: flex;\n  justify-content: center;\n\n  .bd-container__inner {\n    display: flex;\n  }\n}\n\n.bd-page-width {\n  width: 100%;\n  @include media-breakpoint-up(lg) {\n    max-width: $breakpoint-page-width;\n  }\n}\n","// Breakpoint viewport sizes and media queries.\n//\n// Breakpoints are defined as a map of (name: minimum width), order from small to large:\n//\n//    (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px)\n//\n// The map defined in the `$grid-breakpoints` global variable is used as the `$breakpoints` argument by default.\n\n// Name of the next breakpoint, or null for the last breakpoint.\n//\n//    >> breakpoint-next(sm)\n//    md\n//    >> breakpoint-next(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    md\n//    >> breakpoint-next(sm, $breakpoint-names: (xs sm md lg xl xxl))\n//    md\n@function breakpoint-next($name, $breakpoints: $grid-breakpoints, $breakpoint-names: map-keys($breakpoints)) {\n  $n: index($breakpoint-names, $name);\n  @if not $n {\n    @error \"breakpoint `#{$name}` not found in `#{$breakpoints}`\";\n  }\n  @return if($n < length($breakpoint-names), nth($breakpoint-names, $n + 1), null);\n}\n\n// Minimum breakpoint width. Null for the smallest (first) breakpoint.\n//\n//    >> breakpoint-min(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    576px\n@function breakpoint-min($name, $breakpoints: $grid-breakpoints) {\n  $min: map-get($breakpoints, $name);\n  @return if($min != 0, $min, null);\n}\n\n// Maximum breakpoint width.\n// The maximum value is reduced by 0.02px to work around the limitations of\n// `min-` and `max-` prefixes and viewports with fractional widths.\n// See https://www.w3.org/TR/mediaqueries-4/#mq-min-max\n// Uses 0.02px rather than 0.01px to work around a current rounding bug in Safari.\n// See https://bugs.webkit.org/show_bug.cgi?id=178261\n//\n//    >> breakpoint-max(md, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    767.98px\n@function breakpoint-max($name, $breakpoints: $grid-breakpoints) {\n  $max: map-get($breakpoints, $name);\n  @return if($max and $max > 0, $max - .02, null);\n}\n\n// Returns a blank string if smallest breakpoint, otherwise returns the name with a dash in front.\n// Useful for making responsive utilities.\n//\n//    >> breakpoint-infix(xs, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"\"  (Returns a blank string)\n//    >> breakpoint-infix(sm, (xs: 0, sm: 576px, md: 768px, lg: 992px, xl: 1200px, xxl: 1400px))\n//    \"-sm\"\n@function breakpoint-infix($name, $breakpoints: $grid-breakpoints) {\n  @return if(breakpoint-min($name, $breakpoints) == null, \"\", \"-#{$name}\");\n}\n\n// Media of at least the minimum breakpoint width. No query for the smallest breakpoint.\n// Makes the @content apply to the given breakpoint and wider.\n@mixin media-breakpoint-up($name, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($name, $breakpoints);\n  @if $min {\n    @media (min-width: $min) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media of at most the maximum breakpoint width. No query for the largest breakpoint.\n// Makes the @content apply to the given breakpoint and narrower.\n@mixin media-breakpoint-down($name, $breakpoints: $grid-breakpoints) {\n  $max: breakpoint-max($name, $breakpoints);\n  @if $max {\n    @media (max-width: $max) {\n      @content;\n    }\n  } @else {\n    @content;\n  }\n}\n\n// Media that spans multiple breakpoint widths.\n// Makes the @content apply between the min and max breakpoints\n@mixin media-breakpoint-between($lower, $upper, $breakpoints: $grid-breakpoints) {\n  $min: breakpoint-min($lower, $breakpoints);\n  $max: breakpoint-max($upper, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($lower, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($upper, $breakpoints) {\n      @content;\n    }\n  }\n}\n\n// Media between the breakpoint's minimum and maximum widths.\n// No minimum for the smallest breakpoint, and no maximum for the largest one.\n// Makes the @content apply only to the given breakpoint, not viewports any wider or narrower.\n@mixin media-breakpoint-only($name, $breakpoints: $grid-breakpoints) {\n  $min:  breakpoint-min($name, $breakpoints);\n  $next: breakpoint-next($name, $breakpoints);\n  $max:  breakpoint-max($next, $breakpoints);\n\n  @if $min != null and $max != null {\n    @media (min-width: $min) and (max-width: $max) {\n      @content;\n    }\n  } @else if $max == null {\n    @include media-breakpoint-up($name, $breakpoints) {\n      @content;\n    }\n  } @else if $min == null {\n    @include media-breakpoint-down($next, $breakpoints) {\n      @content;\n    }\n  }\n}\n",".bd-header-version-warning,\n.bd-header-announcement {\n  min-height: 3rem;\n  width: 100%;\n  display: flex;\n  position: relative;\n  align-items: center;\n  justify-content: center;\n  text-align: center;\n  padding: 0.5rem 12.5%; // Horizontal padding so the width is 75%\n  // One breakpoint less than $breakpoint-sidebar-primary. See variables/_layout.scss for more info.\n  @include media-breakpoint-down(lg) {\n    // Announcements can take a bit more width on mobile\n    padding: 0.5rem 2%;\n  }\n\n  p {\n    font-weight: bold;\n    margin: 0;\n  }\n\n  &:after {\n    position: absolute;\n    width: 100%;\n    height: 100%;\n    left: 0;\n    top: 0;\n    content: \"\";\n    z-index: -1; // So it doesn't hover over the content\n  }\n\n  &:empty {\n    display: none;\n  }\n\n  // Ensure there is enough contrast against the background\n  a {\n    color: var(--pst-color-inline-code-links);\n  }\n}\n\n// Bg color is now defined in the theme color palette - using our secondary color\n.bd-header-announcement {\n  &:after {\n    background-color: var(--pst-color-secondary-bg);\n  }\n}\n\n.bd-header-version-warning {\n  &:after {\n    background-color: var(--pst-color-danger-bg);\n  }\n}\n","/**\n * Main content area\n */\n.bd-main {\n  flex-grow: 1;\n  flex-direction: column;\n  display: flex;\n  min-width: 0;\n\n  .bd-content {\n    display: flex;\n    justify-content: center;\n    height: 100%;\n    .bd-article-container {\n      justify-content: start;\n      display: flex;\n      flex-direction: column;\n      // Max-width is slightly more than the W3 since our docs often have images.\n      // We shoot for about 100 characters per line instead of 80.\n      // ref: https://www.w3.org/WAI/tutorials/page-structure/styling/#line-length\n      width: 100%;\n      max-width: 60em;\n      overflow-x: auto; // Prevent wide content from pushing off the secondary sidebar\n      padding: 1rem;\n      .bd-article {\n        // Give a bit more verticle spacing on wide screens\n        @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n          padding-top: 1.5rem;\n          padding-left: 2rem;\n        }\n      }\n    }\n  }\n}\n",".bd-footer {\n  width: 100%;\n  border-top: 1px solid var(--pst-color-border);\n\n  .bd-footer__inner {\n    display: flex;\n    flex-grow: 1;\n    padding: 1rem;\n    margin: auto;\n  }\n\n  .footer-items__start,\n  .footer-items__center,\n  .footer-items__end {\n    display: flex;\n    flex-direction: column;\n    gap: 0.5rem;\n    justify-content: center;\n    flex-grow: 1;\n  }\n\n  .footer-items__center {\n    text-align: center;\n  }\n\n  .footer-items__end {\n    text-align: end;\n  }\n\n  // So that paragraphs don't take up extra room\n  .footer-item p {\n    margin-bottom: 0;\n  }\n}\n",".bd-footer-article {\n  margin-top: auto;\n\n  .footer-article-items {\n    display: flex;\n    flex-direction: column;\n  }\n}\n",".bd-footer-content {\n  .footer-content-items {\n    display: flex;\n    flex-direction: column;\n    margin-top: auto;\n  }\n}\n","/**\n * Header at the top of the page\n * It includes the announcement bar and the navigation bar.\n */\n\n// Styling for the Icon links can be found in components/_icon-links.scss\n\n// If we want the shadow to only point downward in the future, set\n// box-shadow to: 0 0.125rem 0.25rem -0.125rem rgba(0, 0, 0, 0.11);\n.bd-header {\n  position: sticky;\n  top: 0;\n  z-index: $zindex-fixed;\n\n  // Overrides bootstrap\n  background: var(--pst-color-on-background) !important;\n  box-shadow: 0 0.125rem 0.25rem 0 var(--pst-color-shadow);\n\n  width: 100%;\n  padding: 0;\n  max-width: 100vw;\n  justify-content: center;\n  .bd-header__inner {\n    display: flex;\n    align-items: center;\n    height: fit-content;\n    padding-left: 1rem;\n    padding-right: 1rem;\n  }\n\n  // These items will define the height of the header\n  .navbar-item {\n    height: var(--pst-header-height);\n    max-height: var(--pst-header-height);\n    display: flex;\n    align-items: center;\n  }\n\n  // Hide the header items on mobile\n  .navbar-header-items {\n    display: none;\n    flex-shrink: 1;\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: flex;\n      flex-grow: 1;\n      padding: 0 0 0 0.5rem;\n    }\n  }\n\n  .navbar-header-items__end,\n  .navbar-header-items__center,\n  .navbar-header-items__start {\n    display: flex;\n    align-items: center;\n    flex-flow: wrap;\n    // In case we wrap our items to multiple rows on small screens\n    row-gap: 0;\n  }\n\n  .navbar-header-items__end,\n  .navbar-header-items__center {\n    column-gap: 1rem;\n  }\n\n  // A little smaller because this is displayed by default on mobile\n  .navbar-header-items__start {\n    flex-shrink: 0;\n    margin-right: auto;\n    gap: 0.5rem;\n  }\n\n  .navbar-header-items__end {\n    // End navbar items should snap to the right\n    justify-content: end;\n  }\n\n  // Contains the navigation links within the navbar\n  .navbar-nav {\n    display: flex;\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      // Center align on wide screens so the dropdown button is centered properly\n      align-items: center;\n    }\n\n    li a.nav-link {\n      @include link-style-text;\n    }\n\n    // Current page is always underlined in the navbar\n    > .current > .nav-link {\n      @include link-navbar-current;\n    }\n\n    // Dropdowns for the extra links\n    .dropdown {\n      button {\n        display: unset;\n        color: var(--pst-color-text-muted);\n        border: none;\n        @include link-style-hover;\n        @include focus-indicator;\n      }\n\n      .dropdown-menu {\n        z-index: $zindex-popover;\n        border: 1px solid var(--pst-color-border);\n        box-shadow: 0 0 0.3rem 0.1rem var(--pst-color-shadow);\n        background-color: var(--pst-color-on-background);\n        padding: 0.5rem 0;\n        margin: 0.5rem 0;\n        min-width: 20rem;\n\n        .dropdown-item {\n          // Give the items in the dropdown some breathing room but let the hit\n          // and hover area of the items extend to the edges of the menu\n          padding: 0.25rem 1.5rem;\n\n          // Override Bootstrap\n          &:focus:not(:hover):not(:active) {\n            background-color: inherit;\n          }\n        }\n\n        // Hide the menu unless show has been clicked\n        &:not(.show) {\n          display: none;\n        }\n      }\n    }\n  }\n}\n\n// inline the element in the navbar as long as they fit and use display block when collapsing\n@include media-breakpoint-up($breakpoint-sidebar-primary) {\n  .navbar-center-items .navbar-item {\n    display: inline-block;\n  }\n}\n\n.nav-link {\n  @include link-style-hover;\n\n  // Override Bootstrap\n  transition: none;\n\n  &.nav-external:after {\n    font: var(--fa-font-solid);\n    content: var(--pst-icon-external-link);\n    font-size: 0.75em;\n    margin-left: 0.3em;\n  }\n}\n\n.bd-navbar-elements li.nav-item i {\n  font-size: 0.7rem;\n  padding-left: 2px;\n  vertical-align: middle;\n}\n\n/**\n * Showing and hiding the sidebar toggle buttons and header items\n */\n\n// Hide the header items on mobile\n.bd-header {\n  // Toggle labels\n  label {\n    &.sidebar-toggle {\n      display: flex;\n      cursor: pointer;\n      font-size: var(--pst-font-size-icon);\n      align-items: center;\n      color: var(--pst-color-muted);\n      margin-bottom: 0;\n      padding-bottom: 0.25rem;\n    }\n\n    &.primary-toggle {\n      margin-right: 1rem;\n      @include media-breakpoint-up($breakpoint-sidebar-primary) {\n        display: none;\n      }\n    }\n\n    &.secondary-toggle {\n      margin-left: 1rem;\n      @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n        display: none;\n      }\n    }\n    @include icon-navbar-hover;\n    @include focus-indicator;\n  }\n\n  // Hide the navbar header items on mobile because they're in the sidebar\n  .navbar-header-items {\n    display: none;\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: inherit;\n    }\n  }\n}\n\n// THe elements next to the hamburger menu only show on narrow screens\n.navbar-persistent--mobile {\n  margin-left: auto;\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    display: none;\n  }\n}\n\n// The navbar-persistent content should only show on wide screens\n.navbar-persistent--container {\n  display: none;\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    display: flex;\n  }\n}\n",".header-article__inner {\n  display: flex;\n  padding: 0 0.5rem;\n\n  // The items define the height so that it disappears if there are no items\n  .header-article-item {\n    min-height: var(--pst-header-article-height);\n    height: var(--pst-header-article-height);\n  }\n\n  .header-article-items__start,\n  .header-article-items__end {\n    display: flex;\n    align-items: start;\n    gap: 0.5rem;\n  }\n\n  .header-article-items__end {\n    margin-left: auto;\n  }\n}\n","/**\n * The primary sidebar on the left.\n * e.g., between-pages navigation.\n */\n\n.bd-sidebar-primary {\n  display: flex;\n  flex-direction: column;\n  gap: 1rem;\n  max-height: calc(100vh - var(--pst-header-height));\n  position: sticky;\n  top: var(--pst-header-height);\n  @include make-col(3);\n\n  // Borders padding and whitespace\n  padding: 2rem 1rem 1rem 1rem;\n  border-right: 1px solid var(--pst-color-border);\n  background-color: var(--pst-color-background);\n  overflow-y: auto;\n\n  font-size: var(--pst-sidebar-font-size-mobile);\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    font-size: var(--pst-sidebar-font-size);\n  }\n\n  // override bootstrap when navlink are displayed in the sidebar\n  .nav-link {\n    font-size: var(--pst-sidebar-font-size-mobile);\n  }\n\n  &.no-sidebar {\n    border-right: 0;\n  }\n\n  &.hide-on-wide {\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: none;\n    }\n  }\n\n  // Headers shouldn't be colored in the sidebars and some extensions add headers\n  h1,\n  h2,\n  h3,\n  h4 {\n    color: var(--pst-color-text-base);\n  }\n\n  .sidebar-primary-items__start,\n  .sidebar-primary-items__end {\n    .sidebar-primary-item {\n      padding: 0.5rem 0;\n    }\n  }\n\n  // Hide the sidebar header items on widescreen since they are visible in the header\n  .sidebar-header-items {\n    display: flex;\n    flex-direction: column;\n\n    .sidebar-header-items__title {\n      font-weight: var(--pst-sidebar-header-font-weight);\n      font-size: var(--pst-sidebar-header-font-size);\n      color: var(--pst-color-text-base);\n      margin-bottom: 0.5rem;\n    }\n\n    // The dropdown toggle for extra links just shows them all instead.\n    .nav-item.dropdown {\n      // On mobile, the dropdown behaves like any other link, no hiding\n      button {\n        display: none;\n      }\n\n      .dropdown-menu {\n        display: flex;\n        flex-direction: column;\n        padding: 0;\n        margin: 0;\n        border: none;\n        background-color: inherit;\n        font-size: inherit;\n      }\n    }\n\n    // Center header items get displayed vertically, end items are displayed horizontally\n    .sidebar-header-items__center {\n      display: flex;\n      flex-direction: column;\n    }\n\n    // Positioning end items\n    .sidebar-header-items__end {\n      display: flex;\n      align-items: center;\n      gap: 0.5rem;\n    }\n\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      display: none;\n    }\n  }\n\n  .sidebar-primary-items__start {\n    // Add a border on mobile to separate it from the header sidebar area\n    border-top: 1px solid var(--pst-color-border);\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      border-top: none;\n    }\n  }\n\n  .sidebar-primary-items__end {\n    margin-top: auto;\n    margin-bottom: 1em;\n  }\n  .list-caption {\n    list-style: none;\n    padding-left: 0px;\n  }\n  li {\n    position: relative;\n    // If it has children, add a bit more padding to wrap the content to avoid\n    // overlapping with the <label>\n    &.has-children {\n      > .reference {\n        padding-right: 30px;\n      }\n    }\n  }\n  // Navigation item chevrons\n  label.toctree-toggle {\n    position: absolute;\n    top: 0;\n    right: 0;\n    height: 30px;\n    width: 30px;\n\n    cursor: pointer;\n\n    display: flex;\n    justify-content: center;\n    align-items: center;\n\n    &:hover {\n      background: var(--pst-color-surface);\n    }\n\n    i {\n      display: inline-block;\n      font-size: 0.75rem;\n      text-align: center;\n      &:hover {\n        color: var(--pst-color-primary);\n      }\n    }\n  }\n  .label-parts {\n    width: 100%;\n    height: 100%;\n    &:hover {\n      background: none;\n    }\n    i {\n      width: 30px;\n      position: absolute;\n      top: 0.3em; // aligning chevron with text\n      right: 0em; // aligning chevron to the right\n    }\n  }\n}\n\n/* Between-page links and captions */\nnav.bd-links {\n  margin-right: -1rem;\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    display: block;\n  }\n\n  ul {\n    list-style: none;\n\n    // Reduce padding of nested `ul` items a bit\n    ul {\n      padding: 0 0 0 1rem;\n    }\n  }\n\n  li > a {\n    display: block;\n    padding: 0.25rem 0.65rem;\n    @include link-sidebar;\n    box-shadow: none;\n\n    &.reference.external {\n      &:after {\n        font: var(--fa-font-solid);\n        content: var(--pst-icon-external-link);\n        font-size: 0.75em;\n        margin-left: 0.3em;\n      }\n    }\n  }\n\n  .current {\n    > a {\n      @include link-sidebar-current;\n      background-color: transparent;\n    }\n  }\n\n  // Title\n  p.bd-links__title {\n    font-size: var(--pst-sidebar-header-font-size);\n    font-weight: var(--pst-sidebar-header-font-weight);\n    margin-bottom: 0.5rem;\n  }\n\n  // Toctree captions\n  p.caption {\n    font-weight: var(--pst-sidebar-header-font-weight);\n    position: relative;\n    margin-top: 1.25rem;\n    margin-bottom: 0.5rem;\n    color: var(--pst-color-text-base);\n    &:first-child {\n      margin-top: 0;\n    }\n\n    font-size: var(--pst-sidebar-font-size-mobile);\n    @include media-breakpoint-up($breakpoint-sidebar-primary) {\n      font-size: var(--pst-sidebar-font-size);\n    }\n  }\n}\n","// Grid system\n//\n// Generate semantic grid columns with these mixins.\n\n@mixin make-row($gutter: $grid-gutter-width) {\n  --#{$prefix}gutter-x: #{$gutter};\n  --#{$prefix}gutter-y: 0;\n  display: flex;\n  flex-wrap: wrap;\n  // TODO: Revisit calc order after https://github.com/react-bootstrap/react-bootstrap/issues/6039 is fixed\n  margin-top: calc(-1 * var(--#{$prefix}gutter-y)); // stylelint-disable-line function-disallowed-list\n  margin-right: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n  margin-left: calc(-.5 * var(--#{$prefix}gutter-x)); // stylelint-disable-line function-disallowed-list\n}\n\n@mixin make-col-ready() {\n  // Add box sizing if only the grid is loaded\n  box-sizing: if(variable-exists(include-column-box-sizing) and $include-column-box-sizing, border-box, null);\n  // Prevent columns from becoming too narrow when at smaller grid tiers by\n  // always setting `width: 100%;`. This works because we set the width\n  // later on to override this initial width.\n  flex-shrink: 0;\n  width: 100%;\n  max-width: 100%; // Prevent `.col-auto`, `.col` (& responsive variants) from breaking out the grid\n  padding-right: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  padding-left: calc(var(--#{$prefix}gutter-x) * .5); // stylelint-disable-line function-disallowed-list\n  margin-top: var(--#{$prefix}gutter-y);\n}\n\n@mixin make-col($size: false, $columns: $grid-columns) {\n  @if $size {\n    flex: 0 0 auto;\n    width: percentage(divide($size, $columns));\n\n  } @else {\n    flex: 1 1 0;\n    max-width: 100%;\n  }\n}\n\n@mixin make-col-auto() {\n  flex: 0 0 auto;\n  width: auto;\n}\n\n@mixin make-col-offset($size, $columns: $grid-columns) {\n  $num: divide($size, $columns);\n  margin-left: if($num == 0, 0, percentage($num));\n}\n\n// Row columns\n//\n// Specify on a parent element(e.g., .row) to force immediate children into NN\n// number of columns. Supports wrapping to new lines, but does not do a Masonry\n// style grid.\n@mixin row-cols($count) {\n  > * {\n    flex: 0 0 auto;\n    width: percentage(divide(1, $count));\n  }\n}\n\n// Framework grid generation\n//\n// Used only by Bootstrap to generate the correct number of grid classes given\n// any value of `$grid-columns`.\n\n@mixin make-grid-columns($columns: $grid-columns, $gutter: $grid-gutter-width, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      // Provide basic `.col-{bp}` classes for equal-width flexbox columns\n      .col#{$infix} {\n        flex: 1 0 0%; // Flexbugs #4: https://github.com/philipwalton/flexbugs#flexbug-4\n      }\n\n      .row-cols#{$infix}-auto > * {\n        @include make-col-auto();\n      }\n\n      @if $grid-row-columns > 0 {\n        @for $i from 1 through $grid-row-columns {\n          .row-cols#{$infix}-#{$i} {\n            @include row-cols($i);\n          }\n        }\n      }\n\n      .col#{$infix}-auto {\n        @include make-col-auto();\n      }\n\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .col#{$infix}-#{$i} {\n            @include make-col($i, $columns);\n          }\n        }\n\n        // `$columns - 1` because offsetting by the width of an entire row isn't possible\n        @for $i from 0 through ($columns - 1) {\n          @if not ($infix == \"\" and $i == 0) { // Avoid emitting useless .offset-0\n            .offset#{$infix}-#{$i} {\n              @include make-col-offset($i, $columns);\n            }\n          }\n        }\n      }\n\n      // Gutters\n      //\n      // Make use of `.g-*`, `.gx-*` or `.gy-*` utilities to change spacing between the columns.\n      @each $key, $value in $gutters {\n        .g#{$infix}-#{$key},\n        .gx#{$infix}-#{$key} {\n          --#{$prefix}gutter-x: #{$value};\n        }\n\n        .g#{$infix}-#{$key},\n        .gy#{$infix}-#{$key} {\n          --#{$prefix}gutter-y: #{$value};\n        }\n      }\n    }\n  }\n}\n\n@mixin make-cssgrid($columns: $grid-columns, $breakpoints: $grid-breakpoints) {\n  @each $breakpoint in map-keys($breakpoints) {\n    $infix: breakpoint-infix($breakpoint, $breakpoints);\n\n    @include media-breakpoint-up($breakpoint, $breakpoints) {\n      @if $columns > 0 {\n        @for $i from 1 through $columns {\n          .g-col#{$infix}-#{$i} {\n            grid-column: auto / span $i;\n          }\n        }\n\n        // Start with `1` because `0` is and invalid value.\n        // Ends with `$columns - 1` because offsetting by the width of an entire row isn't possible.\n        @for $i from 1 through ($columns - 1) {\n          .g-start#{$infix}-#{$i} {\n            grid-column-start: $i;\n          }\n        }\n      }\n    }\n  }\n}\n","/**\n * Secondary sidebar on the right.\n * e.g., in-page table of contents.\n */\n.bd-sidebar-secondary {\n  display: flex;\n  order: 2;\n  flex-shrink: 0;\n  flex-direction: column;\n  position: sticky;\n  top: var(--pst-header-height);\n  max-height: calc(100vh - var(--pst-header-height));\n  padding: 2rem 1rem 1rem 1rem;\n  width: var(--pst-sidebar-secondary);\n  font-size: var(--pst-sidebar-font-size-mobile);\n  @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n    font-size: var(--pst-sidebar-font-size);\n  }\n\n  // Color and border\n  background-color: var(--pst-color-background);\n  overflow-y: auto;\n}\n\n.sidebar-secondary-item {\n  padding: 0.5rem 0.5rem;\n  @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n    border-left: 1px solid var(--pst-color-border);\n    padding-left: 1rem;\n  }\n\n  i {\n    padding-right: 0.5rem;\n  }\n}\n","/*******************************************************************************\n* Rules for the UX/UI of sidebar sliding drawers on mobile\n* Note that this sheet controls styles across many parts of the theme\n* It is aggregated into this one sheet instead of being split across\n* components in order to keep it easier to debug in one place.\n* It is broken up into major sections below.\n*/\n\n/*******************************************************************************\n* Buttons and overlays\n*/\ninput.sidebar-toggle {\n  display: none;\n}\n\n// Background overlays\nlabel.overlay {\n  background-color: black;\n  opacity: 0.5;\n  height: 0;\n  width: 0;\n  position: fixed;\n  top: 0;\n  left: 0;\n  transition: opacity $animation-time ease-out;\n  z-index: $zindex-modal-backdrop;\n}\n\ninput {\n  // Show the correct overlay when its input is checked\n  &#__primary:checked + label.overlay.overlay-primary,\n  &#__secondary:checked + label.overlay.overlay-secondary {\n    height: 100vh;\n    width: 100vw;\n  }\n\n  // Primary sidebar slides in from the left\n  &#__primary:checked ~ .bd-container .bd-sidebar-primary {\n    visibility: visible;\n    margin-left: 0;\n  }\n\n  // Secondary sidebar slides in from the right\n  &#__secondary:checked ~ .bd-container .bd-sidebar-secondary {\n    visibility: visible;\n    margin-right: 0;\n  }\n}\n\n/*******************************************************************************\n* Sidebar drawer behavior\n*/\n\n/**\n * Behavior for sliding drawer elements that will be toggled with an input\n *\n * NOTE: We use this mixin to define the toggle behavior on narrow screens,\n * And the wide-screen behavior of the sections is defined in their own section\n * .scss files.\n */\n@mixin sliding-drawer($side: \"left\") {\n  position: fixed;\n  top: 0;\n  z-index: $zindex-modal;\n  height: 100vh;\n  max-height: 100vh;\n  width: 75%;\n  flex-grow: 0.75;\n  max-width: 350px;\n  transition: visibility $animation-time ease-out,\n    margin $animation-time ease-out;\n  visibility: hidden;\n\n  @if $side == \"right\" {\n    margin-right: -75%;\n    right: 0;\n  } @else {\n    margin-left: -75%;\n    left: 0;\n  }\n}\n\n// Primary sidebar hides/shows at earlier widths\n@include media-breakpoint-up($breakpoint-sidebar-primary) {\n  label.sidebar-toggle.primary-toggle {\n    display: none;\n  }\n\n  input#__primary {\n    &:checked + label.overlay.overlay-primary {\n      height: 0;\n      width: 0;\n    }\n  }\n\n  .bd-sidebar-primary {\n    margin-left: 0;\n    visibility: visible;\n  }\n}\n\n.bd-sidebar-primary {\n  @include media-breakpoint-down($breakpoint-sidebar-primary) {\n    @include sliding-drawer(\"left\");\n  }\n}\n\n.bd-sidebar-secondary {\n  @include media-breakpoint-down($breakpoint-sidebar-secondary) {\n    @include sliding-drawer(\"right\");\n  }\n}\n","/**\n * Breadcrumbs for parent pages meant for the article header\n */\nul.bd-breadcrumbs {\n  list-style: none;\n  padding-left: 0;\n  display: flex;\n  flex-wrap: wrap;\n\n  // Font size slightly smaller to avoid cluttering up space too much\n  font-size: 0.8rem;\n\n  li.breadcrumb-item {\n    display: flex;\n    align-items: center;\n\n    // Style should look like heavier in-page links\n    // keeping visited in the default link colour\n    font-weight: bold;\n    a {\n      @include link-style-text;\n    }\n\n    // Items that aren't the home have a caret to the left\n    &:not(.breadcrumb-home):before {\n      font: var(--fa-font-solid);\n      font-size: 0.8rem;\n      content: var(--pst-breadcrumb-divider);\n      color: var(--pst-color-text-muted);\n      padding: 0 0.5rem;\n    }\n  }\n}\n","/**\n * Icon links in the navbar\n */\n\n.navbar-icon-links {\n  display: flex;\n  flex-direction: row;\n  column-gap: 1rem;\n  flex-wrap: wrap;\n\n  // Remove the padding so that we can define it with flexbox gap above\n  li.nav-item a.nav-link {\n    padding-left: 0;\n    padding-right: 0;\n    @include icon-navbar-hover;\n  }\n\n  // Spacing and centering\n  a span {\n    display: flex;\n    align-items: center;\n  }\n\n  // Icons styling\n  i {\n    &.fa-brands,\n    &.fa-regular,\n    &.fa-solid {\n      vertical-align: middle;\n      font-style: normal;\n      font-size: var(--pst-font-size-icon);\n    }\n\n    /* Social media buttons hard-code the brand color */\n    &.fa-square-twitter:before {\n      color: #55acee;\n    }\n\n    &.fa-square-gitlab:before {\n      color: #548;\n    }\n\n    &.fa-bitbucket:before {\n      color: #0052cc;\n    }\n  }\n\n  // Force images to be icon-sized\n  img.icon-link-image {\n    height: 1.5em;\n    border-radius: 0.2rem;\n  }\n}\n","/**\n * Logo in the navbar\n */\n\n.navbar-brand {\n  position: relative;\n  height: var(--pst-header-height);\n  max-height: var(--pst-header-height);\n  padding: 0.5rem 0;\n  width: auto;\n  margin: 0;\n  display: flex;\n  // Ensure that the logo stays the same length while other content shrinks\n  flex-shrink: 0;\n  align-items: center;\n  gap: 0.5rem;\n\n  // If there's no logo image, we use a p element w/ the site title\n  p {\n    margin-bottom: 0;\n  }\n\n  // If there's a logo, it'll be in an img block\n  img {\n    max-width: 100%;\n    height: 100%;\n    width: auto;\n  }\n\n  // remove underline from brand title on default state\n  a {\n    text-decoration: none;\n  }\n  &:hover {\n    @include link-style-hover;\n  }\n  &:visited {\n    &:hover {\n      @include link-style-hover;\n    }\n  }\n}\n","/**\n * Navigation text links in the navbar\n */\n.navbar-nav {\n  ul {\n    display: block;\n    list-style: none;\n\n    // Reduce padding of nested `ul` items a bit\n    ul {\n      padding: 0 0 0 1rem;\n    }\n  }\n\n  // Navbar links - do not have an underline by default\n  li {\n    display: flex;\n    flex-direction: column;\n\n    a {\n      display: flex;\n      align-items: center;\n      height: 100%;\n      padding-top: 0.25rem;\n      padding-bottom: 0.25rem;\n      @include link-style-text;\n    }\n  }\n\n  /**\n   * Togglable expand/collapse\n   * This is only applicable to the primary sidebar which has these checkboxes\n   */\n  .toctree-checkbox {\n    position: absolute;\n    display: none;\n  }\n\n  .toctree-checkbox {\n    ~ ul {\n      display: none;\n    }\n    ~ label .fa-chevron-down {\n      transform: rotate(0deg);\n    }\n  }\n  .toctree-checkbox:checked {\n    ~ ul {\n      display: block;\n    }\n    ~ label .fa-chevron-down {\n      transform: rotate(180deg);\n    }\n  }\n}\n","/**\n * The list of in-page TOC links\n */\n.page-toc {\n  .section-nav {\n    padding-left: 0;\n    border-bottom: none;\n\n    ul {\n      padding-left: 1rem;\n    }\n  }\n\n  // override bootstrap settings\n  .nav-link {\n    font-size: var(--pst-sidebar-font-size-mobile);\n    @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n      font-size: var(--pst-sidebar-font-size);\n    }\n  }\n\n  .onthispage {\n    color: var(--pst-color-text-base);\n    font-weight: var(--pst-sidebar-header-font-weight);\n    margin-bottom: 0.5rem;\n  }\n}\n","/**\n* Previous / Next navigation buttons\n**/\n.prev-next-area {\n  width: 100%;\n\n  p {\n    margin: 0 0.3em;\n    line-height: 1.3em;\n  }\n\n  i {\n    font-size: 1.2em;\n  }\n\n  a {\n    // So that buttons align with icons\n    display: flex;\n    align-items: center;\n    border: none;\n    padding: 10px;\n    max-width: 45%;\n    overflow-x: hidden;\n    color: var(--pst-color-text-muted);\n    text-decoration: none;\n\n    p.prev-next-title {\n      @include link-style-default;\n      font-weight: var(--pst-admonition-font-weight-heading);\n      font-size: 1.1em;\n    }\n\n    &:hover p.prev-next-title {\n      @include link-style-hover;\n    }\n    // Exception here - keep visited in the default link colour\n    // it still should hover on the default link hover colour\n    &:visited p.prev-next-title {\n      color: var(--pst-color-link);\n      &:hover {\n        color: var(--pst-color-link-hover);\n      }\n    }\n\n    .prev-next-info {\n      flex-direction: column;\n      margin: 0 0.5em;\n\n      .prev-next-subtitle {\n        text-transform: capitalize;\n      }\n    }\n\n    &.left-prev {\n      float: left;\n    }\n\n    &.right-next {\n      float: right;\n\n      div.prev-next-info {\n        text-align: right;\n      }\n    }\n  }\n}\n","/**\n * Search field\n **/\n.bd-search {\n  position: relative;\n  padding-left: 0.5rem;\n  gap: 0.5rem;\n  background-color: var(--pst-color-background);\n  border-radius: $admonition-border-radius;\n  border: 1px solid var(--pst-color-border);\n  color: var(--pst-color-text-base);\n\n  // Background should always be same color regardless of active or not\n  &:active {\n    background-color: var(--pst-color-background);\n    color: var(--pst-color-text-muted);\n  }\n\n  .icon {\n    position: absolute;\n    color: var(--pst-color-border);\n    left: 25px;\n  }\n\n  .fa-solid.fa-magnifying-glass {\n    position: absolute;\n    left: calc((2.5rem - 0.7em) / 2);\n    color: var(--pst-color-text-muted);\n  }\n\n  input {\n    // Inner-text of the search bar\n    &::placeholder {\n      color: var(--pst-color-text-muted);\n    }\n\n    // Remove the little \"x\" that pops up when you start typing\n    &::-webkit-search-cancel-button,\n    &::-webkit-search-decoration {\n      -webkit-appearance: none;\n      appearance: none;\n    }\n  }\n\n  // Shows off the keyboard shortcuts for the button\n  .search-button__kbd-shortcut {\n    display: flex;\n    position: absolute;\n    right: 0.5rem;\n    color: var(--pst-color-border);\n  }\n}\n\n.form-control {\n  background-color: var(--pst-color-background);\n  color: var(--pst-color-text-base);\n\n  &:focus,\n  &:focus-visible {\n    border: none;\n    box-shadow: none;\n    outline: 3px solid var(--pst-color-accent);\n    background-color: var(--pst-color-background);\n    color: var(--pst-color-text-muted);\n  }\n}\n\n/**\n * Search button - located in the navbar\n */\n\n// Search link icon should be a bit bigger since it is separate from icon links\n.search-button {\n  display: flex;\n  align-items: center;\n  align-content: center;\n  color: var(--pst-color-text-muted);\n  border-radius: 0;\n  @include icon-navbar-hover;\n  @include focus-indicator;\n\n  i {\n    font-size: 1.3rem;\n  }\n}\n\n// __search-container will only show up when we use the search pop-up bar\n.search-button__search-container,\n.search-button__overlay {\n  display: none;\n}\n.search-button__wrapper.show {\n  .search-button__search-container {\n    display: flex;\n    // Center in middle of screen just underneath header\n    position: fixed;\n    z-index: $zindex-modal;\n    top: 30%;\n    left: 50%;\n    transform: translate(-50%, -50%);\n    right: 1rem;\n    margin-top: 0.5rem;\n    width: 90%;\n    max-width: 800px;\n  }\n\n  .search-button__overlay {\n    display: flex;\n    position: fixed;\n    z-index: $zindex-modal-backdrop;\n    background-color: black;\n    opacity: 0.5;\n    width: 100%;\n    height: 100%;\n    top: 0px;\n    left: 0px;\n  }\n\n  form.bd-search {\n    flex-grow: 1;\n    padding-top: 0;\n    padding-bottom: 0;\n  }\n\n  // Font and input text a bit bigger\n  svg,\n  input {\n    font-size: var(--pst-font-size-icon);\n  }\n}\n\n/**\n * The search button component that looks like a field.\n * Lives at components/search-button-field.html\n */\n.search-button-field {\n  display: inline-flex;\n  align-items: center;\n  border: var(--pst-color-border) solid 1px;\n  border-radius: 1.5em;\n  color: var(--pst-color-text-muted);\n  padding: 0.5em;\n  background-color: var(--pst-color-surface);\n\n  &:hover {\n    border: 2px solid var(--pst-color-link-hover);\n  }\n  &:focus-visible {\n    border: 2px solid var(--pst-color-accent);\n  }\n\n  // The keyboard shotcut text\n  .search-button__default-text {\n    font-size: var(--bs-nav-link-font-size);\n    font-weight: var(--bs-nav-link-font-weight);\n    margin-right: 0.5em;\n    margin-left: 0.5em;\n  }\n\n  .kbd-shortcut__modifier {\n    font-size: 0.75em;\n  }\n\n  // Ensures that all the text lines up in the middle\n  > * {\n    align-items: center;\n  }\n\n  // Only the icon should be visible on narrow screens\n  > :not(svg) {\n    display: none;\n\n    @include media-breakpoint-up(lg) {\n      display: flex;\n    }\n  }\n}\n","/**\n * The 'Hide Search Matches' button.\n * This only shows up when a person lands on a page after clicking a search result.\n * Clicking it removes the highlighting of the search term from the page.\n * We want it to behave like a button.\n */\ndiv#searchbox {\n  // Leave `#searchbox` rules empty so that it doesn't show at all when it is empty\n  p.highlight-link {\n    margin: 1rem 0;\n    width: fit-content;\n\n    // A bit more margin on wide screens to mimic article behavior\n    @include media-breakpoint-up($breakpoint-sidebar-secondary) {\n      margin-left: 2rem;\n    }\n\n    // Put outer shadow on this one so that we can darken the link w/ an inner shadow\n    @include box-shadow();\n\n    // Style the button to look like a Sphinx Design button\n    a {\n      border-radius: 0.25rem;\n      font-size: 1.25rem;\n      padding: 0.75rem;\n      background-color: var(--pst-color-primary);\n      color: var(--pst-color-primary-text);\n      text-decoration: none;\n\n      // The box shadow is inset so that it darkens the button on hover\n      transition: box-shadow 0.25s ease-out;\n      &:hover {\n        box-shadow: inset 0px 0px 50px 50px rgba(0, 0, 0, 0.25);\n      }\n\n      &:before {\n        content: var(--pst-icon-search-minus);\n        color: unset;\n        font: var(--fa-font-solid);\n        margin-right: 0.5rem;\n      }\n    }\n  }\n}\n","/*********************************************\n* SASS Mixins\n*********************************************/\n\n/**\n * A consistent box shadow style we apply across elements.\n */\n@mixin box-shadow() {\n  box-shadow: 0 0.2rem 0.5rem var(--pst-color-shadow),\n    0 0 0.0625rem var(--pst-color-shadow) !important;\n}\n\n/**\n   * Set background of some cell outputs to white-ish to make sure colors work\n   * This is because many libraries make output that only looks good on white\n   */\n@mixin cell-output-background {\n  color: var(--pst-color-on-background);\n  background-color: var(--pst-color-text-base);\n  border-radius: 0.25rem;\n  padding: 0.5rem;\n}\n","/**\n * Light/dark theme switcher\n */\n\n.theme-switch-button {\n  // overide bootstrap settings\n  margin: 0 -0.5rem;\n  padding: 0; // We pad the `span` not the container\n  color: var(--pst-color-text-muted);\n  border-radius: 0;\n  @include focus-indicator;\n\n  span {\n    display: none;\n    padding: 0.5em;\n\n    @include icon-navbar-hover;\n    &:active {\n      text-decoration: none;\n      color: var(--pst-color-link-hover);\n    }\n  }\n}\n\nhtml[data-mode=\"auto\"] .theme-switch-button span[data-mode=\"auto\"] {\n  display: flex;\n}\nhtml[data-mode=\"light\"] .theme-switch-button span[data-mode=\"light\"] {\n  display: flex;\n}\nhtml[data-mode=\"dark\"] .theme-switch-button span[data-mode=\"dark\"] {\n  display: flex;\n}\n","button.btn.version-switcher__button {\n  border-color: var(--pst-color-border);\n  color: var(--pst-color-text-base);\n  // Add a margin on narrow screens to avoid feeling cramped\n  margin-bottom: 1em;\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    margin-bottom: unset;\n  }\n\n  @include link-style-hover;\n  @include focus-indicator;\n  &:active {\n    color: var(--pst-color-text-base);\n    border-color: var(--pst-color-border);\n  }\n}\n\n.version-switcher__menu {\n  border-color: var(--pst-color-border);\n  border-radius: var(--bs-dropdown-border-radius);\n\n  a.list-group-item {\n    background-color: var(--pst-color-on-background);\n    color: var(--pst-color-text-base);\n    padding: 0.75rem 1.25rem;\n\n    &:not(:last-child) {\n      border-bottom: 1px solid var(--pst-color-border);\n    }\n    @include link-style-hover;\n    &:hover {\n      background-color: var(--pst-color-surface);\n    }\n    &.active {\n      @include link-sidebar-current;\n      position: relative;\n      z-index: 1;\n\n      span:before {\n        content: \"\";\n        width: 100%;\n        height: 100%;\n        position: absolute;\n        z-index: -1;\n        left: 0;\n        top: 0;\n      }\n    }\n  }\n}\n\n// Font behavior on mobile\nbutton.version-switcher__button,\n.version-switcher__menu {\n  font-size: 1.1em; // A bit smaller than other menu font\n  z-index: $zindex-modal; // higher than the sidebars\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    font-size: unset;\n  }\n}\n","/* Collapsing of the TOC sidebar while scrolling */\n/* Nav: hide second level (shown on .active) */\n\nnav.page-toc {\n  // A little extra space before the buttons\n  margin-bottom: 1rem;\n}\n\n.bd-toc .nav,\n.list-caption {\n  .nav {\n    display: none;\n\n    // So we can manually specify a level as visible in the config\n    &.visible {\n      display: block;\n    }\n  }\n\n  > .active > ul {\n    display: block;\n  }\n}\n\n// Each entry of the in-page TOC\n.toc-entry {\n  display: block;\n  a > code {\n    color: var(--pst-color-text-muted);\n  }\n\n  a.nav-link {\n    display: block;\n    padding: 0.125rem 0;\n\n    // Padding w/ negative margin so the top TOC item highlight overlaps w/ the TOC border\n    padding-left: 1rem;\n    margin-left: -1rem;\n\n    @include link-sidebar;\n\n    &.active {\n      @include link-sidebar-current;\n      background-color: transparent;\n\n      &:hover {\n        color: var(--pst-color-link-hover);\n      }\n    }\n  }\n}\n","div.versionadded,\ndiv.versionchanged,\ndiv.deprecated {\n  vertical-align: middle;\n  margin: 1.5625em auto;\n  padding: 0 0.6rem 0 0.6rem;\n  overflow: hidden;\n  page-break-inside: avoid;\n  border-left: 0.2rem solid;\n  border-color: var(--pst-color-info);\n  border-radius: $admonition-border-radius;\n  background-color: var(--pst-color-on-background);\n  @include box-shadow();\n  position: relative;\n\n  > p {\n    margin-bottom: 0.6rem;\n    margin-top: 0.6rem;\n  }\n}\n\ndiv.versionadded {\n  border-color: var(--pst-color-success);\n  background-color: var(--pst-color-success-bg);\n}\n\ndiv.versionchanged {\n  border-color: var(--pst-color-warning);\n  background-color: var(--pst-color-warning-bg);\n}\n\ndiv.deprecated {\n  border-color: var(--pst-color-danger);\n  background-color: var(--pst-color-danger-bg);\n}\n\nspan.versionmodified {\n  font-weight: 600;\n\n  &:before {\n    margin-right: 0.6rem;\n    color: var(--pst-color-info);\n    font: var(--fa-font-solid);\n    content: var(--pst-icon-versionmodified-default);\n  }\n}\n\nspan.versionmodified.added {\n  &:before {\n    color: var(--pst-color-success);\n    content: var(--pst-icon-versionmodified-added);\n  }\n}\n\nspan.versionmodified.changed {\n  &:before {\n    color: var(--pst-color-warning);\n    content: var(--pst-icon-versionmodified-changed);\n  }\n}\n\nspan.versionmodified.deprecated {\n  &:before {\n    color: var(--pst-color-danger);\n    content: var(--pst-icon-versionmodified-deprecated);\n  }\n}\n",".sidebar-indices-items {\n  display: flex;\n  flex-direction: column;\n  border-top: 1px solid var(--pst-color-border);\n\n  @include media-breakpoint-up($breakpoint-sidebar-primary) {\n    border-top: none;\n  }\n\n  .sidebar-indices-items__title {\n    font-weight: var(--pst-sidebar-header-font-weight);\n    font-size: var(--pst-sidebar-header-font-size);\n    color: var(--pst-color-text-base);\n    margin-bottom: 0.5rem;\n  }\n\n  ul.indices-link {\n    margin-right: -1rem;\n    list-style: none;\n    padding: 0;\n\n    li > a {\n      display: block;\n      padding: 0.25rem 0;\n      color: var(--pst-color-text-muted);\n\n      &:hover {\n        color: var(--pst-color-primary);\n        text-decoration: none;\n        background-color: transparent;\n      }\n    }\n  }\n}\n",".bd-sidebar-primary div#rtd-footer-container {\n  position: sticky;\n  bottom: -1rem;\n  margin: -1rem; // ignore sidebar padding\n\n  .rst-versions.rst-badge {\n    position: unset;\n    font-size: 0.9em;\n    font-family: var(--pst-font-family-base);\n    max-width: unset;\n\n    .rst-current-version {\n      display: flex;\n      align-items: center;\n      gap: 0.2rem;\n      height: 2.5rem;\n      transition: background-color 0.2s ease-out;\n      background-color: var(--pst-color-background);\n      color: var(--pst-color-success);\n      border-top: 1px solid var(--pst-color-border);\n    }\n\n    .fa.fa-book {\n      color: var(--pst-color-text-muted);\n      margin-right: auto;\n\n      &::after {\n        color: var(--pst-color-text-base);\n        content: \"Read The Docs\";\n        font-family: var(--pst-font-family-base);\n        font-weight: var(--pst-admonition-font-weight-heading);\n      }\n    }\n    .fa.fa-caret-down {\n      color: var(--pst-color-text-muted);\n    }\n  }\n\n  .rst-versions.rst-badge.shift-up {\n    .rst-current-version {\n      border-bottom: 1px solid var(--pst-color-border);\n    }\n  }\n\n  .rst-other-versions {\n    background-color: var(--pst-color-surface);\n    color: var(--pst-color-text-base);\n\n    dl {\n      dd a {\n        color: var(--pst-color-text-muted);\n      }\n    }\n\n    hr {\n      background-color: var(--pst-color-border);\n    }\n\n    small a {\n      color: var(--pst-color-link);\n    }\n\n    input {\n      padding-left: 0.5rem;\n      border: 1px solid var(--pst-color-border);\n      background-color: var(--pst-color-surface);\n    }\n  }\n}\n","/**\n * Admonitions and blocks of styled content.\n * Admonitions CSS originally inspired by https://squidfunk.github.io/mkdocs-material/getting-started/\n */\n$admonition-border-radius: 0.25rem;\ndiv.admonition,\n.admonition {\n  margin: 1.5625em auto;\n  padding: 0 0.6rem 0.8rem 0.6rem;\n  overflow: hidden;\n  page-break-inside: avoid;\n  border-left: 0.2rem solid;\n  border-color: var(--pst-color-info);\n  border-radius: $admonition-border-radius;\n  background-color: var(--pst-color-on-background);\n  @include box-shadow();\n\n  // Last item should have no spacing since we'll control that w/ padding\n  *:last-child {\n    margin-bottom: 0;\n  }\n\n  // Items after the title should be indented\n  p.admonition-title ~ * {\n    margin-left: 1.4rem;\n    margin-right: 1.4rem;\n  }\n\n  // Lists need to have left margin so they don't spill into it\n  > ol,\n  > ul {\n    margin-left: 1em;\n  }\n\n  // Defaults for all admonitions\n  > .admonition-title {\n    margin: 0 -0.6rem;\n    padding: 0.4rem 0.6rem 0.4rem 2rem;\n    font-weight: var(--pst-admonition-font-weight-heading);\n    position: relative;\n    @include legacy-backdrop-placeholder;\n    background-color: var(--pst-color-info-bg);\n    // now that we use solid colors we want the title on top\n    z-index: 1;\n\n    &:after {\n      position: absolute;\n      left: 0.5rem;\n      width: 1rem;\n      height: 1rem;\n      color: var(--pst-color-info);\n      font: var(--fa-font-solid);\n      line-height: inherit;\n      content: var(--pst-icon-admonition-default);\n      opacity: 1;\n    }\n\n    // Next element after title needs some extra upper-space\n    + * {\n      margin-top: 0.4em;\n    }\n  }\n\n  &.attention {\n    border-color: var(--pst-color-attention);\n    > .admonition-title {\n      background-color: var(--pst-color-attention-bg);\n\n      &:after {\n        color: var(--pst-color-attention);\n        content: var(--pst-icon-admonition-attention);\n      }\n    }\n  }\n\n  &.caution {\n    border-color: var(--pst-color-warning);\n    > .admonition-title {\n      background-color: var(--pst-color-warning-bg);\n\n      &:after {\n        color: var(--pst-color-warning);\n        content: var(--pst-icon-admonition-caution);\n      }\n    }\n  }\n\n  &.warning {\n    border-color: var(--pst-color-warning);\n    > .admonition-title {\n      background-color: var(--pst-color-warning-bg);\n\n      &:after {\n        color: var(--pst-color-warning);\n        content: var(--pst-icon-admonition-warning);\n      }\n    }\n  }\n\n  &.danger {\n    border-color: var(--pst-color-danger);\n    > .admonition-title {\n      background-color: var(--pst-color-danger-bg);\n\n      &:after {\n        color: var(--pst-color-danger);\n        content: var(--pst-icon-admonition-danger);\n      }\n    }\n  }\n\n  &.error {\n    border-color: var(--pst-color-danger);\n    > .admonition-title {\n      background-color: var(--pst-color-danger-bg);\n\n      &:after {\n        color: var(--pst-color-danger);\n        content: var(--pst-icon-admonition-error);\n      }\n    }\n  }\n\n  &.hint {\n    border-color: var(--pst-color-success);\n    > .admonition-title {\n      background-color: var(--pst-color-success-bg);\n\n      &:after {\n        color: var(--pst-color-success);\n        content: var(--pst-icon-admonition-hint);\n      }\n    }\n  }\n\n  &.tip {\n    border-color: var(--pst-color-success);\n    > .admonition-title {\n      background-color: var(--pst-color-success-bg);\n\n      &:after {\n        color: var(--pst-color-success);\n        content: var(--pst-icon-admonition-tip);\n      }\n    }\n  }\n\n  &.important {\n    border-color: var(--pst-color-attention);\n    > .admonition-title {\n      background-color: var(--pst-color-attention-bg);\n\n      &:after {\n        color: var(--pst-color-attention);\n        content: var(--pst-icon-admonition-important);\n      }\n    }\n  }\n\n  &.note {\n    border-color: var(--pst-color-info);\n    > .admonition-title {\n      background-color: var(--pst-color-info-bg);\n\n      &:after {\n        color: var(--pst-color-info);\n        content: var(--pst-icon-admonition-note);\n      }\n    }\n  }\n\n  &.seealso {\n    border-color: var(--pst-color-success);\n    > .admonition-title {\n      background-color: var(--pst-color-success-bg);\n\n      &:after {\n        color: var(--pst-color-success);\n        content: var(--pst-icon-admonition-seealso);\n      }\n    }\n  }\n\n  &.admonition-todo {\n    border-color: var(--pst-color-secondary);\n    > .admonition-title {\n      background-color: var(--pst-color-secondary-bg);\n\n      &:after {\n        color: var(--pst-color-secondary);\n        content: var(--pst-icon-admonition-todo);\n      }\n    }\n  }\n\n  /**\n   * Special-case for a `sidebar` class that makes the admonition float to\n   * the right like the {sidebar} directive.\n   */\n  &.sidebar {\n    max-width: 40%;\n    float: right;\n    clear: both;\n    margin-left: 0.5rem;\n    margin-top: 0;\n\n    // Undo the .sidebar directive border\n    border-width: 0 0 0 0.2rem;\n\n    // TODO: these semantic-color-names border-color rules might no longer be\n    //       needed when we drop support for Sphinx 4 / docutils 0.17\n    &.attention,\n    &.important {\n      border-color: var(--pst-color-attention);\n    }\n\n    &.caution,\n    &.warning {\n      border-color: var(--pst-color-warning);\n    }\n\n    &.danger,\n    &.error {\n      border-color: var(--pst-color-danger);\n    }\n\n    &.hint,\n    &.tip,\n    &.seealso {\n      border-color: var(--pst-color-success);\n    }\n\n    &.note,\n    &.todo {\n      border-color: var(--pst-color-info);\n    }\n\n    // No inner margin since we have less horizontal space w/ the sidebar\n    p.admonition-title ~ * {\n      margin-left: 0rem;\n      margin-right: 0rem;\n    }\n  }\n}\n\n/**************************************************************\n * Similar content blocks that are not technically admonitions.\n */\n\n/**\n * Topics and the {contents} directive\n */\n// Docutils <= 0.17\ndiv.topic,\ndiv.topic.contents,\n// Docutils >= 0.18\nnav.contents,\naside.topic {\n  display: flex;\n  flex-direction: column;\n  background-color: var(--pst-color-surface);\n  border-color: var(--pst-color-border);\n  border-radius: $admonition-border-radius;\n  padding: 1rem 1.25rem;\n  @include box-shadow();\n\n  .topic-title {\n    margin: 0 0 0.5rem 0;\n  }\n\n  // Over-ride text color to ensure enough contrast\n  p {\n    color: var(--pst-color-on-surface) !important;\n  }\n\n  // Over-ride large default padding\n  ul.simple {\n    padding-left: 1rem;\n\n    ul {\n      // So that sub-lists will have a bit less padding\n      padding-left: 2em;\n    }\n  }\n}\n\n/**\n * Sidebar directive\n */\naside.sidebar {\n  border: 1px solid var(--pst-color-border);\n  background-color: var(--pst-color-surface);\n  border-radius: $admonition-border-radius;\n  // to match the admonition-styled sidebars:\n  margin-left: 0.5rem;\n  padding: 0;\n  > *:last-child {\n    padding-bottom: 1rem;\n  }\n\n  p.sidebar-title {\n    position: relative;\n    margin-bottom: 0;\n    padding-top: 0.5rem;\n    padding-bottom: 0.5rem;\n    border-bottom: 1px solid var(--pst-color-border);\n    font-family: var(--pst-font-family-heading);\n    font-weight: var(--pst-admonition-font-weight-heading);\n  }\n\n  // Add margin to the first non-`.sidebar-title` item\n  > *:not(.sidebar-title):first-child,\n  > p.sidebar-title + * {\n    margin-top: 1rem;\n  }\n\n  > * {\n    padding-left: 1rem;\n    padding-right: 1rem;\n  }\n}\n\n/**\n * Rubrics look kind of like section headers\n */\np.rubric {\n  display: flex;\n  flex-direction: column;\n}\n\n/**\n * Seealso is kind of like a vertically-collapsed admonition\n */\n.seealso dd {\n  margin-top: 0;\n  margin-bottom: 0;\n}\n","/**\n * Miscellaneous color functions and mixins\n**/\n\n// loading the math module\n@use \"sass:math\";\n\n// We must add ::before pseudo-element to some theme components (such as admonitions)\n// because users were instructed to customize the background color this way.\n@mixin legacy-backdrop-placeholder {\n  &:before {\n    content: \"\";\n    width: 100%;\n    height: 100%;\n    position: absolute;\n    left: 0;\n    top: 0;\n    z-index: -1;\n    // So that hovering over the text within background is still possible.\n    // Otherwise the background overlays the text and you cannot click or select easily.\n    // ref: https://developer.mozilla.org/en-US/docs/Web/CSS/pointer-events\n    pointer-events: none;\n  }\n}\n\n/**\n* Function to get items from nested maps\n*/\n// @param {Map} $map - Map\n// @param {Arglist} $keys - Keys to fetc\n// @return {*}\n@function map-deep-get($map, $keys...) {\n  @each $key in $keys {\n    $map: map-get($map, $key);\n  }\n  @return $map;\n}\n\n/**\n  * Function to check if the color needs converting to a \"color\" type\n  * if it is a string we cannot use other color manipulation functions\n  * It is used to create the sphinx-design colours as these are often interpolated\n  */\n// @param {String/Color} $color - Color definition from map\n// @return {Color} - Color type (in hex)\n@function check-color($color) {\n  @if type-of($color) == string {\n    $color: from-hex($color);\n  }\n  @return $color;\n}\n\n/**\n  * Function to convert the string representation of a color to a color type (hex)\n  */\n// @param {String} $string - String representation of a color\n@function from-hex($string) {\n  $string-lower: to-lower-case($string);\n  $r: \"\";\n  $g: \"\";\n  $b: \"\";\n  $hex: \"0\" \"1\" \"2\" \"3\" \"4\" \"5\" \"6\" \"7\" \"8\" \"9\" \"a\" \"b\" \"c\" \"d\" \"e\" \"f\";\n  $length: str-length($string);\n  $max: if($length == 4, 1, 2);\n\n  // Check for length accuracy\n  @if $length != 4 and $length != 7 {\n    @return $string;\n  }\n\n  // Loop from the second character (omitting #)\n  @for $i from 2 through $length {\n    $c: str-slice($string-lower, $i, $i);\n\n    // If wrong character, return\n    @if index($hex, $c) == null {\n      @return $string;\n    }\n\n    @if str-length($r) < $max {\n      $r: $r + $c;\n    } @else if str-length($g) < $max {\n      $g: $g + $c;\n    } @else if str-length($b) < $max {\n      $b: $b + $c;\n    }\n  }\n\n  @if $length == 4 {\n    $r: $r + $r;\n    $g: $g + $g;\n    $b: $b + $b;\n  }\n\n  @return rgb(_hex-to-dec($r), _hex-to-dec($g), _hex-to-dec($b));\n}\n\n@function _hex-to-dec($string) {\n  $hex: \"0\" \"1\" \"2\" \"3\" \"4\" \"5\" \"6\" \"7\" \"8\" \"9\" \"a\" \"b\" \"c\" \"d\" \"e\" \"f\";\n  $string: to-lower-case($string);\n  $length: str-length($string);\n\n  $dec: 0;\n  @for $i from 1 through $length {\n    $factor: 1 + (15 * ($length - $i));\n    $index: index($hex, str-slice($string, $i, $i));\n    $dec: $dec + $factor * ($index - 1);\n  }\n\n  @return $dec;\n}\n","// Style API docs from sphinx' autodoc / autosummary\n\n/*******************************************************************************\n* Styling for field lists\n*/\n\n/* grey highlighting of 'parameter' and 'returns' field */\ntable.field-list {\n  border-collapse: separate;\n  border-spacing: 10px;\n  margin-left: 1px;\n\n  th.field-name {\n    padding: 1px 8px 1px 5px;\n    white-space: nowrap;\n    background-color: var(--pst-color-surface);\n  }\n\n  /* italic font for parameter types */\n  td.field-body {\n    p {\n      font-style: italic;\n\n      > strong {\n        font-style: normal;\n      }\n    }\n\n    /* reduced space around parameter description */\n    blockquote {\n      border-left: none;\n      margin: 0em 0em 0.3em;\n      padding-left: 30px;\n    }\n  }\n}\n\n/*******************************************************************************\n* Styling for autosummary tables\n*/\n\n.table.autosummary {\n  // The first column (with the signature) should not wrap\n  td:first-child {\n    white-space: nowrap;\n  }\n}\n\n/* overriding basic.css to use our own monospace font */\n.sig {\n  font-family: var(--pst-font-family-monospace);\n}\n\n/* C++ specific styling - overriding the basic.css to avoid custom colors*/\n\n.sig-inline.c-texpr,\n.sig-inline.cpp-texpr {\n  font-family: unset;\n}\n\n.sig.c .k,\n.sig.c .kt,\n.sig.cpp .k,\n.sig.cpp .kt {\n  color: var(--pst-color-text-base);\n}\n\n.sig.c .m,\n.sig.cpp .m {\n  color: var(--pst-color-text-base);\n}\n\n.sig.c .s,\n.sig.c .sc,\n.sig.cpp .s,\n.sig.cpp .sc {\n  color: var(--pst-color-text-base);\n}\n\n// addition\n\n// .sig.c .sig-name .n,\n// .sig.cpp .sig-name .n {\n//   color: var(--pst-color-inline-code);\n// }\n\n.sig-name {\n  color: var(--pst-color-inline-code);\n}\n\n.sig-param .o,\n.sig-param .default_value {\n  color: var(--pst-color-text-muted);\n  font-weight: normal;\n}\n\n// change target color for dark theme\ndt:target,\nspan.highlighted {\n  background-color: var(--pst-color-target);\n}\n\n.viewcode-back {\n  font-family: var(--pst-font-family-base);\n}\n\n.viewcode-block:target {\n  border-top: 1px solid var(--pst-color-border);\n  border-bottom: 1px solid var(--pst-color-border);\n  position: relative;\n  background-color: var(--pst-color-target);\n}\n\n/*******************************************************************************\n* Styling for autosummary titles like \"parameters\" and \"returns\"\n*/\n\n// the API selector\n// from https://github.com/pradyunsg/furo/blob/main/src/furo/assets/styles/content/_api.sass#L6)\ndl[class]:not(.option-list):not(.field-list):not(.footnote):not(.glossary):not(.simple) {\n  //increase margin bottom after the dl elements\n  margin-bottom: 3rem;\n\n  dd {\n    margin-left: 2rem;\n\n    // Fix until this will be solved to Sphinx https://github.com/sphinx-doc/sphinx/issues/10815\n    & > dl.simple > dt {\n      display: flex;\n    }\n  }\n\n  dl.field-list {\n    display: grid;\n    grid-template-columns: unset;\n  }\n\n  dt.field-odd,\n  dt.field-even {\n    margin-top: 0.2rem;\n    margin-bottom: 0.1rem;\n    background-color: var(--pst-color-surface);\n  }\n}\n","/**\n * Code block styling\n * Note that we inherit a lot of styling from Bootstrap so not many rules here.\n */\n\n// General code block behavior\n// Unset bootstrap behavior\ndiv[class*=\"highlight-\"],\ndiv.highlight,\ndiv.literal-block-wrapper {\n  display: flex;\n  flex-direction: column;\n  width: unset;\n  border-radius: $admonition-border-radius;\n}\n\n// Code blocks with captions\n// There's a wrapper when the code block has a title\ndiv.literal-block-wrapper {\n  border: 1px solid var(--pst-color-border);\n  border-radius: $admonition-border-radius;\n\n  // This is where the title goes\n  div.code-block-caption {\n    margin: 0;\n    border-bottom: 1px solid var(--pst-color-border);\n    padding: 0.5rem;\n    font-size: 1rem;\n    font-weight: var(--pst-font-weight-caption);\n\n    a.headerlink {\n      font-size: inherit;\n    }\n  }\n\n  // Remove the upper border radius since we want it to connect with the title\n  // Remove the box shadow so the wrapper gets the shadow instead\n  div[class*=\"highlight-\"] {\n    margin: 0;\n    border-radius: 0;\n    pre {\n      border: none;\n      box-shadow: none;\n    }\n  }\n}\n\n/**\n * In-line code\n */\ncode.literal {\n  padding: 0.1rem 0.25rem;\n  background-color: var(--pst-color-surface);\n  border: 1px solid var(--pst-color-border);\n  border-radius: 0.25rem;\n}\n\na > code {\n  color: var(--pst-color-inline-code-links);\n}\n\n// Fix for Sphinx's \"highlight\" directive - this is an issue with our accessible pygments theme\n// and the colour we are using for the background of the code blocks.\nhtml[data-theme=\"light\"] .highlight .nf {\n  color: #0078a1 !important;\n}\n\n// Minimum opacity needed for linenos to be WCAG AA conformant\nspan.linenos {\n  opacity: 0.8 !important;\n}\n","figure {\n  a.headerlink {\n    // So that header link doesn't push caption to be off-center.\n    position: absolute;\n    font-size: inherit;\n  }\n  // Default headerlink hover doesn't trigger on figures\n  &:hover a.headerlink {\n    visibility: visible;\n  }\n\n  figcaption {\n    font-family: var(--pst-font-family-heading);\n    font-weight: var(--pst-font-weight-caption);\n    color: var(--pst-color-text-muted);\n    margin-left: auto;\n    margin-right: auto;\n\n    table.table {\n      width: fit-content;\n      margin-left: auto;\n      margin-right: auto;\n    }\n  }\n}\n","// For consistency, add bracket around footnotes/citations which are\n// cited more than once. E.g. [Newton](1,2) instead of Newton(1,2)\ndt.label > span.brackets:not(:only-child):before {\n  content: \"[\";\n}\n\ndt.label > span.brackets:not(:only-child):after {\n  content: \"]\";\n}\n\n// Make footnote as a superscript\na.footnote-reference {\n  vertical-align: super;\n  font-size: small;\n}\n\n// Docutils 0.18 uses an `aside.footnote` container with different internal structure\naside.footnote {\n  margin-bottom: 0.5rem;\n  &:last-child {\n    margin-bottom: 1rem;\n  }\n  span.label,\n  span.backrefs {\n    font-weight: bold;\n  }\n\n  &:target {\n    background-color: var(--pst-color-target);\n  }\n}\n","/**\n * Hacky fixes that don't fit cleanly into other sections\n */\n\n// Ensure user highlighting/selecting behaves properly\n// From https://stackoverflow.com/a/34372191\ntable.highlighttable td.linenos,\nspan.linenos,\ndiv.doctest > div.highlight span.gp {\n  /* gp: Generic.Prompt */\n  user-select: none;\n  -webkit-user-select: text; /* Safari fallback only */\n  -webkit-user-select: none; /* Chrome/Safari */\n  -moz-user-select: none; /* Firefox */\n  -ms-user-select: none; /* IE10+ */\n}\n","// Override bootstrap by restoring the basic theme default.\ndd {\n  margin-top: 3px;\n  margin-bottom: 10px;\n  margin-left: 30px;\n}\n\nol,\nul {\n  padding-inline-start: 2rem;\n  li > p:first-child {\n    margin-bottom: 0.25rem;\n    margin-top: 0.25rem;\n  }\n}\n","// GitHub blockquote style\nblockquote {\n  padding: 1em 1em;\n  color: var(--pst-color-text-muted);\n  border-left: 0.25em solid var(--pst-color-border);\n  border-radius: $admonition-border-radius;\n  position: relative;\n\n  p {\n    color: var(--pst-color-text-base);\n  }\n\n  // remove padding from included line-block to avoid duplication\n  .line-block {\n    margin: 0 0;\n  }\n\n  // remove margin bottom for the last p\n  p:last-child {\n    margin-bottom: 0;\n  }\n\n  @include legacy-backdrop-placeholder;\n  background-color: var(--pst-color-surface);\n\n  //hack to make the text in the blockquote selectable\n  &:before {\n    z-index: -1;\n  }\n}\n","/**\n * Span-level styling within content\n */\n\nspan.guilabel {\n  border: 1px solid var(--pst-color-info);\n  color: var(--pst-color-info);\n  font-size: 80%;\n  font-weight: 700;\n  border-radius: 4px;\n  padding: 2.4px 6px;\n  margin: auto 2px;\n  position: relative;\n  @include legacy-backdrop-placeholder;\n  background-color: var(--pst-color-info-bg);\n}\n\na.reference.download:before {\n  content: var(--pst-icon-download);\n  font: var(--fa-font-solid);\n  font-size: 0.8em;\n  padding: 0 0.25em;\n  color: var(--pst-color-text-muted);\n}\n","/**\n * Tables\n */\n\n// ensure table will fit in the article width and make them y-scrollable\ntable {\n  display: table;\n  overflow: auto;\n\n  // default to table-center\n  margin-left: auto;\n  margin-right: auto;\n\n  &.table-right {\n    margin-right: 0;\n  }\n\n  &.table-left {\n    margin-left: 0;\n  }\n}\n\n// customize table caption from bootstrap\n// to display them on top and centered\ntable caption {\n  text-align: center;\n  caption-side: top;\n  color: var(--pst-color-text-muted);\n}\n\n// MyST Markdown tables use these classes to control alignment\nth,\ntd {\n  &.text-align\\:left {\n    text-align: left;\n  }\n\n  &.text-align\\:right {\n    text-align: right;\n  }\n\n  &.text-align\\:center {\n    text-align: center;\n  }\n}\n\n// override bootstrap background colors\n.table {\n  --bs-table-bg: transparent;\n}\n","/**\n * Style the toctree component in pages (avoid modifying the navbars)\n */\n.toctree-wrapper {\n  p.caption {\n    font-size: 1.5em;\n    margin-bottom: 0em;\n  }\n\n  & > ul {\n    padding-left: 0;\n  }\n\n  li[class^=\"toctree-l\"] {\n    list-style: none;\n    margin-bottom: 0.2em;\n\n    & > a {\n      list-style: none;\n      font-size: 1.1em;\n    }\n\n    & > ul {\n      list-style: none;\n      padding-inline-start: 1.5em;\n    }\n  }\n\n  // slightly bigger font for l1\n  .toctree-l1 > a {\n    font-size: 1.3em;\n  }\n}\n\ndiv.topic.contents,  // Docutils <= 0.17\nnav.contents  // Docutils >= 0.18\n{\n  // Style similarly to toctree\n  ul.simple {\n    list-style: none;\n    padding-left: 0;\n  }\n}\n","/**\n * Mathematics via MathJax.\n *\n * This is designed for MathJax v3\n * ref: https://www.sphinx-doc.org/en/master/usage/extensions/math.html#module-sphinx.ext.mathjax\n */\n\n// Applies to all math elements\nspan.math,\ndiv.math {\n  align-items: center;\n  display: flex;\n  max-width: 100%;\n  // This will be over-ridden for the y-direction and divs\n  overflow: hidden;\n}\n\n// Inline-only\nspan.math {\n  display: inline-flex;\n}\n\n// Block-level only\ndiv.math {\n  gap: 0.5em;\n  // So that the eqno shows up after the equation\n  flex-direction: row-reverse;\n\n  // The equation number / link\n  span.eqno a.headerlink {\n    position: relative;\n    font-size: 1em;\n  }\n\n  // The math container\n  mjx-container {\n    flex-grow: 1;\n    padding-bottom: 0.2rem;\n    overflow: auto;\n\n    // Set height to 0 so that it does not cause scrollbars to appear\n    // ref: https://github.com/mathjax/MathJax/issues/2521\n    mjx-assistive-mml {\n      height: 0;\n    }\n  }\n}\n","/**\n * ABlog\n * ref: https://ablog.readthedocs.io/\n */\n\n/**\n * Sidebar template components\n */\n.ablog-sidebar-item {\n  h2,\n  h3 {\n    font-size: var(--pst-sidebar-header-font-size);\n    // Remove unnecessary vertical whitespace\n    margin-top: 0.5rem;\n\n    // The headers are all links, but this makes them hard to parse\n    // So we change the colors to make them look like headers\n    a {\n      color: var(--pst-color-text-base);\n    }\n  }\n\n  ul {\n    // No bullet points for the primary sidebar items\n    list-style: none;\n    padding-left: 0;\n    // Otherwise a scrollbar randomly shows up\n    overflow-y: hidden;\n    // List of recent post items\n    display: flex;\n    flex-direction: column;\n    gap: 0.5em;\n    margin-bottom: 0;\n\n    // The ablog cloud should move horizontally\n    &.ablog-cloud {\n      flex-direction: row;\n      flex-flow: wrap;\n      gap: 0.5rem;\n\n      // Vertical-align tag clouds\n      li {\n        // Center the tag cloud items\n        display: flex;\n        align-items: center;\n      }\n    }\n  }\n}\n\n/**\n * Previous / next buttons at the bottom\n */\n.ablog__prev-next {\n  font-size: 1.2em;\n  display: flex;\n  padding: 1rem 0;\n\n  // The bottom previous / next arrows\n  > span {\n    // To ensure that the whole thing fits on one line even if there are long titles\n    display: flex;\n    max-width: 45%;\n\n    // Links within each span have the collection of icon + text\n    a {\n      display: flex;\n      align-items: center;\n      margin-left: auto;\n      gap: 1rem;\n      line-height: 1.5rem;\n\n      i:before {\n        color: var(--pst-color-text-base);\n      }\n    }\n  }\n\n  // The first span is for the previous page and aligns to the left\n  span.ablog__prev {\n    i.fa-arrow-circle-left:before {\n      content: var(--pst-icon-angle-left);\n    }\n  }\n\n  // The second span is just an empty space so we remove it because we're\n  // positioning with flex\n  span.ablog__spacer {\n    display: none;\n  }\n\n  // The third span is aligned to the right\n  span.ablog__next {\n    margin-left: auto;\n    text-align: right;\n\n    i.fa-arrow-circle-right:before {\n      content: var(--pst-icon-angle-right);\n    }\n  }\n}\n\n/**\n * {postlist} directive and posts page\n */\n.ablog__collection,\n.postlist {\n  padding-left: 0;\n\n  .ablog-post {\n    list-style: none;\n\n    // Post metadata tags (author, links ,etc) should be a bit smaller\n    .ablog-archive {\n      display: flex;\n      flex-direction: row;\n      flex-wrap: wrap;\n      gap: 1rem;\n      list-style: none;\n      font-size: 0.75rem;\n      padding-left: 0;\n    }\n\n    // Title line should be a bit bigger and bold to stand out\n    .ablog-post-title {\n      margin-top: 0;\n      font-size: 1.25rem;\n\n      a {\n        font-weight: bold;\n      }\n    }\n\n    // Read more button should be a bit bigger\n    .ablog-post-expand {\n      margin-bottom: 0.5rem;\n    }\n  }\n}\n","/**\n * Special cases for Bootstrap functionality\n */\n\n// Bootstrap adds margin to their general container class. However, sphinx/docutils\n// can also generate output with the container class, but in those cases we should\n// not add the margin from bootstrap. Same for max-width.\n.docutils.container {\n  padding-left: unset;\n  padding-right: unset;\n  margin-left: unset;\n  margin-right: unset;\n  max-width: unset;\n  width: unset;\n}\n","/**\n * Sphinx Copybutton\n * ref: https://sphinx-copybutton.readthedocs.io/\n */\n\ndiv.highlight button.copybtn {\n  // Nicer spacing\n  display: flex;\n  align-items: center;\n  justify-content: center;\n\n  // Removes the button background so we more naturally blend with the code cell.\n  background-color: unset;\n\n  // Don't over-ride the success color\n  &:not(.success) {\n    color: var(--pst-color-muted);\n  }\n  border: none;\n  background-color: var(--pst-color-surface);\n\n  &:hover {\n    &:not(.success) {\n      color: var(--pst-color-text);\n      background-color: var(--pst-color-shadow);\n    }\n  }\n\n  // Tooltip styling\n  &.o-tooltip--left:after {\n    color: var(--pst-color-text);\n    background-color: var(--pst-color-surface);\n  }\n}\n","// adapt ethical ad to the theme\n#ethical-ad-placement {\n  .ethical-sidebar a,\n  .ethical-sidebar a:visited,\n  .ethical-sidebar a:hover,\n  .ethical-sidebar a:active,\n  .ethical-footer a,\n  .ethical-footer a:visited,\n  .ethical-footer a:hover,\n  .ethical-footer a:active {\n    color: var(--pst-color-text-base);\n  }\n\n  .ethical-sidebar,\n  .ethical-footer {\n    background-color: var(--pst-color-background);\n    border: 1px solid var(--pst-color-border);\n    border-radius: 5px;\n    color: var(--pst-color-text-base);\n    font-size: 14px;\n    line-height: 20px;\n  }\n}\n","/**\n * Styles for various Sphinx execution libraries.\n * For now, where these define output sections, we simply revert their background\n * to be a \"light theme\" background. This ensures that inputs/outputs behave similarly,\n * because the CSS is often controlled by each package.\n * In the future, we might add dark theme support for specific packages.\n */\n\n/******************************************************************************\n * Jupyter Sphinx\n */\n\n.bd-content div.jupyter_container {\n  // We don't want borders around the whole container, just around code cells\n  border: none;\n  background-color: unset;\n  box-shadow: none;\n\n  // Code cells should have the same style as our other code objects\n  div.output,\n  div.highlight {\n    border-radius: 0.25rem;\n  }\n  div.highlight {\n    background-color: var(--pst-color-surface);\n  }\n\n  // Ensure the style is the same as our code cells. Jupyter Sphinx makes it tiny.\n  .cell_input,\n  .cell_output {\n    border-radius: 0.25rem;\n    pre {\n      padding: 1rem;\n    }\n  }\n}\n","/**\n * Special-cases for packages in the PyData ecosystem\n */\n\n// xarray output display in bootstrap\n.xr-wrap[hidden] {\n  display: block !important;\n}\n","/*******************************************************************************\n* master color map. Only the colors that actually differ between light and dark\n* themes are specified separately.\n*\n* To see the full list of colors see https://www.figma.com/file/rUrrHGhUBBIAAjQ82x6pz9/PyData-Design-system---proposal-for-implementation-(2)?node-id=1234%3A765&t=ifcFT1JtnrSshGfi-1\n*/\n\n/**\n* Function to get items from nested maps\n*/\n// @param {Map} $map - Map\n// @param {Arglist} $keys - Keys to fetch\n// @return {*}\n@function map-deep-get($map, $keys...) {\n  @each $key in $keys {\n    $map: map-get($map, $key);\n  }\n  @return $map;\n}\n\n/* Assign base colors for the PyData theme */\n$color-palette: (\n  // Primary color\n  \"teal\":\n    (\n      \"50\": #f4fbfc,\n      \"100\": #e9f6f8,\n      \"200\": #d0ecf1,\n      \"300\": #abdde6,\n      \"400\": #3fb1c5,\n      \"500\": #0a7d91,\n      \"600\": #085d6c,\n      \"700\": #064752,\n      \"800\": #042c33,\n      \"900\": #021b1f,\n    ),\n  // Secondary color\n  \"violet\":\n    (\n      \"50\": #f4eefb,\n      \"100\": #e0c7ff,\n      \"200\": #d5b4fd,\n      \"300\": #b780ff,\n      \"400\": #9c5ffd,\n      \"500\": #8045e5,\n      \"600\": #6432bd,\n      \"700\": #4b258f,\n      \"800\": #341a61,\n      \"900\": #1e0e39,\n    ),\n  // Neutrals\n  \"gray\":\n    (\n      \"50\": #f9f9fa,\n      \"100\": #f3f4f5,\n      \"200\": #e5e7ea,\n      \"300\": #d1d5da,\n      \"400\": #9ca4af,\n      \"500\": #677384,\n      \"600\": #48566b,\n      \"700\": #29313d,\n      \"800\": #222832,\n      \"900\": #14181e,\n    ),\n  // Accent color\n  \"pink\":\n    (\n      \"50\": #fcf8fd,\n      \"100\": #fcf0fa,\n      \"200\": #f8dff5,\n      \"300\": #f3c7ee,\n      \"400\": #e47fd7,\n      \"500\": #c132af,\n      \"600\": #912583,\n      \"700\": #6e1c64,\n      \"800\": #46123f,\n      \"900\": #2b0b27,\n    ),\n  \"foundation\": (\n    \"white\": #ffffff,\n    // gray-900\n    \"black\": #14181e,\n  )\n);\n\n:root {\n  // Add theme colours to the html root element\n  @each $group-color, $color in $color-palette {\n    @each $color-name, $definition in $color {\n      --pst-#{$group-color}-#{$color-name}: #{$definition};\n    }\n  }\n}\n\n// Static SCSS variables used thoroughout the theme\n// Minimum contrast ratio used for the theme.\n// Acceptable values for WCAG 2.0 are 3, 4.5 and 7.\n// See https://www.w3.org/TR/WCAG20/#visual-audio-contrast-contrast\n// 4.5 - is for text that is 14pt or less\n$min-contrast-ratio-4: 4.5;\n// 3 is for text that is 18pt or bold, or for non-text elements\n$min-contrast-ratio-3: 3;\n\n// Customize the light and dark text colors for use in our color contrast function.\n$foundation-black: #14181e;\n$foundation-white: #ffffff;\n// This is a custom - calculated  color between gray 100 and 200 - to reduce\n// the contrast ratio (avoid a jarring effect)\n$base-light-text: #ced6dd;\n// used in sphinx_design - gray 100\n$foundation-light-gray: #f3f4f5;\n// used in sphinx_design - gray 700\n$foundation-muted-gray: #29313d;\n// used in sphinx_design - gray 800\n$foundation-dark-gray: #222832;\n\n$pst-semantic-colors: (\n  \"primary\": (\n    \"light\": #{map-deep-get($color-palette, \"teal\", \"500\")},\n    \"bg-light\": #{map-deep-get($color-palette, \"teal\", \"200\")},\n    \"dark\": #{map-deep-get($color-palette, \"teal\", \"400\")},\n    \"bg-dark\": #{map-deep-get($color-palette, \"teal\", \"800\")},\n  ),\n  \"secondary\": (\n    \"light\": #{map-deep-get($color-palette, \"violet\", \"500\")},\n    \"bg-light\": #{map-deep-get($color-palette, \"violet\", \"100\")},\n    \"dark\": #{map-deep-get($color-palette, \"violet\", \"400\")},\n    \"bg-dark\": #{map-deep-get($color-palette, \"violet\", \"800\")},\n  ),\n  \"accent\": (\n    \"light\": #{map-deep-get($color-palette, \"pink\", \"500\")},\n    \"bg-light\": #{map-deep-get($color-palette, \"pink\", \"200\")},\n    \"dark\": #{map-deep-get($color-palette, \"pink\", \"400\")},\n    \"bg-dark\": #{map-deep-get($color-palette, \"pink\", \"800\")},\n  ),\n  \"info\": (\n    \"light\": #276be9,\n    \"bg-light\": #dce7fc,\n    \"dark\": #79a3f2,\n    \"bg-dark\": #06245d,\n  ),\n  \"warning\": (\n    \"light\": #f66a0a,\n    \"bg-light\": #f8e3d0,\n    \"dark\": #ff9245,\n    \"bg-dark\": #652a02,\n  ),\n  \"success\": (\n    \"light\": #00843f,\n    \"bg-light\": #d6ece1,\n    \"dark\": #5fb488,\n    \"bg-dark\": #002f17,\n  ),\n  // This is is based on the warning color\n  \"attention\":\n    (\n      \"light\": var(--pst-color-warning),\n      \"bg-light\": var(--pst-color-warning-bg),\n      \"dark\": var(--pst-color-warning),\n      \"bg-dark\": var(--pst-color-warning-bg),\n    ),\n  \"danger\": (\n    \"light\": #d72d47,\n    \"bg-light\": #f9e1e4,\n    \"dark\": #e78894,\n    \"bg-dark\": #4e111b,\n  ),\n  \"text-base\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"800\")},\n    \"dark\": $base-light-text,\n  ),\n  \"text-muted\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"600\")},\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"400\")},\n  ),\n  \"heading-color\": (\n    \"light\": #{$foundation-white},\n    \"dark\": #{$foundation-black},\n  ),\n  \"shadow\": (\n    \"light\": rgba(0, 0, 0, 0.1),\n    \"dark\": rgba(0, 0, 0, 0.2),\n  ),\n  \"border\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"300\")},\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"600\")},\n  ),\n  \"border-muted\": (\n    \"light\": rgba(23, 23, 26, 0.2),\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"700\")},\n  ),\n  \"inline-code\": (\n    \"light\": #{map-deep-get($color-palette, \"pink\", \"600\")},\n    \"dark\": #{map-deep-get($color-palette, \"pink\", \"300\")},\n  ),\n  \"inline-code-links\": (\n    // need to make sure there is enough contrast against the code bg\n    \"light\": #{map-deep-get($color-palette, \"teal\", \"600\")},\n    // keep primary color for dark mode\n    \"dark\": #{map-deep-get($color-palette, \"teal\", \"400\")},\n  ),\n  \"target\": (\n    \"light\": #f3cf95,\n    \"dark\": #675c04,\n  ),\n  // DEPTH COLORS - you can see the elevation colours and shades\n  // in the Figma file https://www.figma.com/file/rUrrHGhUBBIAAjQ82x6pz9/PyData-Design-system---proposal-for-implementation-(2)?node-id=1492%3A922&t=sQeQZehkOzposYEg-1\n  // background: color of the canvas / the furthest back layer\n  \"background\":\n    (\n      \"light\": #{map-deep-get($color-palette, \"foundation\", \"white\")},\n      \"dark\": #{map-deep-get($color-palette, \"foundation\", \"black\")},\n    ),\n  // on-background: provides slight contrast against background\n  // (by use of shadows in light theme)\n  \"on-background\":\n    (\n      \"light\": #{map-deep-get($color-palette, \"foundation\", \"white\")},\n      \"dark\": #{map-deep-get($color-palette, \"gray\", \"800\")},\n    ),\n  \"surface\": (\n    \"light\": #{map-deep-get($color-palette, \"gray\", \"100\")},\n    \"dark\": #{map-deep-get($color-palette, \"gray\", \"700\")},\n  ),\n  // on_surface: object on top of surface object (without shadows)\n  \"on-surface\":\n    (\n      \"light\": #{map-deep-get($color-palette, \"gray\", \"800\")},\n      \"dark\": $foundation-light-gray,\n    ),\n);\n\n/*******************************************************************************\n* write the color rules for each theme (light/dark)\n*/\n\n/* NOTE:\n * Mixins enable us to reuse the same definitions for the different modes\n * https://sass-lang.com/documentation/at-rules/mixin\n * #{something} inserts a variable into a CSS selector or property name\n * https://sass-lang.com/documentation/interpolation\n */\n@mixin theme-colors($mode) {\n  // check if this color is defined differently for light/dark\n  @each $col-name, $definition in $pst-semantic-colors {\n    @if type-of($definition) == map {\n      @each $key, $val in $definition {\n        @if str-index($key, $mode) != null {\n          // since now we define the bg colours in the semantic colours and not\n          // by changing opacity, we need to check if the key contains bg and the\n          // correct mode (light/dark)\n          @if str-index($key, \"bg\") != null {\n            --pst-color-#{$col-name}-bg: #{$val};\n          } @else {\n            --pst-color-#{$col-name}: #{$val};\n          }\n        }\n      }\n    } @else {\n      --pst-color-#{$col-name}: #{$definition};\n    }\n  }\n\n  // assign the \"duplicate\" colors (ones that just reference other variables)\n  & {\n    --pst-color-link: var(--pst-color-primary);\n    --pst-color-link-hover: var(--pst-color-secondary);\n  }\n  // adapt to light/dark-specific content\n  @if $mode == \"light\" {\n    .only-dark,\n    .only-dark ~ figcaption {\n      display: none !important;\n    }\n  } @else {\n    .only-light,\n    .only-light ~ figcaption {\n      display: none !important;\n    }\n    /* Adjust images in dark mode (unless they have class .only-dark or\n     * .dark-light, in which case assume they're already optimized for dark\n     * mode).\n     */\n    img:not(.only-dark):not(.dark-light) {\n      filter: brightness(0.8) contrast(1.2);\n    }\n    /* Give images a light background in dark mode in case they have\n    *  transparency and black text (unless they have class .only-dark or .dark-light, in\n    *  which case assume they're already optimized for dark mode).\n    */\n    .bd-content img:not(.only-dark):not(.dark-light) {\n      background: rgb(255, 255, 255);\n      border-radius: 0.25rem;\n    }\n    // MathJax SVG outputs should be filled to same color as text.\n    .MathJax_SVG * {\n      fill: var(--pst-color-text-base);\n    }\n  }\n}\n\n/* Defaults to light mode if data-theme is not set */\nhtml:not([data-theme]) {\n  @include theme-colors(\"light\");\n}\n\n/* NOTE: @each {...} is like a for-loop\n * https://sass-lang.com/documentation/at-rules/control/each\n */\n@each $mode in (light, dark) {\n  html[data-theme=\"#{$mode}\"] {\n    @include theme-colors($mode);\n    color-scheme: $mode;\n  }\n}\n\n// assign classes too, for runtime use of theme colors\n@each $col-name, $definition in $pst-semantic-colors {\n  .pst-color-#{$col-name} {\n    color: var(--pst-color-#{$col-name});\n  }\n}\n","/*******************************************************************************\n * Special-cases for the sphinx-design library, mainly to make it compatible\n * with the dark/light themes of pydata-sphinx-theme.\n *\n * NOTE: sphinx-design uses !important quite liberally, so here we must do the\n * same for our overrides to have any effect.\n */\n@use \"../variables/color\" as pst-color;\n@use \"sass:meta\";\n@use \"sass:color\";\n\n/*******************************************************************************\n  * Color and variables\n  *\n  * This is a list of the semantic color names from sphinx-design (we only\n  * need to override variables that sphinx-design has actually defined).\n  * https://github.com/executablebooks/sphinx-design/blob/9226a12a/style/_colors.scss#L31-L43\n  */\n$sd-semantic-color-names: (\n  \"primary\",\n  \"secondary\",\n  \"success\",\n  \"info\",\n  \"warning\",\n  \"danger\",\n  \"light\",\n  \"muted\",\n  \"dark\",\n  \"black\",\n  \"white\"\n);\n\n/**\n  * Here we create some extra --pst-color-* variables and use\n  * them to override the value of the corresponding sphinx-design variables.\n  * This is easier than re-writing the sphinx-design rules. Even easier would be\n  * directly assigning our values to the --sd-color-* variables, but then our\n  * downstream users couldn't override *our* colors and have it affect buttons\n  * and badges.\n  *\n  * First, define the extra keys needed to cover the full range of semantic\n  * color names used in sphinx-design, then merge them with the names we\n  * already define for our own needs.\n  * see https://sphinx-design.readthedocs.io/en/latest/css_variables.html\n  */\n$extra-semantic-colors: (\n  \"white\": $foundation-white,\n  \"light\": (\n    light: $foundation-light-gray,\n    bg-light: color.scale($foundation-light-gray, $lightness: 30%),\n    dark: $foundation-light-gray,\n    bg-dark: color.scale($foundation-light-gray, $lightness: -30%),\n  ),\n  \"muted\": (\n    light: $foundation-muted-gray,\n    bg-light: color.scale($foundation-muted-gray, $lightness: 30%),\n    dark: $foundation-light-gray,\n    bg-dark: color.scale($foundation-muted-gray, $lightness: -30%),\n  ),\n  \"dark\": $foundation-dark-gray,\n  \"black\": $foundation-black,\n);\n\n$all-colors: map-merge($pst-semantic-colors, $extra-semantic-colors);\n\n@mixin create-sd-colors($value, $name) {\n  // define the pst variables, so that downstream user overrides will work\n  --pst-color-#{$name}: #{$value};\n  // we are now using a11y-combination to calculate the text color - this is based\n  // on the WCAG color contrast guidelines\n  --pst-color-#{$name}-text: #{a11y-combination($value)};\n  //TODO: highlight seems to be used for buttons @trallard to fix on a11y follow-up work\n  --pst-color-#{$name}-highlight: #{darken($value, 15%)};\n  // override the sphinx-design variables\n  --sd-color-#{$name}: var(--pst-color-#{$name});\n  --sd-color-#{$name}-text: var(--pst-color-#{$name}-text);\n  //TODO: highlight seems to be used for buttons @trallard to fix on a11y follow-up work\n  --sd-color-#{$name}-highlight: var(--pst-color-#{$name}-highlight);\n}\n\n// Now we override the --sd-color-* variables.\n@each $mode in (light, dark) {\n  html[data-theme=\"#{$mode}\"] {\n    // check if this color is defined differently for light/dark\n    @each $name in $sd-semantic-color-names {\n      $definition: map-get($all-colors, $name);\n      @if type-of($definition) == map {\n        @each $key, $value in $definition {\n          @if str-index($key, $mode) != null {\n            // since now we define the bg colours in the semantic colours and not\n            // by changing opacity, we need to check if the key contains bg and the\n            // correct mode (light/dark)\n            @if str-index($key, \"bg\") != null {\n              --sd-color-#{$name}-bg: #{$value};\n              // create local variable\n              $bg-var: --sd-color-#{$name}-bg;\n              $value: check-color($value);\n              --sd-color-#{$name}-bg-text: #{a11y-combination($value)};\n            } @else {\n              $value: check-color($value);\n              @include create-sd-colors($value, $name);\n            }\n          }\n        }\n      } @else {\n        $value: map-get($all-colors, $name);\n        @include create-sd-colors($value, $name);\n      }\n    }\n  }\n}\n\n// Make sure the color border variables are set using our variables\n@each $mode in (light, dark) {\n  html[data-theme=\"#{$mode}\"] {\n    --sd-color-card-border: var(--pst-color-border);\n  }\n}\n\n/*******************************************************************************\n  * shadows\n  */\nhtml[data-theme=\"light\"] {\n  .sd-shadow-xs,\n  .sd-shadow-sm,\n  .sd-shadow-md,\n  .sd-shadow-lg {\n    @include box-shadow();\n  }\n}\n\n/*******************************************************************************\n  * cards\n  */\n\n.bd-content .sd-card {\n  border: 1px solid var(--pst-color-border);\n\n  // TODO - --pst-color-panel-background is not defined... where is this coming from?\n  .sd-card-header {\n    background-color: var(--pst-color-panel-background);\n    border-bottom: 1px solid var(--pst-color-border);\n  }\n  .sd-card-footer {\n    background-color: var(--pst-color-panel-background);\n    border-top: 1px solid var(--pst-color-border);\n  }\n\n  .sd-card-body {\n    background-color: var(--pst-color-panel-background);\n  }\n}\n/*******************************************************************************\n  * tabs\n  */\n\n.bd-content .sd-tab-set {\n  > input {\n    // Active tab label\n    &:checked + label {\n      border-color: transparent transparent var(--pst-color-primary); // top LR bottom\n      color: var(--pst-color-primary);\n    }\n\n    // Hover label\n    &:not(:checked) + label:hover {\n      border-color: var(--pst-color-secondary);\n      color: var(--pst-color-secondary);\n    }\n  }\n\n  // Tab label\n  > label {\n    color: var(--pst-color-text-muted);\n    border-top: 0.125rem solid transparent; // so hover isn't just color change\n    padding-top: 0.5em; // same as bottom padding, so hover overline looks OK\n    // Hovered label\n    html &:hover {\n      color: var(--pst-color-secondary);\n      border-color: var(--pst-color-secondary);\n    }\n  }\n}\n\n/*******************************************************************************\n* Dropdowns\n*/\n\ndetails.sd-dropdown {\n  // Remove all borders to over-ride SD behavior, and we'll add our own later\n  border: 0px !important;\n  summary.sd-card-header {\n    border: 0 !important;\n    & + div.sd-summary-content {\n      border: 0;\n    }\n  }\n  // Drop shadow should behave same as admonitions\n  @include box-shadow();\n\n  // Header is where the \"clickable\" box goes\n  summary.sd-card-header {\n    display: flex;\n    align-items: center;\n    position: relative; // So background color works\n    font-weight: 600;\n    padding-top: 0.5rem;\n    padding-bottom: 0.5rem;\n\n    // Set a variable that we can re-use for colors later\n    // We must set this in the current and content sibling container\n    // so that it is defined in both places\n    --pst-sd-dropdown-color: var(--pst-gray-500);\n    --pst-sd-dropdown-bg-color: var(--pst-color-surface);\n    & + div.sd-summary-content {\n      --pst-sd-dropdown-color: var(--sd-color-card-border);\n    }\n    @each $name in $sd-semantic-color-names {\n      &.sd-bg-#{$name} {\n        --pst-sd-dropdown-color: var(--sd-color-#{$name});\n        --pst-sd-dropdown-bg-color: var(--sd-color-#{$name}-bg);\n        // Otherwise it won't be defined in the sibling element\n        & + div.sd-summary-content {\n          --pst-sd-dropdown-color: var(--sd-color-#{$name});\n          --pst-sd-dropdown-bg-color: var(--sd-color-#{$name}-bg);\n        }\n      }\n      &.sd-bg-text-#{$name} {\n        // Use the WCAG conformant text color\n        color: var(--sd-color-#{$name}-bg-text) !important;\n      }\n    }\n\n    @include legacy-backdrop-placeholder;\n    background-color: var(--pst-sd-dropdown-bg-color) !important;\n\n    // Add a left border with the same structure as our admonitions\n    border-left: 0.2rem solid var(--pst-sd-dropdown-color) !important;\n    & + div.sd-summary-content {\n      border-left: 0.2rem solid var(--pst-sd-dropdown-color) !important;\n      border-bottom-left-radius: calc(0.25rem - 1px);\n      background-color: var(--pst-color-on-background);\n    }\n    span.sd-summary-icon {\n      display: inline-flex;\n      align-items: center;\n      color: var(--pst-sd-dropdown-color) !important;\n      svg {\n        opacity: 1;\n      }\n    }\n\n    // Positioning of the caret\n    .sd-summary-up,\n    .sd-summary-down {\n      top: 0.7rem;\n    }\n  }\n}\n","/**\n * Sphinx togglebutton\n */\n\n.bd-content {\n  // Admonition toggles\n  .admonition {\n    button.toggle-button {\n      color: inherit;\n    }\n  }\n\n  // Details buttons\n  details.toggle-details {\n    // Over-ride border color to re-use our primary color\n    summary {\n      border-left: 3px solid var(--pst-color-primary);\n    }\n  }\n}\n","/**\n * Styles for various Sphinx execution libraries to display pre-executed notebooks.\n * For now, where these define output sections, we simply revert their background\n * to be a \"light theme\" background. This ensures that inputs/outputs behave similarly,\n * because the CSS is often controlled by each package.\n * In the future, we might add dark theme support for specific packages.\n */\n\n/*******************************************************************************\n * nbsphinx\n */\nhtml div.rendered_html\n// NBsphinx ipywidgets output selector\nhtml .jp-RenderedHTMLCommon {\n  table {\n    table-layout: auto;\n  }\n}\n\n// Dark theme special-cases\nhtml[data-theme=\"dark\"] .bd-content {\n  .nboutput {\n    .output_area.rendered_html {\n      @include cell-output-background;\n    }\n\n    .output_area.stderr {\n      background: var(--pst-color-danger);\n    }\n  }\n}\n\n// Add extra padding to the final item in an nbsphinx container\ndiv.nblast.container {\n  margin-bottom: 1rem;\n}\n\n/*******************************************************************************\n * myst NB\n */\n\ndiv.cell_output .output {\n  max-width: 100%;\n  overflow-x: auto;\n}\n\n// Dark theme special-cases\nhtml[data-theme=\"dark\"] .bd-content {\n  div.cell_output {\n    img,\n    .text_html {\n      @include cell-output-background;\n    }\n  }\n}\n\n// Prevent tables from scrunching together\n.bd-content {\n  div.cell_input {\n    display: flex;\n    flex-direction: column;\n    justify-content: stretch;\n  }\n\n  div.cell_input,\n  div.output {\n    border-radius: $admonition-border-radius;\n  }\n\n  div.output {\n    table {\n      table-layout: auto;\n    }\n  }\n}\n","/**\n * style for the various mapping libs based on leaflet (folium, geemap, ipyleaflet)\n * mainly ensure the good display of the maps in both themes and avoid the customization\n * of the tiles\n */\n\n/**\n * avoid border override from pydata-sphinx-theme\n * minimal selctor to get the priority\n */\nhtml[data-theme=\"dark\"] .bd-content img.leaflet-tile.leaflet-tile-loaded {\n  border-radius: 0;\n  padding: 0;\n}\n","/**\n * /search.html page special-cases\n */\n\n.bd-search-container {\n  /*******************************************\n  * Search results\n  */\n  // Whitespace\n  div#search-results {\n    > h2 {\n      font-size: var(--pst-font-size-icon);\n      margin-top: 0;\n    }\n\n    p.search-summary {\n      color: var(--pst-color-text-muted);\n    }\n  }\n\n  ul.search {\n    margin: 0;\n    list-style: none;\n\n    li {\n      background-image: none;\n      padding: 1rem 0;\n      margin: 1rem 0;\n      border-top: 1px solid var(--pst-color-text-muted);\n\n      // First link is the page title, it should be a bit bigger\n      > a {\n        font-size: 1.2em;\n      }\n\n      div.context,\n      p.context {\n        color: var(--pst-color-text-base);\n        margin: 0.5em 0 0 0;\n\n        // Add a # before page section titles to make it clear they are sections\n        a:before {\n          content: \"#\";\n          padding-right: 0.2em;\n          color: var(--pst-color-text-muted);\n        }\n      }\n    }\n  }\n}\n"],"names":[],"sourceRoot":""}
\ No newline at end of file
diff --git a/_static/styles/sphinx-book-theme.css b/_static/styles/sphinx-book-theme.css
new file mode 100644
index 0000000..56c23f1
--- /dev/null
+++ b/_static/styles/sphinx-book-theme.css
@@ -0,0 +1,8 @@
+/*! sphinx-book-theme CSS
+ * BSD 3-Clause License
+ * Copyright (c) 2020, EBP
+ * All rights reserved.
+ *
+ * This follows the 7-1 pattern described here:
+ * https://sass-guidelin.es/#architecture
+ */html[data-theme=dark],html[data-theme=light]{--pst-font-size-base:none;--pst-color-secondary:#e89217}html[data-theme=light]{--sbt-color-announcement:#616161;--pst-color-primary:#176de8}html[data-theme=dark]{--pst-color-primary:#528fe4;--sbt-color-announcement:#616161;--pst-color-background:#121212}html{scroll-padding-top:4rem}.sbt-scroll-pixel-helper{position:absolute;width:0;height:0;top:0;left:0}.d-n,.onlyprint{display:none}@media print{.onlyprint{display:block!important}}@media print{.noprint{display:none!important}}.bd-article-container h1,.bd-article-container h2,.bd-article-container h3,.bd-article-container h4,.bd-article-container h5,.bd-article-container p.caption{color:var(--pst-color-muted)}.bd-article-container h1,.bd-article-container h2{font-weight:500}a.brackets:before{color:inherit;font-family:inherit;margin-right:0}table{position:relative}@media print{.bd-main .bd-content{margin-left:2rem;height:auto}.bd-main .bd-content #jb-print-docs-body{margin-left:0}.bd-main .bd-content #jb-print-docs-body h1{font-size:3em;text-align:center;margin-bottom:0}.bd-main .bd-content .bd-article{padding-top:0}.bd-main .bd-content .bd-article h1:first-of-type{display:none}.bd-main .bd-content .container{min-width:0!important}.bd-main .bd-content h1{margin-top:1em;margin-bottom:1em}.bd-main .bd-content h1,.bd-main .bd-content h2,.bd-main .bd-content h3,.bd-main .bd-content h4{break-after:avoid;color:#000}.bd-main .bd-content table{break-inside:avoid}.bd-main .bd-content pre{word-wrap:break-word}.bd-main .bd-content a.headerlink{display:none}.bd-main .bd-content aside.margin,.bd-main .bd-content aside.sidebar,.bd-main .bd-content blockquote.epigraph{border:none}.bd-main .bd-content .footer{margin-top:1em}.bd-main .bd-content #jb-print-toc{margin-bottom:1.5rem;margin-left:0}.bd-main .bd-content #jb-print-toc .section-nav{border-left:0!important;list-style-type:disc!important;margin-left:3em!important}.bd-main .bd-content #jb-print-toc .section-nav a{text-decoration:none!important}.bd-main .bd-content #jb-print-toc .section-nav li{display:list-item!important}.bd-main .bd-content #jb-print-toc .section-nav .nav{display:none}.bd-main .bd-footer-content{display:none!important}}.bd-header-announcement{background-color:var(--sbt-color-announcement);color:#fff}.bd-main .bd-content{justify-content:left}.bd-main .bd-content .bd-article-container{padding:0;overflow-x:unset;min-width:0}@media (min-width:1200px){.bd-main .bd-content .bd-article-container{max-width:calc(100% - var(--pst-sidebar-secondary))}}.bd-main .bd-content .bd-article-container .bd-article{padding-right:2rem;padding-left:2rem}@media (max-width:768px){.bd-main .bd-content .bd-article-container .bd-article{padding-right:1rem;padding-left:1rem}}.bd-main .bd-content .bd-article-container details.above-input summary,.bd-main .bd-content .bd-article-container details.below-input summary{border-left:3px solid var(--pst-color-primary)}@media (min-width:768px){label.sidebar-toggle.primary-toggle{display:inline-block}}@media (max-width:768px){label.sidebar-toggle.primary-toggle{margin-bottom:0}}@media (min-width:992px){label.sidebar-toggle.secondary-toggle{display:none}}@media (max-width:768px){label.sidebar-toggle.secondary-toggle{margin-bottom:0}}.bd-header-article{display:flex;align-items:center;position:sticky;top:0;background-color:var(--pst-color-background);transition:left .2s;font-size:.9em;padding:0 1rem;z-index:1020}@media (max-width:768px){.bd-header-article{padding:0 .5rem;height:3.5rem}}.scrolled .bd-header-article{box-shadow:0 6px 6px -6px var(--pst-color-shadow)}.bd-header-article .header-article__inner{padding:0}.bd-header-article .header-article-items{display:flex;align-items:center;height:3rem;width:100%}.bd-header-article .header-article-item{display:flex;align-items:center}.bd-header-article .article-header-buttons{display:flex}.bd-header-article .btn{font-size:1.3rem;color:var(--pst-color-text-muted);border:none;padding:0 .5rem;display:flex;align-items:center}.bd-header-article .btn svg{width:1.3rem}.bd-header-article .btn.show,.bd-header-article .btn:hover{color:var(--pst-color-text-base);border:none}.bd-header-article .btn.show+.dropdown-menu,.bd-header-article .btn:hover+.dropdown-menu{display:block}.bd-header-article .btn:focus{box-shadow:none}.bd-header-article .btn.dropdown-toggle:after{display:none}.bd-header-article div.dropdown{display:flex;align-items:center}.bd-header-article .theme-switch-button{margin:0}.bd-header-article .theme-switch-button button,.bd-header-article .theme-switch-button i,.bd-header-article .theme-switch-button span{padding:0}.bd-header-article .theme-switch-button i,.bd-header-article .theme-switch-button span{transition:color .25s ease-out}.bd-header-article .theme-switch-button:active,.bd-header-article .theme-switch-button:hover{background-color:unset!important}.bd-header-article .theme-switch-button:active i,.bd-header-article .theme-switch-button:active span,.bd-header-article .theme-switch-button:hover i,.bd-header-article .theme-switch-button:hover span{color:var(--pst-color-text-base)}.bd-header-article .dropdown-menu{top:2rem;transform:translateX(-75%);box-shadow:0 .2rem .5rem var(--pst-color-shadow),0 0 .0625rem var(--pst-color-shadow)!important;border-color:var(--pst-color-border);background-color:var(--pst-color-background);color:var(--pst-color-text-muted)}.bd-header-article .dropdown-menu:hover{display:block}.bd-header-article .dropdown-menu .dropdown-item{display:inline-flex;align-items:center;padding-left:.5em;font-size:1em}.bd-header-article .dropdown-menu .dropdown-item:hover{text-decoration:none;background-color:initial;color:var(--pst-color-text-base)}.bd-header-article .dropdown-menu .dropdown-item span img{height:1em}.bd-header-article .dropdown-menu .dropdown-item span.btn__icon-container{width:1.7em;align-items:center;display:inline-flex;justify-content:center}.bd-header{position:inherit}.bd-header label.sidebar-toggle{display:none}.bd-sidebar-primary{top:0;max-height:100vh;padding:1rem;transition:margin-left .25s ease 0s,opacity .25s ease 0s,visibility .25s ease 0s}@media (max-width:768px){.bd-sidebar-primary{z-index:1081}}.bd-sidebar-primary .sidebar-primary-items__start{border-top:none}@media (min-width:992px){.bd-sidebar-primary{flex-basis:20%}}@media (min-width:992px){input#__primary:checked~.bd-container .bd-sidebar-primary{margin-left:-20%;visibility:hidden;opacity:0}}.bd-sidebar-secondary{top:0}@media (max-width:992px){.bd-sidebar-secondary{z-index:1081}}.bd-sidebar-secondary .sidebar-secondary-items{padding:0;display:flex;gap:.5rem}.bd-sidebar-secondary .sidebar-secondary-items .sidebar-secondary-item{padding-top:0;padding-bottom:0}.bd-sidebar-secondary .onthispage{height:3rem;min-height:3rem;display:flex;gap:.5rem;align-items:center;margin:0;color:var(--pst-color-muted)}@media (min-width:992px){.bd-sidebar-secondary{background:var(--pst-color-background);height:fit-content;transition:max-height .4s ease;z-index:2;padding:0}.bd-sidebar-secondary .toc-item{border-left-color:var(--pst-color-surface);padding-top:0}.bd-sidebar-secondary .toc-item nav.page-toc{margin-bottom:0;transition:opacity .4s ease}.bd-sidebar-secondary.hide:not(:hover){max-height:3rem;overflow-y:hidden}.scrolled .bd-sidebar-secondary.hide:not(:hover){box-shadow:0 6px 6px -6px rgba(0,0,0,.3)}.bd-sidebar-secondary.hide:not(:hover) .onthispage:after{opacity:1;content:"\f107";font-family:Font Awesome\ 5 Free;font-weight:900;padding-left:.5em;transition:opacity .3s ease}.bd-sidebar-secondary.hide:not(:hover) nav.page-toc{opacity:0}}footer{font-size:var(--sbt-font-size-small-1)}footer.bd-footer-content{display:flex;flex-wrap:wrap;padding:15px;border-top:1px solid #ccc;font-size:87.5%}footer.bd-footer-content .bd-footer-content__inner{padding-left:0}footer.bd-footer-content .bd-footer-content__inner p{margin-bottom:0}.bd-footer-article{padding:0 1rem}@media (max-width:768px){.bd-footer-article{padding:0 .5rem}}#pst-back-to-top{font-size:.8rem}@media (min-width:1200px){#pst-back-to-top{display:none!important}}.bd-sidebar-primary .navbar-icon-links{column-gap:.5rem}.bd-sidebar-primary .navbar-icon-links .nav-link i,.bd-sidebar-primary .navbar-icon-links .nav-link span{font-size:1.2rem}.bd-sidebar-primary .navbar-icon-links .nav-link img{font-size:.8rem}.navbar-brand{height:unset;max-height:unset;flex-direction:column;justify-content:center;gap:.25rem}.navbar-brand:hover{text-decoration:none}.navbar-brand .logo__title{font-size:1.25rem;white-space:normal;text-align:center}.navbar-brand .logo__image{height:unset}.bd-search-container{margin:2em}.bd-search-container #search-results h2:first-child{display:none}.search-button-field{width:100%;font-size:.9rem;display:none}.search-button-field .search-button__kbd-shortcut{margin-left:auto}@media (min-width:992px){.search-button{display:none!important}.search-button-field{display:flex}}div#searchbox{padding-right:2rem;padding-left:2rem}@media (max-width:768px){div#searchbox{padding-right:1rem;padding-left:1rem}}@media (min-width:768px){div#searchbox p.highlight-link{margin-left:0}div#searchbox p.highlight-link a{font-size:1rem}}img{max-width:100%}img.align-center{margin-left:auto;margin-right:auto;display:block}img.align-left{clear:left;float:left;margin-right:1em}img.align-right{clear:right;float:right;margin-left:1em}div.figure{width:100%;margin-bottom:1em;text-align:center}div.figure.align-left{text-align:left}div.figure.align-left p.caption{margin-left:0}div.figure.align-right{text-align:right}div.figure.align-right p.caption{margin-right:0}div.figure p.caption{margin:.5em 10%}div.figure.margin-caption p.caption,div.figure.margin p.caption{margin:.5em 0}div.figure.margin-caption p.caption{text-align:left}div.figure span.caption-number{font-weight:700}div.figure span{font-size:.9rem}label.margin-toggle{margin-bottom:0}label.margin-toggle.marginnote-label{display:none}label.margin-toggle sup{user-select:none}@media (max-width:992px){label.margin-toggle{cursor:pointer;color:#0071bc}label.margin-toggle.marginnote-label{display:inline}label.margin-toggle.marginnote-label:after{content:"\2295"}}input.margin-toggle{display:none}@media (max-width:992px){input.margin-toggle:checked+.marginnote,input.margin-toggle:checked+.sidenote{display:block;float:left;left:1rem;clear:both;width:95%;margin:1rem 2.5%;position:relative}}span.marginnote,span.sidenote{z-index:2;position:relative;width:40%;float:right;background-color:unset;font-size:.9em;margin-left:.5rem;border-left:none}span.marginnote sup,span.sidenote sup{user-select:none}@media (min-width:992px),print{span.marginnote,span.sidenote{width:33%;margin:0 -36% 0 0;clear:right}span.marginnote p.sidebar-title,span.sidenote p.sidebar-title{margin-bottom:-1rem;border-bottom:none;padding-left:0}span.marginnote p.sidebar-title~*,span.sidenote p.sidebar-title~*{padding-left:0;padding-right:0}}@media (max-width:992px){span.marginnote,span.sidenote{display:none}}aside.sidebar .note{margin:1rem;padding:0 0 1rem}aside.sidebar .admonition-title{margin:0 -1rem 0 0}aside.sidebar.margin .sidebar-title:empty{display:none}aside.sidebar.margin .admonition{margin:.5rem;padding-left:0;padding-right:0}aside.sidebar.margin .admonition .admonition-title{margin-left:0;margin-right:0}@media (min-width:992px){aside.sidebar.margin{border:none}aside.sidebar.margin .admonition{margin:1rem 0;padding:0 0 1rem}}.cell.tag_margin,.cell.tag_popout,.margin.docutils.container,aside.margin,div.margin,figure.margin{z-index:2;position:relative;width:40%;float:right;background-color:unset;font-size:.9em;margin-left:.5rem}@media (min-width:992px),print{.cell.tag_margin,.cell.tag_popout,.margin.docutils.container,aside.margin,div.margin,figure.margin{width:33%;margin:0 -36% 0 0;clear:right}.cell.tag_margin p.sidebar-title,.cell.tag_popout p.sidebar-title,.margin.docutils.container p.sidebar-title,aside.margin p.sidebar-title,div.margin p.sidebar-title,figure.margin p.sidebar-title{margin-bottom:-1rem;border-bottom:none;padding-left:0}.cell.tag_margin p.sidebar-title~*,.cell.tag_popout p.sidebar-title~*,.margin.docutils.container p.sidebar-title~*,aside.margin p.sidebar-title~*,div.margin p.sidebar-title~*,figure.margin p.sidebar-title~*{padding-left:0;padding-right:0}}.cell.tag_margin div.cell.tag_margin .cell_output,.cell.tag_popout div.cell.tag_margin .cell_output,.margin.docutils.container div.cell.tag_margin .cell_output,aside.margin div.cell.tag_margin .cell_output,div.margin div.cell.tag_margin .cell_output,figure.margin div.cell.tag_margin .cell_output{padding-left:0}div.figure.margin-caption figcaption,div.figure.margin-caption p.caption,figure.margin-caption figcaption{z-index:2;position:relative;width:40%;float:right;background-color:unset;font-size:.9em;margin-left:.5rem}@media (min-width:992px),print{div.figure.margin-caption figcaption,div.figure.margin-caption p.caption,figure.margin-caption figcaption{width:33%;margin:0 -36% 0 0;clear:right}div.figure.margin-caption figcaption p.sidebar-title,div.figure.margin-caption p.caption p.sidebar-title,figure.margin-caption figcaption p.sidebar-title{margin-bottom:-1rem;border-bottom:none;padding-left:0}div.figure.margin-caption figcaption p.sidebar-title~*,div.figure.margin-caption p.caption p.sidebar-title~*,figure.margin-caption figcaption p.sidebar-title~*{padding-left:0;padding-right:0}}.margin-caption figcaption{text-align:left}div.cell.tag_full-width,div.cell.tag_full_width,div.full-width,div.full_width{z-index:2;position:relative}@media (min-width:992px){div.cell.tag_full-width,div.cell.tag_full_width,div.full-width,div.full_width{max-width:136%;width:136%}}blockquote.epigraph,blockquote.highlights,blockquote.pull-quote{font-size:1.25em;border-left:none;background-color:var(--pst-color-background)}blockquote div>p+p.attribution{font-style:normal;font-size:.9em;text-align:right;color:#6c757d;padding-right:2em}div[class*=highlight-],pre{clear:none}div.cell.tag_output_scroll div.cell_output,div.cell.tag_scroll-input div.cell_input,div.cell.tag_scroll-output div.cell_output{max-height:24em;overflow-y:auto}@media only print{div.utterances,hypothesis-sidebar{display:none}}.thebelab-cell{border:none!important;margin-right:.5em!important}.thebelab-cell .thebelab-input{padding-left:10px!important}.cell.docutils.container{padding-right:0!important}button.thebe-launch-button{height:2.5em;font-size:1em}
\ No newline at end of file
diff --git a/_static/styles/theme.css b/_static/styles/theme.css
new file mode 100644
index 0000000..4519dd9
--- /dev/null
+++ b/_static/styles/theme.css
@@ -0,0 +1,2 @@
+/* Provided by Sphinx's 'basic' theme, and included in the final set of assets */
+@import "../basic.css";
diff --git a/_static/togglebutton.css b/_static/togglebutton.css
new file mode 100644
index 0000000..54a6787
--- /dev/null
+++ b/_static/togglebutton.css
@@ -0,0 +1,160 @@
+/**
+ * Admonition-based toggles
+ */
+
+/* Visibility of the target */
+.admonition.toggle .admonition-title ~ * {
+    transition: opacity .3s, height .3s;
+}
+
+/* Toggle buttons inside admonitions so we see the title */
+.admonition.toggle {
+    position: relative;
+}
+
+/* Titles should cut off earlier to avoid overlapping w/ button */
+.admonition.toggle .admonition-title {
+    padding-right: 25%;
+    cursor: pointer;
+}
+
+/* Hovering will cause a slight shift in color to make it feel interactive */
+.admonition.toggle .admonition-title:hover {
+    box-shadow: inset 0 0 0px 20px rgb(0 0 0 / 1%);
+}
+
+/* Hovering will cause a slight shift in color to make it feel interactive */
+.admonition.toggle .admonition-title:active {
+    box-shadow: inset 0 0 0px 20px rgb(0 0 0 / 3%);
+}
+
+/* Remove extra whitespace below the admonition title when hidden */
+.admonition.toggle-hidden {
+    padding-bottom: 0;
+}
+
+.admonition.toggle-hidden .admonition-title {
+    margin-bottom: 0;
+}
+
+/* hides all the content of a page until de-toggled */
+.admonition.toggle-hidden .admonition-title ~ * {
+    height: 0;
+    margin: 0;
+    opacity: 0;
+    visibility: hidden;
+}
+
+/* General button style and position*/
+button.toggle-button {
+    /**
+     * Background and shape. By default there's no background
+     * but users can style as they wish
+     */  
+    background: none;
+    border: none;
+    outline: none;
+
+    /* Positioning just inside the admonition title */
+    position: absolute;
+    right: 0.5em;
+    padding: 0px;
+    border: none;
+    outline: none;
+}
+
+/* Display the toggle hint on wide screens */
+@media (min-width: 768px) {
+    button.toggle-button.toggle-button-hidden:before {
+        content: attr(data-toggle-hint);  /* This will be filled in by JS */
+        font-size: .8em;
+        align-self: center;
+    }
+}
+
+/* Icon behavior */
+.tb-icon {
+    transition: transform .2s ease-out;
+    height: 1.5em;
+    width: 1.5em;
+    stroke: currentColor;  /* So that we inherit the color of other text */
+}
+
+/* The icon should point right when closed, down when open. */
+/* Open */
+.admonition.toggle button .tb-icon {
+    transform: rotate(90deg);
+}
+
+/* Closed */
+.admonition.toggle button.toggle-button-hidden .tb-icon {
+    transform: rotate(0deg);
+}
+
+/* With details toggles, we don't rotate the icon so it points right */
+details.toggle-details .tb-icon {
+    height: 1.4em;
+    width: 1.4em;
+    margin-top: 0.1em;  /* To center the button vertically */
+}
+
+
+/**
+ * Details-based toggles.
+ * In this case, we wrap elements with `.toggle` in a details block.
+ */
+
+/* Details blocks */
+details.toggle-details {
+    margin: 1em 0;
+}
+
+
+details.toggle-details summary {
+    display: flex;
+    align-items: center;
+    cursor: pointer;
+    list-style: none;
+    border-radius: .2em;
+    border-left: 3px solid #1976d2;
+    background-color: rgb(204 204 204 / 10%);
+    padding: 0.2em 0.7em 0.3em 0.5em; /* Less padding on left because the SVG has left margin */
+    font-size: 0.9em;
+}
+
+details.toggle-details summary:hover {
+    background-color: rgb(204 204 204 / 20%);
+}
+
+details.toggle-details summary:active {
+    background: rgb(204 204 204 / 28%);
+}
+
+.toggle-details__summary-text {
+    margin-left: 0.2em;
+}
+
+details.toggle-details[open] summary {
+    margin-bottom: .5em;
+}
+
+details.toggle-details[open] summary .tb-icon {
+    transform: rotate(90deg);
+}
+
+details.toggle-details[open] summary ~ * {
+    animation: toggle-fade-in .3s ease-out;
+}
+
+@keyframes toggle-fade-in {
+  from {opacity: 0%;}
+  to {opacity: 100%;}
+}
+
+/* Print rules - we hide all toggle button elements at print */
+@media print {
+    /* Always hide the summary so the button doesn't show up */
+    details.toggle-details summary {
+        display: none;
+    }
+}
\ No newline at end of file
diff --git a/_static/togglebutton.js b/_static/togglebutton.js
new file mode 100644
index 0000000..215a7ee
--- /dev/null
+++ b/_static/togglebutton.js
@@ -0,0 +1,187 @@
+/**
+ * Add Toggle Buttons to elements
+ */
+
+let toggleChevron = `
+<svg xmlns="http://www.w3.org/2000/svg" class="tb-icon toggle-chevron-right" width="44" height="44" viewBox="0 0 24 24" stroke-width="1.5" stroke="#000000" fill="none" stroke-linecap="round" stroke-linejoin="round">
+<path stroke="none" d="M0 0h24v24H0z" fill="none"/>
+<polyline points="9 6 15 12 9 18" />
+</svg>`;
+
+var initToggleItems = () => {
+  var itemsToToggle = document.querySelectorAll(togglebuttonSelector);
+  console.log(`[togglebutton]: Adding toggle buttons to ${itemsToToggle.length} items`)
+  // Add the button to each admonition and hook up a callback to toggle visibility
+  itemsToToggle.forEach((item, index) => {
+    if (item.classList.contains("admonition")) {
+      // If it's an admonition block, then we'll add a button inside
+      // Generate unique IDs for this item
+      var toggleID = `toggle-${index}`;
+      var buttonID = `button-${toggleID}`;
+
+      item.setAttribute('id', toggleID);
+      if (!item.classList.contains("toggle")){
+        item.classList.add("toggle");
+      }
+      // This is the button that will be added to each item to trigger the toggle
+      var collapseButton = `
+        <button type="button" id="${buttonID}" class="toggle-button" data-target="${toggleID}" data-button="${buttonID}" data-toggle-hint="${toggleHintShow}" aria-label="Toggle hidden content">
+            ${toggleChevron}
+        </button>`;
+
+      title = item.querySelector(".admonition-title")
+      title.insertAdjacentHTML("beforeend", collapseButton);
+      thisButton = document.getElementById(buttonID);
+
+      // Add click handlers for the button + admonition title (if admonition)
+      admonitionTitle = document.querySelector(`#${toggleID} > .admonition-title`)
+      if (admonitionTitle) {
+        // If an admonition, then make the whole title block clickable
+        admonitionTitle.addEventListener('click', toggleClickHandler);
+        admonitionTitle.dataset.target = toggleID
+        admonitionTitle.dataset.button = buttonID
+      } else {
+        // If not an admonition then we'll listen for the button click
+        thisButton.addEventListener('click', toggleClickHandler);
+      }
+
+      // Now hide the item for this toggle button unless explicitly noted to show
+      if (!item.classList.contains("toggle-shown")) {
+        toggleHidden(thisButton);
+      }
+    } else {
+      // If not an admonition, wrap the block in a <details> block
+      // Define the structure of the details block and insert it as a sibling
+      var detailsBlock = `
+        <details class="toggle-details">
+            <summary class="toggle-details__summary">
+              ${toggleChevron}
+              <span class="toggle-details__summary-text">${toggleHintShow}</span>
+            </summary>
+        </details>`;
+      item.insertAdjacentHTML("beforebegin", detailsBlock);
+
+      // Now move the toggle-able content inside of the details block
+      details = item.previousElementSibling
+      details.appendChild(item)
+      item.classList.add("toggle-details__container")
+
+      // Set up a click trigger to change the text as needed
+      details.addEventListener('click', (click) => {
+        let parent = click.target.parentElement;
+        if (parent.tagName.toLowerCase() == "details") {
+          summary = parent.querySelector("summary");
+          details = parent;
+        } else {
+          summary = parent;
+          details = parent.parentElement;
+        }
+        // Update the inner text for the proper hint
+        if (details.open) {
+          summary.querySelector("span.toggle-details__summary-text").innerText = toggleHintShow;
+        } else {
+          summary.querySelector("span.toggle-details__summary-text").innerText = toggleHintHide;
+        }
+        
+      });
+
+      // If we have a toggle-shown class, open details block should be open
+      if (item.classList.contains("toggle-shown")) {
+        details.click();
+      }
+    }
+  })
+};
+
+// This should simply add / remove the collapsed class and change the button text
+var toggleHidden = (button) => {
+  target = button.dataset['target']
+  var itemToToggle = document.getElementById(target);
+  if (itemToToggle.classList.contains("toggle-hidden")) {
+    itemToToggle.classList.remove("toggle-hidden");
+    button.classList.remove("toggle-button-hidden");
+  } else {
+    itemToToggle.classList.add("toggle-hidden");
+    button.classList.add("toggle-button-hidden");
+  }
+}
+
+var toggleClickHandler = (click) => {
+  // Be cause the admonition title is clickable and extends to the whole admonition
+  // We only look for a click event on this title to trigger the toggle.
+
+  if (click.target.classList.contains("admonition-title")) {
+    button = click.target.querySelector(".toggle-button");
+  } else if (click.target.classList.contains("tb-icon")) {
+    // We've clicked the icon and need to search up one parent for the button
+    button = click.target.parentElement;
+  } else if (click.target.tagName == "polyline") {
+    // We've clicked the SVG elements inside the button, need to up 2 layers
+    button = click.target.parentElement.parentElement;
+  } else if (click.target.classList.contains("toggle-button")) {
+    // We've clicked the button itself and so don't need to do anything
+    button = click.target;
+  } else {
+    console.log(`[togglebutton]: Couldn't find button for ${click.target}`)
+  }
+  target = document.getElementById(button.dataset['button']);
+  toggleHidden(target);
+}
+
+// If we want to blanket-add toggle classes to certain cells
+var addToggleToSelector = () => {
+  const selector = "";
+  if (selector.length > 0) {
+    document.querySelectorAll(selector).forEach((item) => {
+      item.classList.add("toggle");
+    })
+  }
+}
+
+// Helper function to run when the DOM is finished
+const sphinxToggleRunWhenDOMLoaded = cb => {
+  if (document.readyState != 'loading') {
+    cb()
+  } else if (document.addEventListener) {
+    document.addEventListener('DOMContentLoaded', cb)
+  } else {
+    document.attachEvent('onreadystatechange', function() {
+      if (document.readyState == 'complete') cb()
+    })
+  }
+}
+sphinxToggleRunWhenDOMLoaded(addToggleToSelector)
+sphinxToggleRunWhenDOMLoaded(initToggleItems)
+
+/** Toggle details blocks to be open when printing */
+if (toggleOpenOnPrint == "true") {
+  window.addEventListener("beforeprint", () => {
+    // Open the details
+    document.querySelectorAll("details.toggle-details").forEach((el) => {
+      el.dataset["togglestatus"] = el.open;
+      el.open = true;
+    });
+  
+    // Open the admonitions
+    document.querySelectorAll(".admonition.toggle.toggle-hidden").forEach((el) => {
+      console.log(el);
+      el.querySelector("button.toggle-button").click();
+      el.dataset["toggle_after_print"] = "true";
+    });
+  });
+  window.addEventListener("afterprint", () => {
+    // Re-close the details that were closed
+    document.querySelectorAll("details.toggle-details").forEach((el) => {
+      el.open = el.dataset["togglestatus"] == "true";
+      delete el.dataset["togglestatus"];
+    });
+  
+    // Re-close the admonition toggle buttons
+    document.querySelectorAll(".admonition.toggle").forEach((el) => {
+      if (el.dataset["toggle_after_print"] == "true") {
+        el.querySelector("button.toggle-button").click();
+        delete el.dataset["toggle_after_print"];
+      }
+    });
+  });
+}
diff --git a/_static/vendor/fontawesome/6.5.1/LICENSE.txt b/_static/vendor/fontawesome/6.5.1/LICENSE.txt
new file mode 100644
index 0000000..39e18e3
--- /dev/null
+++ b/_static/vendor/fontawesome/6.5.1/LICENSE.txt
@@ -0,0 +1,165 @@
+Fonticons, Inc. (https://fontawesome.com)
+
+--------------------------------------------------------------------------------
+
+Font Awesome Free License
+
+Font Awesome Free is free, open source, and GPL friendly. You can use it for
+commercial projects, open source projects, or really almost whatever you want.
+Full Font Awesome Free license: https://fontawesome.com/license/free.
+
+--------------------------------------------------------------------------------
+
+# Icons: CC BY 4.0 License (https://creativecommons.org/licenses/by/4.0/)
+
+The Font Awesome Free download is licensed under a Creative Commons
+Attribution 4.0 International License and applies to all icons packaged
+as SVG and JS file types.
+
+--------------------------------------------------------------------------------
+
+# Fonts: SIL OFL 1.1 License
+
+In the Font Awesome Free download, the SIL OFL license applies to all icons
+packaged as web and desktop font files.
+
+Copyright (c) 2023 Fonticons, Inc. (https://fontawesome.com)
+with Reserved Font Name: "Font Awesome".
+
+This Font Software is licensed under the SIL Open Font License, Version 1.1.
+This license is copied below, and is also available with a FAQ at:
+http://scripts.sil.org/OFL
+
+SIL OPEN FONT LICENSE
+Version 1.1 - 26 February 2007
+
+PREAMBLE
+The goals of the Open Font License (OFL) are to stimulate worldwide
+development of collaborative font projects, to support the font creation
+efforts of academic and linguistic communities, and to provide a free and
+open framework in which fonts may be shared and improved in partnership
+with others.
+
+The OFL allows the licensed fonts to be used, studied, modified and
+redistributed freely as long as they are not sold by themselves. The
+fonts, including any derivative works, can be bundled, embedded,
+redistributed and/or sold with any software provided that any reserved
+names are not used by derivative works. The fonts and derivatives,
+however, cannot be released under any other type of license. The
+requirement for fonts to remain under this license does not apply
+to any document created using the fonts or their derivatives.
+
+DEFINITIONS
+"Font Software" refers to the set of files released by the Copyright
+Holder(s) under this license and clearly marked as such. This may
+include source files, build scripts and documentation.
+
+"Reserved Font Name" refers to any names specified as such after the
+copyright statement(s).
+
+"Original Version" refers to the collection of Font Software components as
+distributed by the Copyright Holder(s).
+
+"Modified Version" refers to any derivative made by adding to, deleting,
+or substituting — in part or in whole — any of the components of the
+Original Version, by changing formats or by porting the Font Software to a
+new environment.
+
+"Author" refers to any designer, engineer, programmer, technical
+writer or other person who contributed to the Font Software.
+
+PERMISSION & CONDITIONS
+Permission is hereby granted, free of charge, to any person obtaining
+a copy of the Font Software, to use, study, copy, merge, embed, modify,
+redistribute, and sell modified and unmodified copies of the Font
+Software, subject to the following conditions:
+
+1) Neither the Font Software nor any of its individual components,
+in Original or Modified Versions, may be sold by itself.
+
+2) Original or Modified Versions of the Font Software may be bundled,
+redistributed and/or sold with any software, provided that each copy
+contains the above copyright notice and this license. These can be
+included either as stand-alone text files, human-readable headers or
+in the appropriate machine-readable metadata fields within text or
+binary files as long as those fields can be easily viewed by the user.
+
+3) No Modified Version of the Font Software may use the Reserved Font
+Name(s) unless explicit written permission is granted by the corresponding
+Copyright Holder. This restriction only applies to the primary font name as
+presented to the users.
+
+4) The name(s) of the Copyright Holder(s) or the Author(s) of the Font
+Software shall not be used to promote, endorse or advertise any
+Modified Version, except to acknowledge the contribution(s) of the
+Copyright Holder(s) and the Author(s) or with their explicit written
+permission.
+
+5) The Font Software, modified or unmodified, in part or in whole,
+must be distributed entirely under this license, and must not be
+distributed under any other license. The requirement for fonts to
+remain under this license does not apply to any document created
+using the Font Software.
+
+TERMINATION
+This license becomes null and void if any of the above conditions are
+not met.
+
+DISCLAIMER
+THE FONT SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTIES OF
+MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT
+OF COPYRIGHT, PATENT, TRADEMARK, OR OTHER RIGHT. IN NO EVENT SHALL THE
+COPYRIGHT HOLDER BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
+INCLUDING ANY GENERAL, SPECIAL, INDIRECT, INCIDENTAL, OR CONSEQUENTIAL
+DAMAGES, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
+FROM, OUT OF THE USE OR INABILITY TO USE THE FONT SOFTWARE OR FROM
+OTHER DEALINGS IN THE FONT SOFTWARE.
+
+--------------------------------------------------------------------------------
+
+# Code: MIT License (https://opensource.org/licenses/MIT)
+
+In the Font Awesome Free download, the MIT license applies to all non-font and
+non-icon files.
+
+Copyright 2023 Fonticons, Inc.
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of
+this software and associated documentation files (the "Software"), to deal in the
+Software without restriction, including without limitation the rights to use, copy,
+modify, merge, publish, distribute, sublicense, and/or sell copies of the Software,
+and to permit persons to whom the Software is furnished to do so, subject to the
+following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED,
+INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A
+PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
+HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
+OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
+SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+
+--------------------------------------------------------------------------------
+
+# Attribution
+
+Attribution is required by MIT, SIL OFL, and CC BY licenses. Downloaded Font
+Awesome Free files already contain embedded comments with sufficient
+attribution, so you shouldn't need to do anything additional when using these
+files normally.
+
+We've kept attribution comments terse, so we ask that you do not actively work
+to remove them from files, especially code. They're a great way for folks to
+learn about Font Awesome.
+
+--------------------------------------------------------------------------------
+
+# Brand Icons
+
+All brand icons are trademarks of their respective owners. The use of these
+trademarks does not indicate endorsement of the trademark holder by Font
+Awesome, nor vice versa. **Please do not use brand logos for any purpose except
+to represent the company, product, or service to which they refer.**
diff --git a/_static/vendor/fontawesome/6.5.1/css/all.min.css b/_static/vendor/fontawesome/6.5.1/css/all.min.css
new file mode 100644
index 0000000..5f2038f
--- /dev/null
+++ b/_static/vendor/fontawesome/6.5.1/css/all.min.css
@@ -0,0 +1,5 @@
+/*!
+ * Font Awesome Free 6.5.1 by @fontawesome - https://fontawesome.com
+ * License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License)
+ * Copyright 2023 Fonticons, Inc.
+ */.fa{font-family:var(--fa-style-family,"Font Awesome 6 Free");font-weight:var(--fa-style,900)}.fa,.fa-brands,.fa-classic,.fa-regular,.fa-sharp,.fa-solid,.fab,.far,.fas{-moz-osx-font-smoothing:grayscale;-webkit-font-smoothing:antialiased;display:var(--fa-display,inline-block);font-style:normal;font-variant:normal;line-height:1;text-rendering:auto}.fa-classic,.fa-regular,.fa-solid,.far,.fas{font-family:Font Awesome\ 6 Free}.fa-brands,.fab{font-family:Font Awesome\ 6 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diff --git a/_static/vendor/fontawesome/6.5.1/js/all.min.js.LICENSE.txt b/_static/vendor/fontawesome/6.5.1/js/all.min.js.LICENSE.txt
new file mode 100644
index 0000000..ef6dda4
--- /dev/null
+++ b/_static/vendor/fontawesome/6.5.1/js/all.min.js.LICENSE.txt
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+/*!
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+ * License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License)
+ * Copyright 2023 Fonticons, Inc.
+ */
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new file mode 100644
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diff --git a/_static/vendor/fontawesome/6.5.1/webfonts/fa-v4compatibility.ttf b/_static/vendor/fontawesome/6.5.1/webfonts/fa-v4compatibility.ttf
new file mode 100644
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diff --git a/_static/webpack-macros.html b/_static/webpack-macros.html
new file mode 100644
index 0000000..243a858
--- /dev/null
+++ b/_static/webpack-macros.html
@@ -0,0 +1,31 @@
+<!--
+  AUTO-GENERATED from webpack.config.js, do **NOT** edit by hand.
+  These are re-used in layout.html
+-->
+{# Load FontAwesome icons #}
+{% macro head_pre_icons() %}
+  <link href="{{ pathto('_static/vendor/fontawesome/6.5.1/css/all.min.css', 1) }}?digest=8d27b9dea8ad943066ae" rel="stylesheet" />
+  <link rel="preload" as="font" type="font/woff2" crossorigin href="{{ pathto('_static/vendor/fontawesome/6.5.1/webfonts/fa-solid-900.woff2', 1) }}" />
+<link rel="preload" as="font" type="font/woff2" crossorigin href="{{ pathto('_static/vendor/fontawesome/6.5.1/webfonts/fa-brands-400.woff2', 1) }}" />
+<link rel="preload" as="font" type="font/woff2" crossorigin href="{{ pathto('_static/vendor/fontawesome/6.5.1/webfonts/fa-regular-400.woff2', 1) }}" />
+{% endmacro %}
+
+{% macro head_pre_assets() %}
+  <!-- Loaded before other Sphinx assets -->
+  <link href="{{ pathto('_static/styles/theme.css', 1) }}?digest=8d27b9dea8ad943066ae" rel="stylesheet" />
+<link href="{{ pathto('_static/styles/bootstrap.css', 1) }}?digest=8d27b9dea8ad943066ae" rel="stylesheet" />
+<link href="{{ pathto('_static/styles/pydata-sphinx-theme.css', 1) }}?digest=8d27b9dea8ad943066ae" rel="stylesheet" />
+{% endmacro %}
+
+{% macro head_js_preload() %}
+  <!-- Pre-loaded scripts that we'll load fully later -->
+  <link rel="preload" as="script" href="{{ pathto('_static/scripts/bootstrap.js', 1) }}?digest=8d27b9dea8ad943066ae" />
+<link rel="preload" as="script" href="{{ pathto('_static/scripts/pydata-sphinx-theme.js', 1) }}?digest=8d27b9dea8ad943066ae" />
+  <script src="{{ pathto('_static/vendor/fontawesome/6.5.1/js/all.min.js', 1) }}?digest=8d27b9dea8ad943066ae"></script>
+{% endmacro %}
+
+{% macro body_post() %}
+  <!-- Scripts loaded after <body> so the DOM is not blocked -->
+  <script src="{{ pathto('_static/scripts/bootstrap.js', 1) }}?digest=8d27b9dea8ad943066ae"></script>
+<script src="{{ pathto('_static/scripts/pydata-sphinx-theme.js', 1) }}?digest=8d27b9dea8ad943066ae"></script>
+{% endmacro %}
\ No newline at end of file
diff --git a/chapter0/bezier-curves.html b/chapter0/bezier-curves.html
new file mode 100644
index 0000000..1a06809
--- /dev/null
+++ b/chapter0/bezier-curves.html
@@ -0,0 +1,1540 @@
+
+<!DOCTYPE html>
+
+
+<html lang="en" data-content_root="../" >
+
+  <head>
+    <meta charset="utf-8" />
+    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
+
+    <title>Bézier curves &#8212; IGA-Python</title>
+  
+  
+  
+  <script data-cfasync="false">
+    document.documentElement.dataset.mode = localStorage.getItem("mode") || "";
+    document.documentElement.dataset.theme = localStorage.getItem("theme") || "light";
+  </script>
+  
+  <!-- Loaded before other Sphinx assets -->
+  <link href="../_static/styles/theme.css?digest=8d27b9dea8ad943066ae" rel="stylesheet" />
+<link href="../_static/styles/bootstrap.css?digest=8d27b9dea8ad943066ae" rel="stylesheet" />
+<link href="../_static/styles/pydata-sphinx-theme.css?digest=8d27b9dea8ad943066ae" rel="stylesheet" />
+
+  
+  <link href="../_static/vendor/fontawesome/6.5.1/css/all.min.css?digest=8d27b9dea8ad943066ae" rel="stylesheet" />
+  <link rel="preload" as="font" type="font/woff2" crossorigin href="../_static/vendor/fontawesome/6.5.1/webfonts/fa-solid-900.woff2" />
+<link rel="preload" as="font" type="font/woff2" crossorigin href="../_static/vendor/fontawesome/6.5.1/webfonts/fa-brands-400.woff2" />
+<link rel="preload" as="font" type="font/woff2" crossorigin href="../_static/vendor/fontawesome/6.5.1/webfonts/fa-regular-400.woff2" />
+
+    <link rel="stylesheet" type="text/css" href="../_static/pygments.css?v=fa44fd50" />
+    <link rel="stylesheet" type="text/css" href="../_static/styles/sphinx-book-theme.css?v=384b581d" />
+    <link rel="stylesheet" type="text/css" href="../_static/togglebutton.css?v=13237357" />
+    <link rel="stylesheet" type="text/css" href="../_static/copybutton.css?v=76b2166b" />
+    <link rel="stylesheet" type="text/css" href="../_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css" />
+    <link rel="stylesheet" type="text/css" href="../_static/sphinx-thebe.css?v=4fa983c6" />
+    <link rel="stylesheet" type="text/css" href="../_static/proof.css" />
+    <link rel="stylesheet" type="text/css" href="../_static/design-style.1e8bd061cd6da7fc9cf755528e8ffc24.min.css?v=0a3b3ea7" />
+  
+  <!-- Pre-loaded scripts that we'll load fully later -->
+  <link rel="preload" as="script" href="../_static/scripts/bootstrap.js?digest=8d27b9dea8ad943066ae" />
+<link rel="preload" as="script" href="../_static/scripts/pydata-sphinx-theme.js?digest=8d27b9dea8ad943066ae" />
+  <script src="../_static/vendor/fontawesome/6.5.1/js/all.min.js?digest=8d27b9dea8ad943066ae"></script>
+
+    <script src="../_static/documentation_options.js?v=9eb32ce0"></script>
+    <script src="../_static/doctools.js?v=888ff710"></script>
+    <script src="../_static/sphinx_highlight.js?v=dc90522c"></script>
+    <script src="../_static/clipboard.min.js?v=a7894cd8"></script>
+    <script src="../_static/copybutton.js?v=f281be69"></script>
+    <script src="../_static/scripts/sphinx-book-theme.js?v=efea14e4"></script>
+    <script>let toggleHintShow = 'Click to show';</script>
+    <script>let toggleHintHide = 'Click to hide';</script>
+    <script>let toggleOpenOnPrint = 'true';</script>
+    <script src="../_static/togglebutton.js?v=4a39c7ea"></script>
+    <script>var togglebuttonSelector = '.toggle, .admonition.dropdown';</script>
+    <script src="../_static/design-tabs.js?v=36754332"></script>
+    <script>const THEBE_JS_URL = "https://unpkg.com/thebe@0.8.2/lib/index.js"; const thebe_selector = ".thebe,.cell"; const thebe_selector_input = "pre"; const thebe_selector_output = ".output, .cell_output"</script>
+    <script async="async" src="../_static/sphinx-thebe.js?v=c100c467"></script>
+    <script>var togglebuttonSelector = '.toggle, .admonition.dropdown';</script>
+    <script>const THEBE_JS_URL = "https://unpkg.com/thebe@0.8.2/lib/index.js"; const thebe_selector = ".thebe,.cell"; const thebe_selector_input = "pre"; const thebe_selector_output = ".output, .cell_output"</script>
+    <script>window.MathJax = {"options": {"processHtmlClass": "tex2jax_process|mathjax_process|math|output_area"}}</script>
+    <script defer="defer" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
+    <script>DOCUMENTATION_OPTIONS.pagename = 'chapter0/bezier-curves';</script>
+    <link rel="index" title="Index" href="../genindex.html" />
+    <link rel="search" title="Search" href="../search.html" />
+    <link rel="next" title="B-Splines" href="bsplines.html" />
+    <link rel="prev" title="Bernstein polynomials" href="bezier.html" />
+  <meta name="viewport" content="width=device-width, initial-scale=1"/>
+  <meta name="docsearch:language" content="en"/>
+  </head>
+  
+  
+  <body data-bs-spy="scroll" data-bs-target=".bd-toc-nav" data-offset="180" data-bs-root-margin="0px 0px -60%" data-default-mode="">
+
+  
+  
+  <a id="pst-skip-link" class="skip-link" href="#main-content">Skip to main content</a>
+  
+  <div id="pst-scroll-pixel-helper"></div>
+  
+  <button type="button" class="btn rounded-pill" id="pst-back-to-top">
+    <i class="fa-solid fa-arrow-up"></i>
+    Back to top
+  </button>
+
+  
+  <input type="checkbox"
+          class="sidebar-toggle"
+          name="__primary"
+          id="__primary"/>
+  <label class="overlay overlay-primary" for="__primary"></label>
+  
+  <input type="checkbox"
+          class="sidebar-toggle"
+          name="__secondary"
+          id="__secondary"/>
+  <label class="overlay overlay-secondary" for="__secondary"></label>
+  
+  <div class="search-button__wrapper">
+    <div class="search-button__overlay"></div>
+    <div class="search-button__search-container">
+<form class="bd-search d-flex align-items-center"
+      action="../search.html"
+      method="get">
+  <i class="fa-solid fa-magnifying-glass"></i>
+  <input type="search"
+         class="form-control"
+         name="q"
+         id="search-input"
+         placeholder="Search this book..."
+         aria-label="Search this book..."
+         autocomplete="off"
+         autocorrect="off"
+         autocapitalize="off"
+         spellcheck="false"/>
+  <span class="search-button__kbd-shortcut"><kbd class="kbd-shortcut__modifier">Ctrl</kbd>+<kbd>K</kbd></span>
+</form></div>
+  </div>
+  
+    <header class="bd-header navbar navbar-expand-lg bd-navbar">
+    </header>
+  
+
+  <div class="bd-container">
+    <div class="bd-container__inner bd-page-width">
+      
+      
+      
+      <div class="bd-sidebar-primary bd-sidebar">
+        
+
+  
+  <div class="sidebar-header-items sidebar-primary__section">
+    
+    
+    
+    
+  </div>
+  
+    <div class="sidebar-primary-items__start sidebar-primary__section">
+        <div class="sidebar-primary-item">
+
+  
+
+<a class="navbar-brand logo" href="../intro.html">
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+  
+  
+  
+  
+    
+    
+      
+    
+    
+    <img src="../_static/logo.png" class="logo__image only-light" alt="IGA-Python - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="IGA-Python - Home"/>`);</script>
+  
+  
+</a></div>
+        <div class="sidebar-primary-item">
+
+ <script>
+ document.write(`
+   <button class="btn navbar-btn search-button-field search-button__button" title="Search" aria-label="Search" data-bs-placement="bottom" data-bs-toggle="tooltip">
+    <i class="fa-solid fa-magnifying-glass"></i>
+    <span class="search-button__default-text">Search</span>
+    <span class="search-button__kbd-shortcut"><kbd class="kbd-shortcut__modifier">Ctrl</kbd>+<kbd class="kbd-shortcut__modifier">K</kbd></span>
+   </button>
+ `);
+ </script></div>
+        <div class="sidebar-primary-item"><nav class="bd-links bd-docs-nav" aria-label="Main">
+    <div class="bd-toc-item navbar-nav active">
+        
+        <ul class="nav bd-sidenav bd-sidenav__home-link">
+            <li class="toctree-l1">
+                <a class="reference internal" href="../intro.html">
+                    IGA-Python
+                </a>
+            </li>
+        </ul>
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Introduction</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="howto.html">What to expect from IGA-Python</a></li>
+<li class="toctree-l1 current active has-children"><a class="reference internal" href="cad.html">Computer Aided Design</a><input checked="" class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-1"><i class="fa-solid fa-chevron-down"></i></label><ul class="current">
+<li class="toctree-l2 current active has-children"><a class="reference internal" href="bezier.html">Bernstein polynomials</a><input checked="" class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul class="current">
+<li class="toctree-l3 current active"><a class="current reference internal" href="#">Bézier curves</a></li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="bsplines.html">B-Splines</a><input class="toctree-checkbox" id="toctree-checkbox-3" name="toctree-checkbox-3" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-3"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="bsplines-curves.html">B-Splines curves</a></li>
+<li class="toctree-l3"><a class="reference internal" href="bsplines-surfaces.html">B-Splines surfaces</a></li>
+<li class="toctree-l3"><a class="reference internal" href="bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="iga.html">Isogeometric Analysis</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="data-structure.html">Data Structure</a></li>
+
+<li class="toctree-l2"><a class="reference internal" href="poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/install.html">Installation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter1/sympde.html">SymPDE concepts and their mathematical meaning</a><input class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-5"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/topology.html">Topology concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/space.html">Function Space concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/sympde.html">SymPDE API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-13" name="toctree-checkbox-13" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-13"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.algebra.html">sympde.core.algebra</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.basic.html">sympde.core.basic</a></li>
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+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.utils.html">sympde.core.utils</a></li>
+</ul>
+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.analytical_mapping.html">sympde.topology.analytical_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.basic.html">sympde.topology.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.callable_mapping.html">sympde.topology.callable_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.datatype.html">sympde.topology.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.derivatives.html">sympde.topology.derivatives</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.domain.html">sympde.topology.domain</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.html">sympde.topology.tests</a><input class="toctree-checkbox" id="toctree-checkbox-17" name="toctree-checkbox-17" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-17"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_callable_mapping.html">sympde.topology.tests.test_callable_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_derivatives.html">sympde.topology.tests.test_derivatives</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_field_2d.html">sympde.topology.tests.test_field_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.html">sympde.expr</a><input class="toctree-checkbox" id="toctree-checkbox-18" name="toctree-checkbox-18" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-18"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.basic.html">sympde.expr.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.equation.html">sympde.expr.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_equation_2d.html">sympde.expr.tests.test_equation_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.html">sympde.printing</a><input class="toctree-checkbox" id="toctree-checkbox-20" name="toctree-checkbox-20" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-20"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.latex.html">sympde.printing.latex</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.tests.test_latex.html">sympde.printing.tests.test_latex</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.html">sympde.exterior</a><input class="toctree-checkbox" id="toctree-checkbox-22" name="toctree-checkbox-22" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-22"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.calculus.html">sympde.exterior.calculus</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.form.html">sympde.exterior.form</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_compiler.html">sympde.exterior.tests.test_compiler</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
+</ul>
+</li>
+</ul>
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+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.utilities.utils.html">sympde.utilities.utils</a></li>
+</ul>
+</li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Bézier curves</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-a-point-on-bezier-curve">Evaluation of a point on Bézier curve</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#examples">Examples</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivatives-of-a-bezier-curve">Derivatives of a Bezier curve</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#integration-of-bezier-curves">Integration of Bezier curves</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#properties-of-bezier-curves">Properties of Bezier curves</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#decasteljau-algorithm">DeCasteljau Algorithm</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#conversion-from-monomial-form">Conversion from monomial form</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#conversion-to-monomial-form">Conversion to monomial form</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rational-bezier-curves">Rational Bézier curves</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#example">Example</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#composite-bezier-curve">Composite Bézier curve</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="bezier-curves">
+<h1>Bézier curves<a class="headerlink" href="#bezier-curves" title="Link to this heading">#</a></h1>
+<p>We recall the definition of a Bézier curve:</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \mathcal{C}(t) = \sum_{k=0}^n \mathbf{P}_k B_k^n(t)
+\end{align}
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(\left( \mathbf{P} \right)_{0 \le k \le n}\)</span> are the control points or Bézier points.</p>
+<section id="evaluation-of-a-point-on-bezier-curve">
+<h2>Evaluation of a point on Bézier curve<a class="headerlink" href="#evaluation-of-a-point-on-bezier-curve" title="Link to this heading">#</a></h2>
+<p>The following function evaluates a Bézier curve, given the control points <span class="math notranslate nohighlight">\(P\)</span> at <span class="math notranslate nohighlight">\(x\)</span>. The degree of the Bernstein polynomials is computed from the length of <span class="math notranslate nohighlight">\(P\)</span>.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">point_on_bezier_curve</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">x</span><span class="p">):</span>
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">P</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">b</span> <span class="o">=</span> <span class="n">all_bernstein</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+    <span class="n">c</span> <span class="o">=</span> <span class="mf">0.</span>
+    <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">c</span> <span class="o">+=</span> <span class="n">b</span><span class="p">[</span><span class="n">k</span><span class="p">]</span><span class="o">*</span><span class="n">P</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
+    <span class="k">return</span> <span class="n">c</span>
+</pre></div>
+</div>
+</section>
+<section id="examples">
+<h2>Examples<a class="headerlink" href="#examples" title="Link to this heading">#</a></h2>
+<div class="proof example admonition" id="bezier-curve-ex-1">
+<p class="admonition-title"><span class="caption-number">Example 1 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">nt</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nt</span><span class="p">)</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">t</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ts</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bezier_curve</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_16_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bezier-curve-ex-2">
+<p class="admonition-title"><span class="caption-number">Example 2 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">nt</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nt</span><span class="p">)</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">t</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ts</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bezier_curve</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_17_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bezier-curve-ex-3">
+<p class="admonition-title"><span class="caption-number">Example 3 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">nt</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nt</span><span class="p">)</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">]</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">t</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ts</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bezier_curve</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+<span class="n">i</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_18_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bezier-curve-ex-4">
+<p class="admonition-title"><span class="caption-number">Example 4 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">nt</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nt</span><span class="p">)</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">]</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">t</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ts</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bezier_curve</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+<span class="n">i</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.7</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_19_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bezier-curve-ex-5">
+<p class="admonition-title"><span class="caption-number">Example 5 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">nt</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nt</span><span class="p">)</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">]</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">t</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ts</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bezier_curve</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+<span class="n">i</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">-</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_20_0.png" /></p>
+</section>
+</div></section>
+<section id="derivatives-of-a-bezier-curve">
+<h2>Derivatives of a Bezier curve<a class="headerlink" href="#derivatives-of-a-bezier-curve" title="Link to this heading">#</a></h2>
+<p>Using the formulae \ref{eq:bernstein-der}, we have</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \mathcal{C}^\prime(t) = \sum_{k=0}^n \mathbf{P}_k {B_k^n}^\prime(t) 
+                        = \sum_{k=0}^n n \mathbf{P}_k \left(B_{k-1}^{n-1}(x) - B_k^{n-1}(x) \right)  
+\end{align}
+\]</div>
+<p>by reordering the indices, we get \ref{eq:bezier-curve-der}</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \mathcal{C}^\prime(t) &amp;= \sum_{k=0}^{n-1} n \left( \mathbf{P}_{k+1} - \mathbf{P}_k \right) B_k^{n-1}(t) 
+  \label{eq:bezier-curve-der}
+\end{align}
+\]</div>
+<p>Therefor, we get the direct acces to the first order derivatives at the extremeties of a Bézier curve using the formulae \ref{eq:bezier-curve-der-1-ext}</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+  \begin{cases}
+    \mathcal{C}^\prime(0) = n \left( \mathbf{P}_{1} - \mathbf{P}_0 \right)  
+    \\ 
+    \mathcal{C}^\prime(1) = n \left( \mathbf{P}_{n} - \mathbf{P}_{n-1} \right)  
+  \end{cases}
+  \label{eq:bezier-curve-der-1-ext}
+\end{align}
+\end{split}\]</div>
+<p>Second derivatives can also be computed directly from the control points using the formulae  \ref{eq:bezier-curve-der-2-ext}</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+  \begin{cases}
+    \mathcal{C}^{\prime\prime}(0) = n(n-1) \left( \mathbf{P}_{0} - 2 \mathbf{P}_1 + \mathbf{P}_{2} \right)  
+    \\ 
+    \mathcal{C}^{\prime\prime}(1) = n(n-1) \left( \mathbf{P}_{n} - 2 \mathbf{P}_{n-1} + \mathbf{P}_{n-2} \right)  
+  \end{cases}
+  \label{eq:bezier-curve-der-2-ext}
+\end{align}
+\end{split}\]</div>
+<div class="proof definition admonition" id="definition-5">
+<p class="admonition-title"><span class="caption-number">Definition 2 </span> (<span class="math notranslate nohighlight">\(r^{th}\)</span> forward difference)</p>
+<section class="definition-content" id="proof-content">
+<p>Let us consider a set of (control) points <span class="math notranslate nohighlight">\(\mathbf{P}\)</span>.
+The <span class="math notranslate nohighlight">\(r^{th}\)</span> forward difference of <span class="math notranslate nohighlight">\(\mathbf{P}\)</span> is defined as<br />
+$<span class="math notranslate nohighlight">\(
+  \begin{align}
+    \triangle^r \mathbf{P}_i := \triangle^{r-1} \mathbf{P}_{i+1} - \triangle^{r-1} \mathbf{P}_i  
+  \end{align}
+\)</span><span class="math notranslate nohighlight">\(
+  with 
+\)</span><span class="math notranslate nohighlight">\(
+  \begin{align}
+    \triangle \mathbf{P}_i = \triangle^1 \mathbf{P}_i := \mathbf{P}_{i+1} - \mathbf{P}_i  
+  \end{align}
+\)</span>$</p>
+</section>
+</div><div class="proof proposition admonition" id="proposition-6">
+<p class="admonition-title"><span class="caption-number">Proposition 2 </span> (High order dirivatives)</p>
+<section class="proposition-content" id="proof-content">
+<div class="math notranslate nohighlight">
+\[
+  \begin{align}
+    \mathcal{C}^{(r)}(t) = \frac{n!}{\left( n-r \right)!} \sum\limits_{i=0}^{n-r} \triangle^r \mathbf{P}_i B_i^{n-r}(t) 
+    \label{eq:bezier-high-deriv}
+  \end{align}
+\]</div>
+</section>
+</div><p>Eqs (\ref{eq:bezier-curve-der-1-ext}) and (\ref{eq:bezier-curve-der-2-ext}) can be generelaized for high order derivatives. We have in fact the following result:</p>
+<div class="proof remark admonition" id="remark-7">
+<p class="admonition-title"><span class="caption-number">Remark 1 </span></p>
+<section class="remark-content" id="proof-content">
+<p>The derivatives of a Bézier curve at its extremeties up to order <span class="math notranslate nohighlight">\(r\)</span> depend only on the first (or last) <span class="math notranslate nohighlight">\(r+1\)</span> control points, and vice versa.</p>
+</section>
+</div><div class="proof proposition admonition" id="proposition-8">
+<p class="admonition-title"><span class="caption-number">Proposition 3 </span></p>
+<section class="proposition-content" id="proof-content">
+<p>\label{prop:bezier-forward-difference}
+$<span class="math notranslate nohighlight">\(\triangle^r \mathbf{P}_0 = \sum\limits_{i=0}^{r} \left( -1 \right)^{r-i} \binom{r}{i} \mathbf{P}_i \)</span>$</p>
+</section>
+</div></section>
+<section id="integration-of-bezier-curves">
+<h2>Integration of Bezier curves<a class="headerlink" href="#integration-of-bezier-curves" title="Link to this heading">#</a></h2>
+<div class="proof proposition admonition" id="proposition-9">
+<p class="admonition-title"><span class="caption-number">Proposition 4 </span></p>
+<section class="proposition-content" id="proof-content">
+<p>A primitive of A Bézier curve <span class="math notranslate nohighlight">\(\mathcal{C}(t) = \sum_{k=0}^n \mathbf{P}_k B_k^n(t)\)</span> has the Bézier representation</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \left( \int \mathcal{C} \right) (t) = \sum_{k=0}^{n+1} \mathbf{Q}_k B_k^{n+1}(t)
+  \label{eq:bezier-primitive}
+\end{align}
+\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \mathbf{Q}_k := \mathbf{Q}_{k-1} + \frac{1}{n+1} \mathbf{P}_{k-1} 
+                = \mathbf{Q}_{0}  + \frac{1}{n+1} \left( \sum\limits_{i=0}^{k-1} \mathbf{P}_{i}  \right)
+\end{align}
+\]</div>
+</section>
+</div><p>Here, <span class="math notranslate nohighlight">\(\mathbf{Q}_{0}\)</span> denotes an arbitrary integration constant.</p>
+<div class="proof remark admonition" id="remark-10">
+<p class="admonition-title"><span class="caption-number">Remark 2 </span></p>
+<section class="remark-content" id="proof-content">
+<p>Using Eq. (\ref{eq:bezier-primitive}), we have <span class="math notranslate nohighlight">\(\int_0^1 \mathcal{C}(t)~dt = \frac{1}{n+1}\left( \sum\limits_{i=0}^n \mathbf{P}_i \right)\)</span>.</p>
+</section>
+</div></section>
+<section id="properties-of-bezier-curves">
+<h2>Properties of Bezier curves<a class="headerlink" href="#properties-of-bezier-curves" title="Link to this heading">#</a></h2>
+<p>Because of the <em>symmetry</em> of the Bernstein polynomials, we have</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\sum_{k=0}^n \mathbf{P}_k B_k^n = \sum_{k=0}^n \mathbf{P}_k B_{n-k}^n\)</span></p></li>
+</ul>
+<p>Thanks to the interpolation property, for <span class="math notranslate nohighlight">\(B_0^n\)</span> and <span class="math notranslate nohighlight">\(B_n^n\)</span>, of the Bernstein polynomials at the endpoints, we get</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\mathcal{C}(0) = \mathbf{P}_0\)</span>  and <span class="math notranslate nohighlight">\(\mathcal{C}(1) = \mathbf{P}_n\)</span></p></li>
+</ul>
+<p>Using the partition unity property of the Bernstein polynomials, we get</p>
+<ul class="simple">
+<li><p>any point <span class="math notranslate nohighlight">\(\mathcal{C}(t)\)</span> is an <em>affine combination</em> of the control points</p></li>
+</ul>
+<p>Therefor,</p>
+<ul class="simple">
+<li><p>Bézier curves are affinely invariant; <em>i.e.</em> the image curve <span class="math notranslate nohighlight">\(\Phi(\sum_{k=0}^n \mathbf{P}_k B_k^n)\)</span> of a Bézier curve, by an affine mapping <span class="math notranslate nohighlight">\(\Phi\)</span>, is the Bézier curve having <span class="math notranslate nohighlight">\(\left( \Phi(\mathbf{P}_i) \right)_{0 \le i \le n}\)</span> as control points.</p></li>
+</ul>
+<p>due to the partition unity of the Bernstein polynomials and their non-negativity,</p>
+<ul class="simple">
+<li><p>any point <span class="math notranslate nohighlight">\(\mathcal{C}(t)\)</span> is a <em>convex combination</em> of the control points</p></li>
+</ul>
+<p>finally, we get the <em>convex hull property</em>,</p>
+<ul class="simple">
+<li><p>A Bézier curve lies in the convex hull of its control points</p></li>
+</ul>
+</section>
+<section id="decasteljau-algorithm">
+<h2>DeCasteljau Algorithm<a class="headerlink" href="#decasteljau-algorithm" title="Link to this heading">#</a></h2>
+<p>In this section, we introduce the deCasteljau algorithm Algo. (\ref{algo:decasteljau}). The basic idea comes from the following remark;
+using the reccurence formulae Eq. (\ref{eq:bernstein-rec}), we have</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+  \mathcal{C}(t) &amp;= \sum_{k=0}^n \mathbf{P}_k B_k^n(t) = \sum_{k=0}^n \mathbf{P}_k \left( (1-t) B_k^{n-1}(t) + t B_{k-1}^{n-1}(t) \right) \\ 
+                 &amp;= \sum_{k=0}^{n-1} \mathbf{P}_k^1  B_k^{n-1}(t) \quad\quad\quad\quad\quad [\mathbf{P}_k^{1} := (1-t)\mathbf{P}_k + t \mathbf{P}_{k+1}] \\ 
+                 &amp;= \sum_{k=0}^{n-2} \mathbf{P}_k^2  B_k^{n-2}(t) \quad\quad\quad\quad\quad [\mathbf{P}_k^{2} := (1-t)\mathbf{P}_k^1 + t \mathbf{P}_{k+1}^1] \\ 
+                 &amp;= \ldots \\
+                 &amp;= \sum_{k=0}^{0} \mathbf{P}_k^n  B_k^{0}(t) \quad\quad\quad\quad\quad [\mathbf{P}_k^{n} := (1-t)\mathbf{P}_k^{n-1} + t \mathbf{P}_{k+1}^{n-1}] \\ 
+                 &amp;= \mathbf{P}_0^n 
+\end{align}
+\end{split}\]</div>
+<div class="proof algorithm admonition" id="algo-decasteljau">
+<p class="admonition-title"><span class="caption-number">Algorithm 1 </span> (Evaluation of a Bézier curve, defined by its control points <span class="math notranslate nohighlight">\(\mathbf{P}\)</span> at <span class="math notranslate nohighlight">\(x\)</span>)</p>
+<section class="algorithm-content" id="proof-content">
+<p><strong>Input</strong> <span class="math notranslate nohighlight">\(\mathbf{P}, x\)</span>
+<strong>Output</strong> <span class="math notranslate nohighlight">\(value\)</span></p>
+<p><span class="math notranslate nohighlight">\(n \gets { len } \left( \mathbf{P} \right) - 1\)</span></p>
+<p><span class="math notranslate nohighlight">\(\mathbf{Q} \gets \mathbf{P}\)</span></p>
+<p>For <span class="math notranslate nohighlight">\(j \gets 0\)</span> : <span class="math notranslate nohighlight">\(n\)</span></p>
+<p>For <span class="math notranslate nohighlight">\(i \gets 0\)</span> : <span class="math notranslate nohighlight">\(n-j\)</span></p>
+<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>$Q[i] \gets (1-x) Q[i] + x Q[i+1]$
+</pre></div>
+</div>
+<p>Return <span class="math notranslate nohighlight">\(Q[0]\)</span></p>
+</section>
+</div><div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">decasteljau</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">P</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">Q</span> <span class="o">=</span> <span class="n">P</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
+    <span class="n">x1</span> <span class="o">=</span> <span class="mf">1.</span><span class="o">-</span><span class="n">x</span>
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+            <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">x1</span><span class="o">*</span><span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">x</span><span class="o">*</span><span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span>
+    <span class="n">c</span> <span class="o">=</span> <span class="n">Q</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+    <span class="k">return</span> <span class="n">c</span>
+</pre></div>
+</div>
+</section>
+<section id="conversion-from-monomial-form">
+<h2>Conversion from monomial form<a class="headerlink" href="#conversion-from-monomial-form" title="Link to this heading">#</a></h2>
+<p>Let us consider a monomial representation of a curve</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\mathcal{C}(t) = \sum_{k=0}^n \mathbf{Q}_k t^k 
+\end{align}
+\]</div>
+<p>Since</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+  t^k &amp;= \frac{1}{\binom{n}{k}} \binom{n}{k} t^k \left(1-t+t \right)^{n-k} 
+      = \frac{1}{\binom{n}{k}} \left( \sum\limits_{i=0}^{n-k} \binom{n}{k} \binom{n-k}{n-k-i} t^{i+k} \left( 1-t \right)^{n-k-i} \right) \\
+      &amp;= \frac{1}{\binom{n}{k}} \left(  \sum\limits_{i=0}^{n-k} \binom{k+i}{k} B_{k+i}^n(t) \right)
+      = \frac{1}{\binom{n}{k}} \left(  \sum\limits_{j=0}^{n} \binom{j}{k} B_{j}^n(t) \right) 
+\end{align}
+\end{split}\]</div>
+<p>we get,</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\mathcal{C}(t) = \sum_{k=0}^n \mathbf{Q}_k t^k = \sum_{k=0}^n \mathbf{Q}_k \frac{1}{\binom{n}{k}} \left(  \sum\limits_{j=0}^{n} \binom{j}{k} B_{j}^n(t) \right)  
+\end{align}
+\]</div>
+<p>hence,</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\mathcal{C}(t) = \sum_{j=0}^n \left(\sum\limits_{k=0}^{n} \frac{\binom{j}{k}}{\binom{n}{k}} \mathbf{Q}_k \right) B_{j}^n(t)  
+\end{align}
+\]</div>
+<p>which is a Bézier curve of the form</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\mathcal{C}(t) = \sum_{k=0}^n \mathbf{P}_k B_k^n(t)
+\end{align}
+\]</div>
+<p>with</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\mathbf{P}_k =  \sum\limits_{k=0}^{n} \frac{\binom{j}{k}}{\binom{n}{k}} \mathbf{Q}_k 
+\end{align}
+\]</div>
+<div class="proof remark admonition" id="remark-12">
+<p class="admonition-title"><span class="caption-number">Remark 3 </span></p>
+<section class="remark-content" id="proof-content">
+<p>If <span class="math notranslate nohighlight">\(\mathbf{Q}_k = 0\)</span> for all <span class="math notranslate nohighlight">\(k \geq 2\)</span>, then the control points are given by <span class="math notranslate nohighlight">\(\mathbf{P}_k = \mathbf{Q}_0 + k \mathbf{Q}_1\)</span>.</p>
+</section>
+</div><div class="proof proposition admonition" id="proposition-13">
+<p class="admonition-title"><span class="caption-number">Proposition 5 </span> (Linear Precision)</p>
+<section class="proposition-content" id="proof-content">
+<p>Conversly, if the <span class="math notranslate nohighlight">\(n+1\)</span> control points <span class="math notranslate nohighlight">\(\mathbf{P}\)</span> lie equidistantly on a line, then <span class="math notranslate nohighlight">\(\mathcal{C}\)</span> is a linear polynomial, which can be written as <span class="math notranslate nohighlight">\(\mathcal{C}(t) = \left(1-t\right) \mathbf{P}_0 + t \mathbf{P}_n\)</span>.</p>
+</section>
+</div></section>
+<section id="conversion-to-monomial-form">
+<h2>Conversion to monomial form<a class="headerlink" href="#conversion-to-monomial-form" title="Link to this heading">#</a></h2>
+<p>Given a Bézier curve <span class="math notranslate nohighlight">\(\mathcal{C}(t) = \sum_{k=0}^n \mathbf{P}_k B_k^n(t)\)</span>, we can derive its monomial representation using the Taylor expansion,</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+  \mathcal{C}(t) &amp;= \sum\limits_{k=0}^n \frac{1}{k!}\mathcal{C}^{(k)}(0) t^k \\
+                 &amp;= \sum\limits_{k=0}^n \binom{n}{k} \triangle^k \mathbf{P}_0 t^k  
+\end{align}
+\end{split}\]</div>
+</section>
+<section id="rational-bezier-curves">
+<h2>Rational Bézier curves<a class="headerlink" href="#rational-bezier-curves" title="Link to this heading">#</a></h2>
+<p>We first introduce the notion of Rational Bernstein polynomials defined as,  and <span class="math notranslate nohighlight">\(w_i &gt; 0, \forall i \in \left[0, n \right]\)</span></p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  R_k^n(t) := \frac{w_i B_k^n(t)}{\sum_{i=0}^n w_i B_i^n(t)}
+  \label{eq:rational-bernstein}
+\end{align}
+\]</div>
+<div class="proof proposition admonition" id="proposition-14">
+<p class="admonition-title"><span class="caption-number">Proposition 6 </span> (Properties of Rational Bernstein polynomials)</p>
+<section class="proposition-content" id="proof-content">
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(R_k^n(x) \ge 0\)</span>, for all <span class="math notranslate nohighlight">\(k \in \left[0, n \right]\)</span> and <span class="math notranslate nohighlight">\(x \in \left[ 0, 1 \right]\)</span> \hfill [positivity]</p></li>
+<li><p><span class="math notranslate nohighlight">\(\sum_{k=0}^n R_k^n(x) = 1\)</span>, for all <span class="math notranslate nohighlight">\(x \in \left[ 0, 1 \right]\)</span> \hfill [partition of unity]</p></li>
+<li><p><span class="math notranslate nohighlight">\(R_0^n(0) = R_n^n(1) = 1\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(R_k^n\)</span> has exactly one maximum on the interval <span class="math notranslate nohighlight">\(\left[ 0, 1 \right]\)</span>,</p></li>
+<li><p>Bernstein polynomials are Rational Bernstein polynomials when all weights are equal</p></li>
+</ul>
+</section>
+</div><div class="proof definition admonition" id="definition-15">
+<p class="admonition-title"><span class="caption-number">Definition 3 </span> (Rational Bézier curve)</p>
+<section class="definition-content" id="proof-content">
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \mathcal{C}(t) = \sum_{k=0}^n \mathbf{P}_k R_k^n(t) 
+  \label{eq:bezier-curve}
+\end{align}
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(R_k^n(t) := \frac{w_i B_k^n(t)}{\sum_{i=0}^n w_i B_i^n(t)}\)</span> and <span class="math notranslate nohighlight">\(w_i &gt; 0, \forall i \in \left[0, n \right]\)</span></p>
+</section>
+</div><section id="example">
+<h3>Example<a class="headerlink" href="#example" title="Link to this heading">#</a></h3>
+<div class="proof example admonition" id="bezier-curve-ex-7">
+<p class="admonition-title"><span class="caption-number">Example 6 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">nt</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nt</span><span class="p">)</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+
+<span class="c1"># weights</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">])</span>
+
+<span class="c1"># weithed control points in 3D</span>
+<span class="n">Pw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,:</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>  <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+<span class="n">Qw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">t</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ts</span><span class="p">):</span>
+    <span class="n">Qw</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bezier_curve</span><span class="p">(</span><span class="n">Pw</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_25_0.png" /></p>
+</section>
+</div></section>
+</section>
+<section id="composite-bezier-curve">
+<h2>Composite Bézier curve<a class="headerlink" href="#composite-bezier-curve" title="Link to this heading">#</a></h2>
+<div class="proof example admonition" id="bezier-curve-ex-6">
+<p class="admonition-title"><span class="caption-number">Example 7 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">n</span> <span class="o">=</span> <span class="mi">2</span>
+
+<span class="n">nt</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nt</span><span class="p">)</span>
+
+<span class="c1"># ...</span>
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">t</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ts</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bezier_curve</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-r&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--or&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+<span class="c1"># ...</span>
+
+<span class="c1"># ...</span>
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">t</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ts</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bezier_curve</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ob&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+<span class="c1"># ...</span>
+
+<span class="c1"># ...</span>
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nt</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">t</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ts</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bezier_curve</span><span class="p">(</span><span class="n">P</span><span class="p">,</span><span class="n">t</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-g&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--og&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+<span class="c1"># ...</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_22_0.png" /></p>
+</section>
+</div></section>
+</section>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-a-point-on-bezier-curve">Evaluation of a point on Bézier curve</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#examples">Examples</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivatives-of-a-bezier-curve">Derivatives of a Bezier curve</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#integration-of-bezier-curves">Integration of Bezier curves</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#properties-of-bezier-curves">Properties of Bezier curves</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#decasteljau-algorithm">DeCasteljau Algorithm</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#conversion-from-monomial-form">Conversion from monomial form</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#conversion-to-monomial-form">Conversion to monomial form</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rational-bezier-curves">Rational Bézier curves</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#example">Example</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#composite-bezier-curve">Composite Bézier curve</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="data-structure.html">Data Structure</a></li>
+
+<li class="toctree-l2"><a class="reference internal" href="poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/install.html">Installation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter1/sympde.html">SymPDE concepts and their mathematical meaning</a><input class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-5"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/topology.html">Topology concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/space.html">Function Space concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/sympde.html">SymPDE API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-13" name="toctree-checkbox-13" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-13"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.algebra.html">sympde.core.algebra</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.basic.html">sympde.core.basic</a></li>
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+</li>
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+</ul>
+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.analytical_mapping.html">sympde.topology.analytical_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.basic.html">sympde.topology.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.callable_mapping.html">sympde.topology.callable_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.datatype.html">sympde.topology.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.derivatives.html">sympde.topology.derivatives</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.domain.html">sympde.topology.domain</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_derivatives.html">sympde.topology.tests.test_derivatives</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_field_2d.html">sympde.topology.tests.test_field_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.html">sympde.expr</a><input class="toctree-checkbox" id="toctree-checkbox-18" name="toctree-checkbox-18" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-18"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.equation.html">sympde.expr.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_equation_2d.html">sympde.expr.tests.test_equation_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.latex.html">sympde.printing.latex</a></li>
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+</ul>
+</li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.calculus.html">sympde.exterior.calculus</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.form.html">sympde.exterior.form</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_compiler.html">sympde.exterior.tests.test_compiler</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
+</ul>
+</li>
+</ul>
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+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.utilities.utils.html">sympde.utilities.utils</a></li>
+</ul>
+</li>
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+    <h1>Bernstein polynomials</h1>
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+                <h2> Contents </h2>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-bernstein-polynomials">Evaluation of Bernstein polynomials</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bernstein-polynomials-properties">Bernstein polynomials properties</a></li>
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+  <section class="tex2jax_ignore mathjax_ignore" id="bernstein-polynomials">
+<h1>Bernstein polynomials<a class="headerlink" href="#bernstein-polynomials" title="Link to this heading">#</a></h1>
+<p>Without loss of generality, we restrict to the case of the unit interval, namely <span class="math notranslate nohighlight">\(a=0\)</span> and <span class="math notranslate nohighlight">\(b=1\)</span>.
+In figure (Fig. \ref{fig:bernstein-polynomials}), we plot the first sixth Bernstein polynomials.</p>
+<div class="proof definition admonition" id="definition-0">
+<p class="admonition-title"><span class="caption-number">Definition 1 </span></p>
+<section class="definition-content" id="proof-content">
+<p>Fro <span class="math notranslate nohighlight">\(n \in \mathbb{N}\)</span>, the <span class="math notranslate nohighlight">\(n\)</span>-th degree Bernstein polynomials <span class="math notranslate nohighlight">\(B_{j,n}\,:\, [0,1] \longrightarrow \mathbb{R}\)</span> are defined by</p>
+<div class="math notranslate nohighlight">
+\[
+B_{j,n}(x)=\frac{n!}{(n-j)!j!} x^j (1-x)^{n-j}, \quad x \in [0,1], \quad j=0, \ldots,n.
+\]</div>
+</section>
+</div><p>Bernstein polynomials exhibit interesting properties highlighted in the following proposition,</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># needed imports</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+</pre></div>
+</div>
+<p>We first consider the evaluation of the Bernstein polynomials.</p>
+<section id="evaluation-of-bernstein-polynomials">
+<h2>Evaluation of Bernstein polynomials<a class="headerlink" href="#evaluation-of-bernstein-polynomials" title="Link to this heading">#</a></h2>
+<p>The following function evaluates all Bernstein polynomials of degree <span class="math notranslate nohighlight">\(n\)</span> at <span class="math notranslate nohighlight">\(x\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">all_bernstein</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
+    <span class="n">b</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.</span>
+    <span class="n">x1</span> <span class="o">=</span> <span class="mf">1.</span><span class="o">-</span><span class="n">x</span>
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">saved</span> <span class="o">=</span> <span class="mf">0.</span>
+        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">j</span><span class="p">):</span>
+            <span class="n">tmp</span> <span class="o">=</span> <span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+            <span class="n">b</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">saved</span> <span class="o">+</span> <span class="n">x1</span><span class="o">*</span><span class="n">tmp</span>
+            <span class="n">saved</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">tmp</span>
+        <span class="n">b</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">saved</span>
+    <span class="k">return</span> <span class="n">b</span>
+</pre></div>
+</div>
+</section>
+<section id="bernstein-polynomials-properties">
+<h2>Bernstein polynomials properties<a class="headerlink" href="#bernstein-polynomials-properties" title="Link to this heading">#</a></h2>
+<div class="proof proposition admonition" id="proposition-1">
+<p class="admonition-title"><span class="caption-number">Proposition 1 </span></p>
+<section class="proposition-content" id="proof-content">
+<p>We have the following properties of Bernstein Polynomials,</p>
+<ul class="simple">
+<li><p>Positivity: <span class="math notranslate nohighlight">\(B_{j,n}(x) \ge 0\)</span>, for all <span class="math notranslate nohighlight">\(x \in \left[ 0, 1 \right]\)</span> and <span class="math notranslate nohighlight">\(0 \leq j \leq n\)</span>.</p></li>
+<li><p>Partition of unity: <span class="math notranslate nohighlight">\(\sum_{j=0}^n B_{j,n}(x) = 1\)</span>, for all <span class="math notranslate nohighlight">\(x \in \left[ 0, 1 \right]\)</span>.</p></li>
+<li><p><span class="math notranslate nohighlight">\(B_{0,n}(0) = B_{n,n}(1) = 1\)</span>.</p></li>
+<li><p><span class="math notranslate nohighlight">\(B_{j,n}\)</span> has exactly one maximum on the interval <span class="math notranslate nohighlight">\(\left[ 0, 1 \right]\)</span>, at <span class="math notranslate nohighlight">\(\frac{j}{n}\)</span>.</p></li>
+<li><p>Symmetry: <span class="math notranslate nohighlight">\(B_{j,n}\)</span> is symmetric with respect to <span class="math notranslate nohighlight">\(x = \frac{1}{2}\)</span> for <span class="math notranslate nohighlight">\(j=0,\ldots,n\)</span>.</p></li>
+<li><p>Bernstein polynomials can be defined recursively using the formula</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  B_{j,n}(x) = (1-x) B_{j,n-1}(x) + x B_{k-1,n-1}(x),
+\end{align}
+\]</div>
+<p>where we assume <span class="math notranslate nohighlight">\(B_{j,n}(x) = 0\)</span> if <span class="math notranslate nohighlight">\(j &lt; 0\)</span> or <span class="math notranslate nohighlight">\(j &gt; n\)</span>.</p>
+<ul class="simple">
+<li><p>Bernstein derivatives can be computed using the formulae</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  {B_{j,n}}^\prime(x) = n \left(B_{j-1,n-1}(x) - B_{j,n-1}(x) \right).
+\end{align}
+\]</div>
+<p>using the same assumption as before.</p>
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+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="howto.html">What to expect from IGA-Python</a></li>
+<li class="toctree-l1 current active has-children"><a class="reference internal" href="cad.html">Computer Aided Design</a><input checked="" class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-1"><i class="fa-solid fa-chevron-down"></i></label><ul class="current">
+<li class="toctree-l2 has-children"><a class="reference internal" href="bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="bezier-curves.html">Bézier curves</a></li>
+</ul>
+</li>
+<li class="toctree-l2 current active has-children"><a class="reference internal" href="bsplines.html">B-Splines</a><input checked="" class="toctree-checkbox" id="toctree-checkbox-3" name="toctree-checkbox-3" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-3"><i class="fa-solid fa-chevron-down"></i></label><ul class="current">
+<li class="toctree-l3 current active"><a class="current reference internal" href="#">B-Splines curves</a></li>
+<li class="toctree-l3"><a class="reference internal" href="bsplines-surfaces.html">B-Splines surfaces</a></li>
+<li class="toctree-l3"><a class="reference internal" href="bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="iga.html">Isogeometric Analysis</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="data-structure.html">Data Structure</a></li>
+
+<li class="toctree-l2"><a class="reference internal" href="poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/install.html">Installation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter1/sympde.html">SymPDE concepts and their mathematical meaning</a><input class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-5"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/topology.html">Topology concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/space.html">Function Space concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/sympde.html">SymPDE API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-13" name="toctree-checkbox-13" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-13"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.basic.html">sympde.core.basic</a></li>
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+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.utils.html">sympde.core.utils</a></li>
+</ul>
+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.analytical_mapping.html">sympde.topology.analytical_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.basic.html">sympde.topology.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.callable_mapping.html">sympde.topology.callable_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.datatype.html">sympde.topology.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.derivatives.html">sympde.topology.derivatives</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.domain.html">sympde.topology.domain</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_callable_mapping.html">sympde.topology.tests.test_callable_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_derivatives.html">sympde.topology.tests.test_derivatives</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_field_2d.html">sympde.topology.tests.test_field_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.html">sympde.expr</a><input class="toctree-checkbox" id="toctree-checkbox-18" name="toctree-checkbox-18" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-18"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.basic.html">sympde.expr.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.equation.html">sympde.expr.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_equation_2d.html">sympde.expr.tests.test_equation_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.html">sympde.printing</a><input class="toctree-checkbox" id="toctree-checkbox-20" name="toctree-checkbox-20" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-20"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.latex.html">sympde.printing.latex</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.tests.test_latex.html">sympde.printing.tests.test_latex</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.html">sympde.exterior</a><input class="toctree-checkbox" id="toctree-checkbox-22" name="toctree-checkbox-22" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-22"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.calculus.html">sympde.exterior.calculus</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>B-Splines curves</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#properties-of-b-splines-curves">Properties of B-Splines curves</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derivative-of-a-b-spline-curve">Derivative of a B-spline curve</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#examples">Examples</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rational-b-splines-nurbs-curves">Rational B-Splines (NURBS) curves</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Examples</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="b-splines-curves">
+<h1>B-Splines curves<a class="headerlink" href="#b-splines-curves" title="Link to this heading">#</a></h1>
+<p>Let <span class="math notranslate nohighlight">\((\mathbf{P}_i)_{ 0 \leqslant i \leqslant n}\in \mathbb{R}^d\)</span>  be a sequence of control points. Following the same approach as for Bézier curves, we define B-Splines curves as</p>
+<div class="proof definition admonition" id="definition-0">
+<p class="admonition-title"><span class="caption-number">Definition 5 </span> (B-Spline curve)</p>
+<section class="definition-content" id="proof-content">
+<p>The B-spline curve in <span class="math notranslate nohighlight">\(\mathbb{R}^d\)</span> associated to <span class="math notranslate nohighlight">\(T=(t_i)_{0\leqslant i \leqslant n + p + 1}\)</span> and <span class="math notranslate nohighlight">\((\mathbf{P}_i)_{ 0 \leqslant i \leqslant n}\)</span> is defined by :</p>
+<div class="math notranslate nohighlight">
+\[
+\mathcal{C}(t) = \sum_{i=0}^n N_i^p(t) \textbf{P}_i
+\]</div>
+</section>
+</div><div class="proof remark admonition" id="remark-1">
+<p class="admonition-title"><span class="caption-number">Remark 9 </span></p>
+<section class="remark-content" id="proof-content">
+<p>The use of open knot vector leads to an <strong>interpolating curve</strong> at the endpoints.</p>
+</section>
+</div><section id="properties-of-b-splines-curves">
+<h2>Properties of B-Splines curves<a class="headerlink" href="#properties-of-b-splines-curves" title="Link to this heading">#</a></h2>
+<p>We have the following properties for a B-spline curve:</p>
+<ul class="simple">
+<li><p>If <span class="math notranslate nohighlight">\(n=p\)</span>, then <span class="math notranslate nohighlight">\(\mathcal{C}\)</span> is a Bézier-curve,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathcal{C}\)</span> is a piecewise polynomial curve,</p></li>
+<li><p>The curve interpolates its extremas if the associated multiplicity of the first and the last knot are maximum (<em>i.e.</em> equal to <span class="math notranslate nohighlight">\(p+1\)</span>),</p></li>
+<li><p>Invariance with respect to affine transformations,</p></li>
+<li><p>B-Spline curves are affinely invariant; <em>i.e.</em> the image curve <span class="math notranslate nohighlight">\(\Phi(\sum_{i=0}^n \mathbf{P}_i N_i^p)\)</span> of a B-Spline curve, by an affine mapping <span class="math notranslate nohighlight">\(\Phi\)</span>, is the B-Spline curve having <span class="math notranslate nohighlight">\(\left( \Phi(\mathbf{P}_i) \right)_{0 \le i \le n}\)</span> as control points and the same knot vector,</p></li>
+<li><p>strong convex-hull property:
+if <span class="math notranslate nohighlight">\(t_j \leq t \leq t_{j+1}\)</span>, then <span class="math notranslate nohighlight">\(\mathcal{C}(t)\)</span> is inside the convex-hull associated to the control points <span class="math notranslate nohighlight">\(\mathbf{P}_{j-p},\cdots,\mathbf{P}_{j}\)</span>,</p></li>
+<li><p>local modification : moving <span class="math notranslate nohighlight">\(\mathbf{P}_{j}\)</span> affects <span class="math notranslate nohighlight">\(\mathcal{C}(t)\)</span>, only in the interval <span class="math notranslate nohighlight">\([t_j,t_{j+p+1}]\)</span>,</p></li>
+<li><p>the control polygon is a linear approximation of the curve. We will see later that the control polygon converges to the curve under knot insertion and degree elevation (with different <em>speeds</em>).</p></li>
+<li><p><em>Variation diminishing</em>: no plane intersects the curve more than the control polygon.</p></li>
+</ul>
+</section>
+<section id="derivative-of-a-b-spline-curve">
+<h2>Derivative of a B-spline curve<a class="headerlink" href="#derivative-of-a-b-spline-curve" title="Link to this heading">#</a></h2>
+<p>Using the derivative formula for <em>B-spline</em>, we can compute the derivative of a B-Spline curve:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\mathcal{C}^{\prime}(t) &amp;= \sum_{i=0}^{n} \left( {N_{i}^{p}}^{\prime}(t) \mathbf{P}_i \right) 
+\\
+                        &amp;= \sum_{i=0}^{n} \left( \frac{p}{t_{i+p}-t_{i}}N_{i}^{p-1}(t) \mathbf{P}_i - \frac{p}{t_{i+1+p}-t_{i+1}}N_{i+1}^{p-1}(t) \mathbf{P}_i \right)
+\\
+                        &amp;= \sum_{i=0}^{n} \frac{p}{t_{i+p}-t_{i}}N_{i}^{p-1}(t) \mathbf{P}_i - \sum_{i=1}^{n+1} \frac{p}{t_{i+p}-t_{i}}N_{i}^{p-1}(t) \mathbf{P}_{i-1}
+\\
+                        &amp;= \sum\limits_{i} {N_{i}^{p-1}}(t)\frac{p}{t_{i+p}-t_{i}} \left( \mathbf{P}_i - \mathbf{P}_{i-1} \right) 
+\\
+                        &amp;= \sum\limits_{i} {N_{i}^{p-1}}(t) \frac{p}{t_{i+p}-t_{i}} \nabla \mathbf{P}_i
+\end{align}
+\end{split}\]</div>
+<p>where we introduced the <strong>first backward difference</strong> operator <span class="math notranslate nohighlight">\(\nabla \mathbf{P}_i := \mathbf{P}_i - \mathbf{P}_{i-1}\)</span>.</p>
+<div class="proof example admonition" id="bsplines-curves-ex-1">
+<p class="admonition-title"><span class="caption-number">Example 25 </span></p>
+<section class="example-content" id="proof-content">
+<p>we consider a quadratic B-Spline curve with the knot vector <span class="math notranslate nohighlight">\(T=\{000~\frac{2}{5}~\frac{3}{5}~111 \}\)</span>.</p>
+<p>We have <span class="math notranslate nohighlight">\(\mathcal{C}(t) = \sum_{i=0}^{4} {N_{i}^{2}}^{\prime}(t) \mathbf{P}_i \)</span>, then</p>
+<div class="math notranslate nohighlight">
+\[
+\mathcal{C}^{\prime}(t) = \sum\limits_{i} {N_{i}^{1}}(t) \mathbf{Q}_i
+\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[
+\mathbf{Q}_0 = 5 \{\mathbf{P}_{2} - \mathbf{P}_1\} ~~~~\mathbf{Q}_1 = \frac{10}{3} \{ \mathbf{P}_{3} - \mathbf{P}_2\}
+\]</div>
+<div class="math notranslate nohighlight">
+\[
+\mathbf{Q}_2 = \frac{10}{3} \{ \mathbf{P}_{4} - \mathbf{P}_3\}~~~~\mathbf{Q}_3 = 5 \{\mathbf{P}_{5} - \mathbf{P}_4\}
+\]</div>
+<p>the <em>B-splines</em> <span class="math notranslate nohighlight">\(\{ N_{i}^{1},~0 \leq i \leq 3\}\)</span> are associated to the knot vector <span class="math notranslate nohighlight">\(T^{\ast}=\{00~\frac{2}{5}~\frac{3}{5}~11 \}\)</span>.</p>
+</section>
+</div><div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># needed imports</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># importing bsplines utilities</span>
+<span class="kn">from</span> <span class="nn">bsplines_utilities</span> <span class="kn">import</span> <span class="n">find_span</span><span class="p">,</span> <span class="n">all_bsplines</span><span class="p">,</span> <span class="n">point_on_bspline_curve</span>
+</pre></div>
+</div>
+</section>
+<section id="examples">
+<h2>Examples<a class="headerlink" href="#examples" title="Link to this heading">#</a></h2>
+<div class="proof example admonition" id="bsplines-curves-ex-2">
+<p class="admonition-title"><span class="caption-number">Example 26 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>    
+
+<span class="n">nx</span> <span class="o">=</span> <span class="mi">101</span>
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_5_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-curves-ex-3">
+<p class="admonition-title"><span class="caption-number">Example 27 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">]</span>  
+
+<span class="n">nx</span> <span class="o">=</span> <span class="mi">101</span>
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_6_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-curves-ex-4">
+<p class="admonition-title"><span class="caption-number">Example 28 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>   
+
+<span class="n">nx</span> <span class="o">=</span> <span class="mi">101</span>
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">2.2</span><span class="p">,</span> <span class="mf">1.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">1.4</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_7_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-curves-ex-5">
+<p class="admonition-title"><span class="caption-number">Example 29 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>   
+
+<span class="n">nx</span> <span class="o">=</span> <span class="mi">101</span>
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">2.2</span><span class="p">,</span> <span class="mf">1.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">2.2</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_8_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-curves-ex-6">
+<p class="admonition-title"><span class="caption-number">Example 30 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>   
+
+<span class="n">nx</span> <span class="o">=</span> <span class="mi">101</span>
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">2.2</span><span class="p">,</span> <span class="mf">1.3</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_9_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-curves-ex-7">
+<p class="admonition-title"><span class="caption-number">Example 31 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">6</span>
+<span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">7.</span><span class="p">,</span> <span class="mf">6.</span><span class="p">,</span> <span class="mf">5.</span><span class="p">,</span>  <span class="mf">4.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">.5</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>   
+
+<span class="n">nx</span> <span class="o">=</span> <span class="mi">101</span>
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+    <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mf">0.6</span><span class="p">,</span> <span class="mf">7.4</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.4</span><span class="p">,</span> <span class="mf">1.4</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_10_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-curves-ex-8">
+<p class="admonition-title"><span class="caption-number">Example 32 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span>    <span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">.2</span><span class="p">,</span> <span class="mf">.4</span><span class="p">,</span> <span class="mf">.6</span><span class="p">,</span> <span class="mf">.8</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">degree</span> <span class="o">=</span> <span class="mi">2</span>
+    <span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+    
+    <span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">7.</span><span class="p">,</span> <span class="mf">6.</span><span class="p">,</span> <span class="mf">5.</span><span class="p">,</span>  <span class="mf">4.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">.5</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>   
+    
+    <span class="n">nx</span> <span class="o">=</span> <span class="mi">101</span>
+    <span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
+    
+    <span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+        <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+    
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+        <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mf">0.6</span><span class="p">,</span> <span class="mf">7.4</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.4</span><span class="p">,</span> <span class="mf">1.4</span><span class="p">])</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_11_0.png" /></p>
+</section>
+</div></section>
+<section id="rational-b-splines-nurbs-curves">
+<h2>Rational B-Splines (NURBS) curves<a class="headerlink" href="#rational-b-splines-nurbs-curves" title="Link to this heading">#</a></h2>
+<p>We recall that NURBS are defined as</p>
+<div class="math notranslate nohighlight">
+\[
+  R_i^p(t) := \frac{w_i N_i^p(t)}{\sum_{j=0}^n w_j N_j^p(t)}
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(w_i &gt; 0, \forall i \in \left[0, n \right]\)</span> are non-negative real numbers, called the weights.</p>
+<p>NURBS curves are then defined as</p>
+<div class="math notranslate nohighlight">
+\[
+  \mathcal{C}(t) = \sum_{i=0}^n \mathbf{P}_i R_i^p(t) 
+\]</div>
+<p>Notice that a NURBS curve in <span class="math notranslate nohighlight">\(\mathbb{R}^d\)</span> can be described as a NURBS curve in <span class="math notranslate nohighlight">\(\mathbb{R}^{d+1}\)</span> using the control points:</p>
+<p><span class="math notranslate nohighlight">\(\textbf{P}^{\omega}_i = ( \omega_i \textbf{P}_i, \omega_i )\)</span></p>
+<p>This remark is used for the evaluation and also to extend most of the B-Splines fundamental geometric operations to NURBS curves.</p>
+<div class="proof remark admonition" id="remark-10">
+<p class="admonition-title"><span class="caption-number">Remark 10 </span></p>
+<section class="remark-content" id="proof-content">
+<p>Although there is a function <strong>point_on_nurbs_curve</strong> to evaluate NURBS curves, we make use of an explicit call to the spline version in this notebook.</p>
+</section>
+</div><section id="id1">
+<h3>Examples<a class="headerlink" href="#id1" title="Link to this heading">#</a></h3>
+<div class="proof example admonition" id="nurbs-curves-ex-1">
+<p class="admonition-title"><span class="caption-number">Example 33 </span> (Quarter circle)</p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+
+<span class="c1"># weights</span>
+<span class="n">s2</span> <span class="o">=</span> <span class="mf">1.</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
+<span class="n">W</span><span class="p">[:]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+
+<span class="c1"># weithed control points in 3D</span>
+<span class="n">Pw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,:</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>  <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+<span class="n">nx</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>    
+
+<span class="n">Qw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+    <span class="n">Qw</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">Pw</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>    
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_15_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="nurbs-curves-ex-2">
+<p class="admonition-title"><span class="caption-number">Example 34 </span> (Circular arc of <span class="math notranslate nohighlight">\(120°\)</span>)</p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mf">6.</span><span class="p">)</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span> <span class="n">a</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">.5</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mf">.5</span><span class="p">]</span>
+
+<span class="c1"># weights</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
+<span class="n">W</span><span class="p">[:]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+
+<span class="c1"># weithed control points in 3D</span>
+<span class="n">Pw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,:</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>  <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+<span class="n">nx</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>    
+
+<span class="n">Qw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+    <span class="n">Qw</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">Pw</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="mf">.0</span><span class="p">,</span> <span class="mf">2.2</span><span class="p">])</span>    
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_17_01.png" /></p>
+</section>
+</div><div class="proof example admonition" id="nurbs-curves-ex-3">
+<p class="admonition-title"><span class="caption-number">Example 35 </span> (half circle)</p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span>  <span class="mf">2.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">]</span>
+
+<span class="c1"># weights</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
+<span class="n">W</span><span class="p">[:]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+
+<span class="c1"># weithed control points in 3D</span>
+<span class="n">Pw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,:</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>  <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+<span class="n">nx</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>    
+
+<span class="n">Qw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+    <span class="n">Qw</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">Pw</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">.2</span><span class="p">,</span> <span class="mf">2.2</span><span class="p">])</span>    
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_19_01.png" /></p>
+</section>
+</div><div class="proof example admonition" id="nurbs-curves-ex-4">
+<p class="admonition-title"><span class="caption-number">Example 36 </span> (Circle as 4 arcs)</p>
+<section class="example-content" id="proof-content">
+<p>We already saw how to construct a circular arc (with angle = <span class="math notranslate nohighlight">\(\frac{\pi}{2}\)</span>, using Bézier curves.</p>
+<p>Constructing a circular arc can be done in different ways. In this example, we consider a B-Spline curve as a composite of 4 Bézier curves.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">.25</span><span class="p">,</span> <span class="mf">.25</span><span class="p">,</span> <span class="mf">.5</span><span class="p">,</span> <span class="mf">.5</span><span class="p">,</span> <span class="mf">.75</span><span class="p">,</span> <span class="mf">.75</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">]</span>
+
+<span class="c1"># weights</span>
+<span class="n">s2</span> <span class="o">=</span> <span class="mf">1.</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
+<span class="n">W</span><span class="p">[:]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">s2</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+
+<span class="c1"># weithed control points in 3D</span>
+<span class="n">Pw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,:</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>  <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+<span class="n">nx</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>    
+
+<span class="n">Qw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+    <span class="n">Qw</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">Pw</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+    
+<span class="c1"># plot P_0 = P_8</span>
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_0 = \mathbf</span><span class="si">{P}</span><span class="s1">_8$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">])</span>    
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_21_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="nurbs-curves-ex-5">
+<p class="admonition-title"><span class="caption-number">Example 37 </span> (Circle as 3 arcs)</p>
+<section class="example-content" id="proof-content">
+<p>We already saw how to construct a circular arc (with angle = <span class="math notranslate nohighlight">\(120°\)</span>, using Bézier curves.</p>
+<p>In this example, we consider a B-Spline curve as a composite of 3 Bézier curves.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">2.</span><span class="o">/</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">2.</span><span class="o">/</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mf">6.</span><span class="p">)</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span> <span class="n">a</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">a</span><span class="p">,</span>  <span class="n">a</span><span class="p">]</span>
+<span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">.5</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mf">.5</span><span class="p">,</span>  <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">.5</span><span class="p">]</span>
+
+<span class="c1"># weights</span>
+<span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
+<span class="n">W</span><span class="p">[:]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="o">/</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+
+<span class="c1"># weithed control points in 3D</span>
+<span class="n">Pw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,:</span><span class="mi">2</span><span class="p">]</span> <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">Pw</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>  <span class="o">=</span> <span class="n">W</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
+
+<span class="n">nx</span> <span class="o">=</span> <span class="mi">200</span>
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>    
+
+<span class="n">Qw</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
+<span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+    <span class="n">Qw</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">Pw</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
+
+<span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">Qw</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span><span class="o">/</span><span class="n">Qw</span><span class="p">[:,</span><span class="mi">2</span><span class="p">]</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-b&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">P</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">P</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;--ok&#39;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">0.7</span><span class="p">)</span>
+
+<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+    <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>
+    
+<span class="c1"># plot P_0 = P_6</span>
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,:]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_0 = \mathbf</span><span class="si">{P}</span><span class="s1">_8$&#39;</span><span class="p">)</span>
+
+<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="o">-</span><span class="mf">2.2</span><span class="p">,</span> <span class="mf">2.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.2</span><span class="p">,</span> <span class="mf">2.2</span><span class="p">])</span>    
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_23_0.png" /></p>
+</section>
+</div></section>
+</section>
+</section>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.fem.html">psydac.api.fem</a></li>
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+    <h1>Fundamental geometric operations for B-Splines</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#knot-insertion">Knot insertion</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#order-elevation">Order elevation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#translation">Translation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rotation">Rotation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#homothetic-transformation">Homothetic transformation</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="fundamental-geometric-operations-for-b-splines">
+<h1>Fundamental geometric operations for B-Splines<a class="headerlink" href="#fundamental-geometric-operations-for-b-splines" title="Link to this heading">#</a></h1>
+<p>Having more control on a curve, adding new control points, can be done in two different ways:</p>
+<ol class="arabic simple">
+<li><p>insert new knots</p></li>
+<li><p>elevate the polynomial degree</p></li>
+</ol>
+<p>In general, one way need to use both approaches.
+For each strategy, we shall consider one algorithm.</p>
+<p><strong>Remark</strong>:
+In this notebook we only use B-Splines curves, but similar functions are also available for NURBS.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># needed imports</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+</pre></div>
+</div>
+<section id="knot-insertion">
+<h2>Knot insertion<a class="headerlink" href="#knot-insertion" title="Link to this heading">#</a></h2>
+<p>Assuming an initial B-Spline curve defined by:</p>
+<ul class="simple">
+<li><p>its degree <span class="math notranslate nohighlight">\(p\)</span></p></li>
+<li><p>knot vector <span class="math notranslate nohighlight">\(T=(t_i)_{0\leqslant i \leqslant n + p + 1}\)</span></p></li>
+<li><p>control points <span class="math notranslate nohighlight">\((\mathbf{P}_i)_{ 0 \leqslant i \leqslant n}\)</span></p></li>
+</ul>
+<p>We are interested in the new B-Spline curve as the result of the insertion of the knot <span class="math notranslate nohighlight">\(t\)</span>, <span class="math notranslate nohighlight">\(m\)</span> times (with a span <span class="math notranslate nohighlight">\(j\)</span>, i.e. <span class="math notranslate nohighlight">\(t_j \leqslant t &lt; t_{j+1}\)</span>).</p>
+<p>After such operation, the degree is remain unchanged, while the knot vector is enriched by <span class="math notranslate nohighlight">\(t\)</span>, <span class="math notranslate nohighlight">\(m\)</span> times. The aim is then to compute the new control points <span class="math notranslate nohighlight">\((\mathbf{Q}_i)_{ 0 \leqslant i \leqslant n + m}\)</span></p>
+<p>For this purpose we use the DeBoor algorithm:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+   n &amp;:= n+m
+   \\
+   p &amp;:= p
+   \\
+   T &amp;:= \{ t_0,.., t_j, \underbrace{t, ..., t}_{m}, t_{j+1},.., t_{n+k}\}
+   \\
+   \textbf{Q}_i &amp;:= \textbf{Q}_i^{m}
+\end{align}
+\end{split}\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+   \textbf{Q}_i^0 &amp;= \textbf{P}_i
+   \\
+   \textbf{Q}_i^r &amp;= \alpha_i^r \textbf{Q}_i^{r-1} + (1-\alpha_i^r) \textbf{Q}_{i-1}^{r-1}
+\end{align}
+\end{split}\]</div>
+<p>with</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+   \alpha_i^r &amp;= 
+      \left\{\begin{array}{cc}
+         1 &amp;  i \leqslant j-p+r-1 \\
+         \frac{t-t_i}{t_{i+p-r+1}-t_i} &amp; j-p+r \leqslant i \leqslant j-m \\
+         0 &amp; j-m+1 \leqslant i 
+      \end{array}\right.   
+\end{align}
+\end{split}\]</div>
+<p>The following python function implements the previous algorithm.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># import the knot insertion function</span>
+<span class="kn">from</span> <span class="nn">bsplines_utilities</span> <span class="kn">import</span> <span class="n">insert_knot_bspline_curve</span>
+</pre></div>
+</div>
+<p>From now on, we shall also use the following function to plot curves and their control points:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">bsplines_utilities</span> <span class="kn">import</span> <span class="n">plot_curve</span>
+</pre></div>
+</div>
+<p><strong>Examples</strong></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">example_1</span><span class="p">(</span><span class="n">times</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
+    <span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+    <span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+    <span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">]</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">with_ctrl_pts</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+    
+    <span class="c1"># we don&#39;t plot the first and last points</span>
+    <span class="c1"># they will be plotted later</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>    
+
+    <span class="n">t</span> <span class="o">=</span> <span class="mf">0.5</span>
+    <span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">insert_knot_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">times</span><span class="o">=</span><span class="n">times</span><span class="p">)</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Q&#39;</span><span class="p">,</span> <span class="n">with_ctrl_pts</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+    
+    <span class="c1"># we don&#39;t plot the first and last points</span>
+    <span class="c1"># they will be plotted later</span>
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{Q}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>    
+     
+    <span class="c1"># plot endpoints</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="p">;</span> <span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">Pi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+        <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span>
+            <span class="n">Pi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+            
+        <span class="n">Qi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{Q}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+        <span class="n">text</span> <span class="o">=</span> <span class="s1">&#39;$</span><span class="si">{Pi}</span><span class="s1"> = </span><span class="si">{Qi}</span><span class="s1">$&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">Pi</span><span class="o">=</span><span class="n">Pi</span><span class="p">,</span> <span class="n">Qi</span><span class="o">=</span><span class="n">Qi</span><span class="p">)</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">text</span><span class="p">)</span>     
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">example_1</span><span class="p">(</span><span class="n">times</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_9_01.png" /></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">example_1</span><span class="p">(</span><span class="n">times</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_10_01.png" /></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">example_1</span><span class="p">(</span><span class="n">times</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_11_01.png" /></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">example_2a</span><span class="p">(</span><span class="n">nt</span> <span class="o">=</span> <span class="mi">3</span><span class="p">):</span>
+    <span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+    <span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+    <span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">]</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">with_ctrl_pts</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+    
+    <span class="c1"># we don&#39;t plot the first and last points</span>
+    <span class="c1"># they will be plotted later</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>    
+
+    <span class="n">ts</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nt</span><span class="o">+</span><span class="mi">2</span><span class="p">)[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+    <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ts</span><span class="p">:</span>
+        <span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">insert_knot_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">times</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+        
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Q&#39;</span><span class="p">,</span> <span class="n">with_ctrl_pts</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+    
+    <span class="c1"># we don&#39;t plot the first and last points</span>
+    <span class="c1"># they will be plotted later</span>
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{Q}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>    
+     
+    <span class="c1"># plot endpoints</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="p">;</span> <span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">Pi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+        <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span>
+            <span class="n">Pi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+            
+        <span class="n">Qi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{Q}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+        <span class="n">text</span> <span class="o">=</span> <span class="s1">&#39;$</span><span class="si">{Pi}</span><span class="s1"> = </span><span class="si">{Qi}</span><span class="s1">$&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">Pi</span><span class="o">=</span><span class="n">Pi</span><span class="p">,</span> <span class="n">Qi</span><span class="o">=</span><span class="n">Qi</span><span class="p">)</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">text</span><span class="p">)</span>     
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># curve after inserting [0.25 0.5  0.75]</span>
+<span class="n">example_2a</span><span class="p">(</span><span class="n">nt</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span> 
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_13_0.png" /></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># curve after inserting [0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9]</span>
+<span class="n">example_2a</span><span class="p">(</span><span class="n">nt</span><span class="o">=</span><span class="mi">9</span><span class="p">)</span> 
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_14_0.png" /></p>
+</section>
+<section id="order-elevation">
+<h2>Order elevation<a class="headerlink" href="#order-elevation" title="Link to this heading">#</a></h2>
+<p>There are also many algorithms for degree elevation of a B-Spline curve. In the sequel, we will be using the one developped by Huand, Hu and Martin (2005).</p>
+<p>The previous algorithm is implemented by the following function</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># import the degree elevation function</span>
+<span class="kn">from</span> <span class="nn">bsplines_utilities</span> <span class="kn">import</span> <span class="n">elevate_degree_bspline_curve</span>
+</pre></div>
+</div>
+<p><strong>Example 1.</strong></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">example_1</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
+    <span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+    <span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+    <span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">.5</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">.5</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">]</span>
+
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">with_ctrl_pts</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+    
+    <span class="c1"># we don&#39;t plot the first and last points</span>
+    <span class="c1"># they will be plotted later</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>    
+    
+    <span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">elevate_degree_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Q&#39;</span><span class="p">,</span> <span class="n">with_ctrl_pts</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+        
+    <span class="c1"># we don&#39;t plot the first and last points</span>
+    <span class="c1"># they will be plotted later</span>
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{Q}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>    
+
+    <span class="c1"># plot endpoints</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="p">;</span> <span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">Pi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+        <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span>
+            <span class="n">Pi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+            
+        <span class="n">Qi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{Q}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+        <span class="n">text</span> <span class="o">=</span> <span class="s1">&#39;$</span><span class="si">{Pi}</span><span class="s1"> = </span><span class="si">{Qi}</span><span class="s1">$&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">Pi</span><span class="o">=</span><span class="n">Pi</span><span class="p">,</span> <span class="n">Qi</span><span class="o">=</span><span class="n">Qi</span><span class="p">)</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">text</span><span class="p">)</span>     
+    
+<span class="n">example_1</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_18_01.png" /></p>
+<p><strong>Example 2.</strong></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">example_2</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
+    <span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+    <span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">n_initial</span> <span class="o">=</span> <span class="n">n</span>
+
+    <span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">260.</span><span class="p">,</span> <span class="mf">100.</span><span class="p">,</span> <span class="mf">260.</span><span class="p">,</span> <span class="mf">420.</span><span class="p">,</span> <span class="mf">580.</span><span class="p">,</span> <span class="mf">420.</span><span class="p">]</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">100.</span><span class="p">,</span> <span class="mf">260.</span><span class="p">,</span> <span class="mf">420.</span><span class="p">,</span> <span class="mf">420.</span><span class="p">,</span> <span class="mf">260.</span><span class="p">,</span> <span class="mf">100.</span><span class="p">]</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">with_ctrl_pts</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+    
+    <span class="c1"># we don&#39;t plot the first and last points</span>
+    <span class="c1"># they will be plotted later</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>        
+
+    <span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">elevate_degree_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">m</span><span class="o">=</span><span class="n">m</span><span class="p">)</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Q&#39;</span><span class="p">,</span> <span class="n">with_ctrl_pts</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>    
+        
+    <span class="c1"># we don&#39;t plot the first and last points</span>
+    <span class="c1"># they will be plotted later</span>
+    <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
+        <span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="s1">&#39;$\mathbf</span><span class="si">{Q}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}$&#39;</span><span class="p">)</span>    
+
+    <span class="c1"># plot endpoints</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
+        <span class="n">x</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="p">;</span> <span class="n">y</span> <span class="o">=</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
+        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
+            <span class="n">Pi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+        <span class="k">elif</span> <span class="n">i</span> <span class="o">==</span> <span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">:</span>
+            <span class="n">Pi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{P}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">n_initial</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+            
+        <span class="n">Qi</span> <span class="o">=</span> <span class="s1">&#39;\mathbf</span><span class="si">{Q}</span><span class="s1">_{&#39;</span> <span class="o">+</span> <span class="nb">str</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="o">+</span> <span class="s1">&#39;}&#39;</span>
+        <span class="n">text</span> <span class="o">=</span> <span class="s1">&#39;$</span><span class="si">{Pi}</span><span class="s1"> = </span><span class="si">{Qi}</span><span class="s1">$&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">Pi</span><span class="o">=</span><span class="n">Pi</span><span class="p">,</span> <span class="n">Qi</span><span class="o">=</span><span class="n">Qi</span><span class="p">)</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">y</span><span class="o">+</span><span class="mf">0.05</span><span class="p">,</span><span class="n">text</span><span class="p">)</span>         
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">example_2</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_21_01.png" /></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">example_2</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_22_01.png" /></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">example_2</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_23_01.png" /></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">example_2</span><span class="p">(</span><span class="n">m</span><span class="o">=</span><span class="mi">8</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_24_0.png" /></p>
+</section>
+<section id="translation">
+<h2>Translation<a class="headerlink" href="#translation" title="Link to this heading">#</a></h2>
+<p>Since B-Spline curves are invariant under affine transformations, whan translating a curve with a given displacement, the new curve has the same knot vector while the control points are obtained by applying the same translation to the initial ones.</p>
+<p>The following function implements this utility</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># import the translation of a B-Spline curve</span>
+<span class="kn">from</span> <span class="nn">bsplines_utilities</span> <span class="kn">import</span> <span class="n">translate_bspline_curve</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">example_1</span><span class="p">():</span>
+    <span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+    <span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+    <span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">]</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
+    
+    <span class="n">knots</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">translate_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">])</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Q&#39;</span><span class="p">)</span>
+    
+<span class="n">example_1</span><span class="p">()</span>    
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_27_0.png" /></p>
+</section>
+<section id="rotation">
+<h2>Rotation<a class="headerlink" href="#rotation" title="Link to this heading">#</a></h2>
+<p>Rotating a B-Spline curve with an angle <span class="math notranslate nohighlight">\(\theta\)</span> is done by applying it on the control points, as for the translation algorithm.</p>
+<p>The following function implements this utility.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># import the rotation of a B-Spline curve</span>
+<span class="kn">from</span> <span class="nn">bsplines_utilities</span> <span class="kn">import</span> <span class="n">rotate_bspline_curve</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">example_1</span><span class="p">():</span>
+    <span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+    <span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+    <span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">]</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
+    
+    <span class="n">knots</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">rotate_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mf">3.</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mf">1.</span><span class="p">,</span><span class="mf">0.</span><span class="p">])</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Q&#39;</span><span class="p">)</span>
+    
+<span class="n">example_1</span><span class="p">()</span> 
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_30_01.png" /></p>
+</section>
+<section id="homothetic-transformation">
+<h2>Homothetic transformation<a class="headerlink" href="#homothetic-transformation" title="Link to this heading">#</a></h2>
+<p>The following function implements a hmothetic transformation for B-Spline curves.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># import the homothetic transformation of a B-Spline curve</span>
+<span class="kn">from</span> <span class="nn">bsplines_utilities</span> <span class="kn">import</span> <span class="n">homothetic_bspline_curve</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">example_1</span><span class="p">():</span>
+    <span class="n">knots</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+    <span class="n">n</span>      <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+    <span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.</span><span class="p">]</span>
+    <span class="n">P</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span>  <span class="mf">0.</span><span class="p">]</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">)</span>
+    
+    <span class="n">knots</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">homothetic_bspline_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="p">[</span><span class="mf">0.</span><span class="p">,</span><span class="mf">0.</span><span class="p">])</span>
+    <span class="n">plot_curve</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;r&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;Q&#39;</span><span class="p">)</span>
+    
+<span class="n">example_1</span><span class="p">()</span> 
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_33_01.png" /></p>
+</section>
+</section>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#knot-insertion">Knot insertion</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#order-elevation">Order elevation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#translation">Translation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rotation">Rotation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#homothetic-transformation">Homothetic transformation</a></li>
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+    <h1>B-Splines surfaces</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+        </div>
+    </div>
+</div>
+
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+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="b-splines-surfaces">
+<h1>B-Splines surfaces<a class="headerlink" href="#b-splines-surfaces" title="Link to this heading">#</a></h1>
+<p>The B-spline surface in <span class="math notranslate nohighlight">\(\mathbb{R}^d\)</span> associated to knots <span class="math notranslate nohighlight">\((T_u, T_v)\)</span> where <span class="math notranslate nohighlight">\(T_u=(u_i)_{0\leqslant i \leqslant n_u + p_u + 1}\)</span> and <span class="math notranslate nohighlight">\(T_v=(v_i)_{0\leqslant i \leqslant n_v + p_v + 1}\)</span>, and control points <span class="math notranslate nohighlight">\((\mathbf{P}_{ij})_{ 0 \leqslant i \leqslant n_u,  0 \leqslant j \leqslant n_v}\)</span> is defined by :</p>
+<div class="math notranslate nohighlight">
+\[
+\mathcal{C}(u,v) = \sum_{i=0}^{n_u} \sum_{j=0}^{n_v} N_i^{p_u}(u) N_j^{p_v}(v) \textbf{P}_{i,j}
+\]</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># needed imports</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">Axes3D</span>
+<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">cm</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># importing bsplines utilities</span>
+<span class="kn">from</span> <span class="nn">bsplines_utilities</span> <span class="kn">import</span> <span class="n">find_span</span><span class="p">,</span> <span class="n">all_bsplines</span><span class="p">,</span> <span class="n">point_on_nurbs_surface</span><span class="p">,</span> <span class="n">point_on_bspline_surface</span><span class="p">,</span> <span class="n">insert_knot_bspline_surface</span><span class="p">,</span> <span class="n">insert_knot_nurbs_surface</span>
+</pre></div>
+</div>
+<p><strong>Example 1</strong>:</p>
+<p>In this first example, we consider a surface that approximate the function <span class="math notranslate nohighlight">\((u,v) \mapsto \sin(\pi u) \sin(\pi v)\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">example_1</span><span class="p">():</span>    
+    <span class="n">Tu</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">pu</span> <span class="o">=</span> <span class="mi">2</span>
+    <span class="n">nu</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Tu</span><span class="p">)</span> <span class="o">-</span> <span class="n">pu</span> <span class="o">-</span> <span class="mi">1</span>
+
+    <span class="n">Tv</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">pv</span> <span class="o">=</span> <span class="mi">2</span>
+    <span class="n">nv</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Tv</span><span class="p">)</span> <span class="o">-</span> <span class="n">pv</span> <span class="o">-</span> <span class="mi">1</span>
+
+    <span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nu</span><span class="p">,</span> <span class="n">nv</span><span class="p">,</span><span class="mi">1</span><span class="p">))</span>
+    
+    <span class="n">gridu</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nu</span><span class="p">)</span>
+    <span class="n">gridv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nv</span><span class="p">)</span>
+    
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">u</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">gridu</span><span class="p">):</span>
+        <span class="k">for</span> <span class="n">j</span><span class="p">,</span><span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">gridv</span><span class="p">):</span>
+            <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">u</span><span class="p">)</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">v</span><span class="p">)</span>
+    
+    <span class="n">nx</span> <span class="o">=</span> <span class="mi">101</span>
+    <span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">nx</span><span class="p">)</span>
+    
+    <span class="n">ny</span> <span class="o">=</span> <span class="mi">101</span>
+    <span class="n">ys</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">ny</span><span class="p">)</span>
+    
+    <span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nx</span><span class="p">,</span> <span class="n">ny</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
+    <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+        <span class="k">for</span> <span class="n">j</span><span class="p">,</span><span class="n">y</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">ys</span><span class="p">):</span>
+            <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_surface</span><span class="p">(</span><span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
+            
+    <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
+    <span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">gca</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">&#39;3d&#39;</span><span class="p">)</span>
+  
+    <span class="n">X</span><span class="p">,</span> <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">)</span>
+    <span class="n">Z</span> <span class="o">=</span> <span class="n">Q</span><span class="p">[:,:,</span><span class="mi">0</span><span class="p">]</span>
+
+    <span class="c1"># Plot the surface.</span>
+    <span class="n">surf</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">Z</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">cm</span><span class="o">.</span><span class="n">coolwarm</span><span class="p">,</span>
+                           <span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">antialiased</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+    
+    <span class="c1"># Add a color bar which maps values to colors.</span>
+    <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">surf</span><span class="p">,</span> <span class="n">shrink</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">aspect</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
+
+<span class="n">example_1</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_5_01.png" /></p>
+<p><strong>Example 2</strong>:</p>
+<p>This example shows the construction of a square in 2D.
+The B-Spline surface is created using the function <strong>make_square</strong></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">make_square</span><span class="p">(</span><span class="n">origin</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> <span class="n">length</span><span class="o">=</span><span class="mf">1.</span><span class="p">):</span>
+    <span class="n">Tu</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">Tv</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">pu</span> <span class="o">=</span> <span class="mi">1</span>
+    <span class="n">pv</span> <span class="o">=</span> <span class="mi">1</span>
+    <span class="n">nu</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Tu</span><span class="p">)</span> <span class="o">-</span> <span class="n">pu</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">nv</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Tv</span><span class="p">)</span> <span class="o">-</span> <span class="n">pv</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">gridu</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">Tu</span><span class="p">)</span>
+    <span class="n">gridv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">Tv</span><span class="p">)</span>
+
+    <span class="n">origin</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">origin</span><span class="p">)</span>
+
+    <span class="n">P</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([[[</span><span class="mf">0.</span><span class="p">,</span><span class="mf">0.</span><span class="p">],[</span><span class="mf">0.</span><span class="p">,</span><span class="mf">1.</span><span class="p">]],[[</span><span class="mf">1.</span><span class="p">,</span><span class="mf">0.</span><span class="p">],[</span><span class="mf">1.</span><span class="p">,</span><span class="mf">1.</span><span class="p">]]])</span>
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
+        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">):</span>
+            <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">origin</span> <span class="o">+</span> <span class="n">P</span><span class="p">[</span><span class="n">i</span><span class="p">,</span><span class="n">j</span><span class="p">,:]</span><span class="o">*</span><span class="n">length</span>
+
+    <span class="k">return</span> <span class="p">(</span><span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span><span class="p">),</span> <span class="p">(</span><span class="n">pu</span><span class="p">,</span> <span class="n">pv</span><span class="p">),</span> <span class="n">P</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_surface</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degrees</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">weights</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">Nu</span><span class="o">=</span><span class="mi">101</span><span class="p">,</span> <span class="n">Nv</span><span class="o">=</span><span class="mi">101</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">):</span>
+    <span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span> <span class="o">=</span> <span class="n">knots</span>
+    <span class="n">pu</span><span class="p">,</span> <span class="n">pv</span> <span class="o">=</span> <span class="n">degrees</span>
+
+    <span class="n">nu</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Tu</span><span class="p">)</span> <span class="o">-</span> <span class="n">pu</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">nv</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Tv</span><span class="p">)</span> <span class="o">-</span> <span class="n">pv</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">gridu</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">Tu</span><span class="p">)</span>
+    <span class="n">gridv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">Tv</span><span class="p">)</span>
+
+    <span class="n">us</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">Nu</span><span class="p">)</span>
+    <span class="n">vs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">Nv</span><span class="p">)</span>
+
+    <span class="c1"># ...</span>
+    <span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">Nv</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="k">if</span> <span class="n">weights</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">u</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">gridu</span><span class="p">):</span>
+            <span class="k">for</span> <span class="n">j</span><span class="p">,</span><span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">vs</span><span class="p">):</span>
+                <span class="n">Q</span><span class="p">[</span><span class="n">j</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_surface</span><span class="p">(</span><span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
+                
+            <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-&#39;</span><span class="o">+</span><span class="n">color</span><span class="p">)</span>
+
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">u</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">gridu</span><span class="p">):</span>
+            <span class="k">for</span> <span class="n">j</span><span class="p">,</span><span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">vs</span><span class="p">):</span>
+                <span class="n">Q</span><span class="p">[</span><span class="n">j</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_nurbs_surface</span><span class="p">(</span><span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">weights</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
+                
+            <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-&#39;</span><span class="o">+</span><span class="n">color</span><span class="p">)</span>
+    <span class="c1"># ...</span>
+
+    <span class="c1"># ...</span>
+    <span class="n">Q</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">Nu</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
+    <span class="k">if</span> <span class="n">weights</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
+        <span class="k">for</span> <span class="n">j</span><span class="p">,</span><span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">gridv</span><span class="p">):</span>
+            <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">u</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">us</span><span class="p">):</span>
+                <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_bspline_surface</span><span class="p">(</span><span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
+
+            <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-&#39;</span><span class="o">+</span><span class="n">color</span><span class="p">)</span>                
+
+    <span class="k">else</span><span class="p">:</span>
+        <span class="k">for</span> <span class="n">j</span><span class="p">,</span><span class="n">v</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">gridv</span><span class="p">):</span>
+            <span class="k">for</span> <span class="n">i</span><span class="p">,</span><span class="n">u</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">us</span><span class="p">):</span>
+                <span class="n">Q</span><span class="p">[</span><span class="n">i</span><span class="p">,:]</span> <span class="o">=</span> <span class="n">point_on_nurbs_surface</span><span class="p">(</span><span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">weights</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
+                
+            <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">Q</span><span class="p">[:,</span><span class="mi">0</span><span class="p">],</span> <span class="n">Q</span><span class="p">[:,</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;-&#39;</span><span class="o">+</span><span class="n">color</span><span class="p">)</span>                
+    <span class="c1"># ...</span>
+<span class="c1"># ...</span>
+
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">example_2</span><span class="p">():</span>    
+    <span class="n">knots</span><span class="p">,</span> <span class="n">degrees</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">make_square</span><span class="p">(</span><span class="n">origin</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">),</span> 
+                                    <span class="n">length</span><span class="o">=</span><span class="mf">1.</span><span class="p">)</span>
+    
+    <span class="n">t</span> <span class="o">=</span> <span class="mf">0.5</span>
+    <span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span><span class="p">,</span> <span class="n">pu</span><span class="p">,</span> <span class="n">pv</span><span class="p">,</span> <span class="n">P</span> <span class="o">=</span> <span class="n">insert_knot_bspline_surface</span><span class="p">(</span><span class="o">*</span><span class="n">knots</span><span class="p">,</span> <span class="o">*</span><span class="n">degrees</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> 
+                                <span class="n">times</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> 
+                                <span class="n">axis</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+    
+    <span class="n">plot_surface</span><span class="p">((</span><span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span><span class="p">),</span> <span class="p">(</span><span class="n">pu</span><span class="p">,</span> <span class="n">pv</span><span class="p">),</span> <span class="n">P</span><span class="p">,</span> 
+                 <span class="n">weights</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> 
+                 <span class="n">Nu</span><span class="o">=</span><span class="mi">101</span><span class="p">,</span> <span class="n">Nv</span><span class="o">=</span><span class="mi">101</span><span class="p">,</span> 
+                 <span class="n">color</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">)</span>
+    
+<span class="n">example_2</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_9_02.png" /></p>
+<p><strong>Example 3</strong>:</p>
+<p>The following example shows how to create and plots a circle using NURBS.</p>
+<p>We shall need the following function:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">make_circle</span><span class="p">(</span><span class="n">center</span><span class="o">=</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span><span class="mf">0.</span><span class="p">),</span> <span class="n">radius</span><span class="o">=</span><span class="mf">1.</span><span class="p">):</span>
+    <span class="n">Tu</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">Tv</span>  <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]</span>
+    <span class="n">pu</span> <span class="o">=</span> <span class="mi">2</span>
+    <span class="n">pv</span> <span class="o">=</span> <span class="mi">2</span>
+    <span class="n">nu</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Tu</span><span class="p">)</span> <span class="o">-</span> <span class="n">pu</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">nv</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">Tv</span><span class="p">)</span> <span class="o">-</span> <span class="n">pv</span> <span class="o">-</span> <span class="mi">1</span>
+    <span class="n">gridu</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">Tu</span><span class="p">)</span>
+    <span class="n">gridv</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">unique</span><span class="p">(</span><span class="n">Tv</span><span class="p">)</span>
+
+
+    <span class="n">s</span> <span class="o">=</span> <span class="mf">1.</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
+    <span class="n">P</span>          <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">nu</span><span class="p">,</span><span class="n">nv</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
+    <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,:]</span>   <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="o">-</span><span class="n">s</span>   <span class="p">,</span> <span class="o">-</span><span class="n">s</span>   <span class="p">])</span>
+    <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,:]</span>   <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">s</span> <span class="p">,</span> <span class="mf">0.</span>   <span class="p">])</span>
+    <span class="n">P</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,:]</span>   <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="o">-</span><span class="n">s</span>   <span class="p">,</span> <span class="n">s</span>    <span class="p">])</span>
+    <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,:]</span>   <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="mf">0.</span>   <span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">s</span> <span class="p">])</span>
+    <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,:]</span>   <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="mf">0.</span>   <span class="p">,</span> <span class="mf">0.0</span>  <span class="p">])</span>
+    <span class="n">P</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,:]</span>   <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="mf">0.</span>   <span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">s</span>  <span class="p">])</span>
+    <span class="n">P</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,:]</span>   <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="n">s</span>    <span class="p">,</span> <span class="o">-</span><span class="n">s</span>   <span class="p">])</span>
+    <span class="n">P</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,:]</span>   <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="mi">2</span><span class="o">*</span><span class="n">s</span>  <span class="p">,</span> <span class="mf">0.</span>   <span class="p">])</span>
+    <span class="n">P</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,:]</span>   <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="n">s</span>    <span class="p">,</span> <span class="n">s</span>    <span class="p">])</span>
+
+    <span class="n">P</span> <span class="o">*=</span> <span class="n">radius</span>
+    <span class="n">P</span><span class="p">[:,:,</span><span class="mi">0</span><span class="p">]</span> <span class="o">+=</span> <span class="n">center</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
+    <span class="n">P</span><span class="p">[:,:,</span><span class="mi">1</span><span class="p">]</span> <span class="o">+=</span> <span class="n">center</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+
+    <span class="n">W</span>       <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">))</span>
+    <span class="n">W</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>  <span class="o">=</span> <span class="mf">1.</span>
+    <span class="n">W</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>  <span class="o">=</span> <span class="n">s</span>
+    <span class="n">W</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span>  <span class="o">=</span> <span class="mf">1.</span>
+    <span class="n">W</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>  <span class="o">=</span> <span class="n">s</span>
+    <span class="n">W</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>  <span class="o">=</span> <span class="mf">1.</span>
+    <span class="n">W</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>  <span class="o">=</span> <span class="n">s</span>
+    <span class="n">W</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>  <span class="o">=</span> <span class="mf">1.</span>
+    <span class="n">W</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>  <span class="o">=</span> <span class="n">s</span>
+    <span class="n">W</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>  <span class="o">=</span> <span class="mf">1.</span>
+
+    <span class="k">return</span> <span class="p">(</span><span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span><span class="p">),</span> <span class="p">(</span><span class="n">pu</span><span class="p">,</span> <span class="n">pv</span><span class="p">),</span> <span class="n">P</span><span class="p">,</span> <span class="n">W</span>
+
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">example_3</span><span class="p">():</span>    
+    <span class="n">knots</span><span class="p">,</span> <span class="n">degrees</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">W</span> <span class="o">=</span> <span class="n">make_circle</span><span class="p">(</span><span class="n">center</span><span class="o">=</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span><span class="mf">0.</span><span class="p">),</span> <span class="n">radius</span><span class="o">=</span><span class="mf">1.</span><span class="p">)</span>
+    
+    <span class="n">t</span> <span class="o">=</span> <span class="mf">0.5</span>
+    <span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span><span class="p">,</span> <span class="n">pu</span><span class="p">,</span> <span class="n">pv</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">W</span> <span class="o">=</span> <span class="n">insert_knot_nurbs_surface</span><span class="p">(</span><span class="o">*</span><span class="n">knots</span><span class="p">,</span> <span class="o">*</span><span class="n">degrees</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">W</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> 
+                                <span class="n">times</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> 
+                                <span class="n">axis</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+    
+    <span class="n">plot_surface</span><span class="p">((</span><span class="n">Tu</span><span class="p">,</span> <span class="n">Tv</span><span class="p">),</span> <span class="p">(</span><span class="n">pu</span><span class="p">,</span> <span class="n">pv</span><span class="p">),</span> <span class="n">P</span><span class="p">,</span> 
+                 <span class="n">weights</span><span class="o">=</span><span class="n">W</span><span class="p">,</span> 
+                 <span class="n">Nu</span><span class="o">=</span><span class="mi">101</span><span class="p">,</span> <span class="n">Nv</span><span class="o">=</span><span class="mi">101</span><span class="p">,</span> 
+                 <span class="n">color</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">)</span>
+    
+<span class="n">example_3</span><span class="p">()</span>
+</pre></div>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.datatype.html">sympde.topology.datatype</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
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+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
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+</li>
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+</ul>
+</li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.expr.html">psydac.api.ast.expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.glt.html">psydac.api.ast.glt</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.utilities.html">psydac.api.ast.utilities</a></li>
+</ul>
+</li>
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+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#b-splines-properties">B-Splines properties</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-b-splines">Evaluation of B-Splines</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#knots-vector-families">Knots vector families</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#clamped-knots-open-knots-vector">Clamped knots (open knots vector)</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#uniform">uniform</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#non-uniform">non-uniform</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#unclamped-knots">Unclamped knots</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">uniform</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">non-uniform</a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="b-splines">
+<h1>B-Splines<a class="headerlink" href="#b-splines" title="Link to this heading">#</a></h1>
+<p>Given a subdivision <span class="math notranslate nohighlight">\(\{x_0 &lt;  x_1 &lt; \cdots &lt; x_r\}\)</span> of the interval <span class="math notranslate nohighlight">\(I = [x_0, x_r]\)</span>, the \textbf{Schoenberg space} is the space of piecewise polynomials of degree <span class="math notranslate nohighlight">\(p\)</span>, on the interval <span class="math notranslate nohighlight">\(I\)</span> and given regularities <span class="math notranslate nohighlight">\(\{k_1, k_2, \cdots, k_{r-1}\}\)</span> at the internal points <span class="math notranslate nohighlight">\(\{x_1, x_2, \cdots, x_{r-1}\}\)</span>.</p>
+<p>Given <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(p\)</span> as natural numbers, let us consider a sequence of non decreasing real numbers <span class="math notranslate nohighlight">\(T=\{t_i\}_{0\leqslant i \leqslant m}\)</span>. <span class="math notranslate nohighlight">\(T\)</span> is called <strong>knots sequence</strong>. From a knots sequence, we can generate a B-splines family using the reccurence formula \ref{eq:bspline-reccurence}.</p>
+<div class="proof definition admonition" id="def-bspline">
+<p class="admonition-title"><span class="caption-number">Definition 4 </span> (B-Splines using Cox-DeBoor Formula)</p>
+<section class="definition-content" id="proof-content">
+<p>The j-th B-spline of degree <span class="math notranslate nohighlight">\(p\)</span> is defined by the recurrence relation:</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  N_j^p = \frac{t-t_j}{t_{j+p}-t_{j}} N_j^{p-1} + \frac{t_{j+p+1}-t}{t_{j+p+1}-t_{j+1}} N_{j+1}^{p-1},
+  \label{eq:bspline-reccurence}
+\end{align}
+\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[
+N_j^0(t) = \chi_{ \left[ t_j, t_{j+1} \right[ }(t)
+\]</div>
+<p>for <span class="math notranslate nohighlight">\(0 \leq j \leq m-p-1\)</span>.</p>
+</section>
+</div><div class="proof remark admonition" id="remark-1">
+<p class="admonition-title"><span class="caption-number">Remark 4 </span></p>
+<section class="remark-content" id="proof-content">
+<p>When working with Bernstein polynomials, we introduced the convention <span class="math notranslate nohighlight">\(B_i^n=0\)</span> for all <span class="math notranslate nohighlight">\(i&lt;0\)</span> or <span class="math notranslate nohighlight">\(i&gt;n\)</span>. For B-Splines, we have a similar convetion <span class="math notranslate nohighlight">\(N_j^p=0\)</span> if <span class="math notranslate nohighlight">\(j&lt;0\)</span> or <span class="math notranslate nohighlight">\(j&gt;n\)</span>. In addition, we also assume <span class="math notranslate nohighlight">\(\frac{0}{0} = 0\)</span>, when using the formula \ref{eq:bspline-reccurence} and <span class="math notranslate nohighlight">\(N_j^0 = 0\)</span> if <span class="math notranslate nohighlight">\(t_j = t_{j+1}\)</span>.</p>
+</section>
+</div><div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># needed imports</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">empty</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+</pre></div>
+</div>
+<div class="proof example admonition" id="bsplines-ex-1">
+<p class="admonition-title"><span class="caption-number">Example 8 </span></p>
+<section class="example-content" id="proof-content">
+<p>We consider a linear B-Spline with the knot vector <span class="math notranslate nohighlight">\(T = [0, 1, 2]\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">N1_0</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="n">t</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span> <span class="k">return</span> <span class="mi">2</span><span class="o">-</span><span class="n">t</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N1_0</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_0^1$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_5_1.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-2">
+<p class="admonition-title"><span class="caption-number">Example 9 </span></p>
+<section class="example-content" id="proof-content">
+<p>We consider a linear B-Spline with the knot vector <span class="math notranslate nohighlight">\(T = [0, 0, 1]\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">N1_0</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="mi">1</span><span class="o">-</span><span class="n">t</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N1_0</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_0^1$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_7_1.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-3">
+<p class="admonition-title"><span class="caption-number">Example 10 </span></p>
+<section class="example-content" id="proof-content">
+<p>We consider a linear B-Spline with the knot vector <span class="math notranslate nohighlight">\(T = [0, 1, 1]\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">N1_0</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="n">t</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mi">201</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N1_0</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_0^1$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_9_1.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-4">
+<p class="admonition-title"><span class="caption-number">Example 11 </span></p>
+<section class="example-content" id="proof-content">
+<p>We consider linear B-Splines with the knot vector <span class="math notranslate nohighlight">\(T = [0, 0, 1, 1]\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">N1_0</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="mi">1</span><span class="o">-</span><span class="n">t</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="k">def</span> <span class="nf">N1_1</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="n">t</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mi">201</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N1_0</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_0^1$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N1_1</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_1^1$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_11_1.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-5">
+<p class="admonition-title"><span class="caption-number">Example 12 </span></p>
+<section class="example-content" id="proof-content">
+<p>We consider linear B-Splines with the knot vector <span class="math notranslate nohighlight">\(T = [0, 0, 1, 2]\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">N1_0</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="mi">1</span><span class="o">-</span><span class="n">t</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="k">def</span> <span class="nf">N1_1</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="n">t</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span> <span class="k">return</span> <span class="mi">2</span><span class="o">-</span><span class="n">t</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mi">201</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N1_0</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_0^1$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N1_1</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_1^1$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_13_1.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-6">
+<p class="admonition-title"><span class="caption-number">Example 13 </span></p>
+<section class="example-content" id="proof-content">
+<p>We consider a quadratic B-Spline with the knot vector <span class="math notranslate nohighlight">\(T = [0, 0, 1, 1]\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">N2_0</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">t</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="p">)</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mi">201</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N2_0</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_0^2$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_15_1.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-7">
+<p class="admonition-title"><span class="caption-number">Example 14 </span></p>
+<section class="example-content" id="proof-content">
+<p>We consider a quadratic B-Spline with the knot vector <span class="math notranslate nohighlight">\(T = [0, 0, 1, 2]\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">T</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
+<span class="k">def</span> <span class="nf">N2_0</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">t</span><span class="o">-</span><span class="mf">3.</span><span class="o">/</span><span class="mf">2.</span><span class="o">*</span><span class="n">t</span><span class="o">**</span><span class="mi">2</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span> <span class="k">return</span> <span class="mf">0.5</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mi">200</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N2_0</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_0^2$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_17_1.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-8">
+<p class="admonition-title"><span class="caption-number">Example 15 </span></p>
+<section class="example-content" id="proof-content">
+<p>We consider linear B-Splines with the knot vector <span class="math notranslate nohighlight">\(T = [0, 0, 1, 2, 3, 3]\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">N1_0</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="mi">1</span><span class="o">-</span><span class="n">t</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="k">def</span> <span class="nf">N1_1</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="n">t</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span> <span class="k">return</span> <span class="mi">2</span><span class="o">-</span><span class="n">t</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="k">def</span> <span class="nf">N1_2</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">2</span><span class="p">:</span> <span class="k">return</span> <span class="n">t</span><span class="o">-</span><span class="mi">1</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">3</span><span class="p">:</span> <span class="k">return</span> <span class="mi">3</span><span class="o">-</span><span class="n">t</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="k">def</span> <span class="nf">N1_3</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">2</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">3</span><span class="p">:</span> <span class="k">return</span> <span class="n">t</span><span class="o">-</span><span class="mi">2</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">,</span> <span class="mi">201</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N1_0</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_0^1$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N1_1</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_1^1$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N1_2</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_2^1$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N1_3</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_3^1$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_19_1.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-9">
+<p class="admonition-title"><span class="caption-number">Example 16 </span></p>
+<section class="example-content" id="proof-content">
+<p>We consider quadratic B-Splines with the knot vector <span class="math notranslate nohighlight">\(T = [0, 0, 0, 1, 1, 1]\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">N2_0</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="k">def</span> <span class="nf">N2_1</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="mi">2</span><span class="o">*</span><span class="n">t</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">t</span><span class="p">)</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="k">def</span> <span class="nf">N2_2</span><span class="p">(</span><span class="n">t</span><span class="p">):</span>
+    <span class="k">if</span> <span class="n">t</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">t</span><span class="o">&lt;</span> <span class="mi">1</span><span class="p">:</span> <span class="k">return</span> <span class="n">t</span><span class="o">**</span><span class="mi">2</span>
+    <span class="k">return</span> <span class="mf">0.</span>
+
+<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mi">201</span><span class="p">)[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N2_0</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_0^2$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N2_1</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_1^2$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span> <span class="p">[</span><span class="n">N2_2</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">xs</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_2^2$&#39;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_21_1.png" /></p>
+</section>
+</div><section id="b-splines-properties">
+<h2>B-Splines properties<a class="headerlink" href="#b-splines-properties" title="Link to this heading">#</a></h2>
+<p>B-Splines have many interesting properties that are listed below. In general, most of the proofs are done by induction on the B-Spline degree.</p>
+<div class="proof proposition admonition" id="proposition-11">
+<p class="admonition-title"><span class="caption-number">Proposition 7 </span></p>
+<section class="proposition-content" id="proof-content">
+<p>B-splines are piecewise polynomial of degree <span class="math notranslate nohighlight">\(p\)</span></p>
+</section>
+</div><div class="proof remark admonition" id="remark-12">
+<p class="admonition-title"><span class="caption-number">Remark 5 </span></p>
+<section class="remark-content" id="proof-content">
+<p>The B-splines functions associated to open knots sequences without internal knots, \textit{i.e.} the length of the knots sequence is exactly <span class="math notranslate nohighlight">\(2p+2\)</span>, are the Bernstein polynomials of degree<span class="math notranslate nohighlight">\(p\)</span>.</p>
+</section>
+</div><div class="proof proposition admonition" id="proposition-13">
+<p class="admonition-title"><span class="caption-number">Proposition 8 </span> (compact support)</p>
+<section class="proposition-content" id="proof-content">
+<p><span class="math notranslate nohighlight">\(N_j^p(t) = 0\)</span> for all <span class="math notranslate nohighlight">\(t \notin [t_j, t_{j+p+1})\)</span>.</p>
+</section>
+</div><div class="proof proposition admonition" id="proposition-14">
+<p class="admonition-title"><span class="caption-number">Proposition 9 </span></p>
+<section class="proposition-content" id="proof-content">
+<p>\label{prop:non-vanishing-bsplines}
+If <span class="math notranslate nohighlight">\(t \in~ [ t_j,t_{j+1} ) \)</span>, then only the \textit{B-splines} <span class="math notranslate nohighlight">\(\{ N_{j-p}^p,\cdots,N_{j}^p \}\)</span> are non vanishing at <span class="math notranslate nohighlight">\(t\)</span>.</p>
+</section>
+</div><div class="proof proposition admonition" id="proposition-15">
+<p class="admonition-title"><span class="caption-number">Proposition 10 </span> (non-negativity)</p>
+<section class="proposition-content" id="proof-content">
+<p><span class="math notranslate nohighlight">\(N_j^p(t) \ge 0, \quad \forall t \in [ t_j, t_{j+p+1} ) \)</span></p>
+</section>
+</div><div class="proof proposition admonition" id="proposition-16">
+<p class="admonition-title"><span class="caption-number">Proposition 11 </span> (Partition of unity)</p>
+<section class="proposition-content" id="proof-content">
+<p><span class="math notranslate nohighlight">\(\sum N_i^{p}(t) = 1, \forall t \in \mathbb{R}\)</span></p>
+</section>
+</div><div class="proof remark admonition" id="remark-17">
+<p class="admonition-title"><span class="caption-number">Remark 6 </span></p>
+<section class="remark-content" id="proof-content">
+<p>The previous sum <span class="math notranslate nohighlight">\(\sum N_i^{p}(t)\)</span> has a meaning since for every <span class="math notranslate nohighlight">\(t \in \mathbb{R}\)</span>, only <span class="math notranslate nohighlight">\(p+1\)</span> B-Splines are non-vanishing.</p>
+</section>
+</div><div class="proof remark admonition" id="remark-18">
+<p class="admonition-title"><span class="caption-number">Remark 7 </span></p>
+<section class="remark-content" id="proof-content">
+<p>For the sake of simplicity, we shall avoid using summation indices on linear expansion of B-Splines.</p>
+</section>
+</div><div class="proof lemma admonition" id="lemma-19">
+<p class="admonition-title"><span class="caption-number">Lemma 1 </span></p>
+<section class="lemma-content" id="proof-content">
+<div class="math notranslate nohighlight">
+\[
+  \begin{align}
+  \sum a_j N_j^p = \sum \left(\frac{t_{j+p}-t}{t_{j+p}-t_{j}}a_{j-1} + \frac{t-t_j}{t_{j+p}-t_{j}} a_j\right) N_j^{p-1}  
+  \end{align}
+\]</div>
+</section>
+</div><div class="proof proposition admonition" id="proposition-20">
+<p class="admonition-title"><span class="caption-number">Proposition 12 </span> (Marsden’s idenity)</p>
+<section class="proposition-content" id="proof-content">
+<div class="math notranslate nohighlight">
+\[
+  \begin{align}
+    \left( x-y \right)^p = \sum\limits_{j} \rho_j^p(y) N_j^p(x)
+    \label{eq:marsden-spline}
+  \end{align}
+\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[
+  \begin{align}
+    \rho_j^p(y) = \prod\limits_{i=j+1}^{j+p} (t_{i} -y)   
+  \end{align}
+\]</div>
+</section>
+</div><p>Thanks to Marsden’s identity and the proposition \ref{prop:non-vanishing-bsplines}, we now are able to prove the local linear independence of the B-Splines basis functions.</p>
+<div class="proof proposition admonition" id="proposition-21">
+<p class="admonition-title"><span class="caption-number">Proposition 13 </span> (Local linear independence)</p>
+<section class="proposition-content" id="proof-content">
+<p>On each interval <span class="math notranslate nohighlight">\([t_j, t_{j+1})\)</span>, the B-Splines are lineary independent.</p>
+</section>
+</div><div class="proof remark admonition" id="remark-22">
+<p class="admonition-title"><span class="caption-number">Remark 8 </span></p>
+<section class="remark-content" id="proof-content">
+<p>We will see in the Approximation theory part, that the B-Splines family has also a global linear independence property.</p>
+</section>
+</div><div class="proof proposition admonition" id="proposition-23">
+<p class="admonition-title"><span class="caption-number">Proposition 14 </span> (Derivatives of B-Splines)</p>
+<section class="proposition-content" id="proof-content">
+<p>The derivative of B-Splines can be computed reccursivly by deriving the formula \ref{eq:bspline-reccurence}, which gives</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  {N_j^p}^\prime = \frac{p}{t_{j+p}-t_{j}} N_j^{p-1} - \frac{p}{t_{j+p+1}-t_{j+1}} N_{j+1}^{p-1}
+  \label{eq:bspline-ders-reccurence}
+\end{align}
+\]</div>
+</section>
+</div></section>
+<section id="evaluation-of-b-splines">
+<h2>Evaluation of B-Splines<a class="headerlink" href="#evaluation-of-b-splines" title="Link to this heading">#</a></h2>
+<p>Given a knot sequence <span class="math notranslate nohighlight">\(T=\{t_i\}_{0\leqslant i \leqslant n + p}\)</span>, we are interested in the algorithmic evaluation of B-Splines of degree <span class="math notranslate nohighlight">\(p\)</span>.</p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Add evaluation diagram</p>
+</div>
+<p>For a given real point <span class="math notranslate nohighlight">\(x\)</span>, it is done in two steps:</p>
+<ol class="arabic simple">
+<li><p>find the knot span index <span class="math notranslate nohighlight">\(j\)</span>, such that <span class="math notranslate nohighlight">\(x \in~ ] t_j,t_{j+1} [ \)</span></p></li>
+<li><p>evaluate all non-vanishing B-Splines <span class="math notranslate nohighlight">\(N_{j-p}^p, \cdots, N_j^p\)</span></p></li>
+</ol>
+<p>The first point is achieved by the function implemented by the following function:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">find_span</span><span class="p">(</span> <span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">x</span> <span class="p">):</span>
+    <span class="c1"># Knot index at left/right boundary</span>
+    <span class="n">low</span>  <span class="o">=</span> <span class="n">degree</span>
+    <span class="n">high</span> <span class="o">=</span> <span class="mi">0</span>
+    <span class="n">high</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">degree</span>
+
+    <span class="c1"># Check if point is exactly on left/right boundary, or outside domain</span>
+    <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;=</span> <span class="n">knots</span><span class="p">[</span><span class="n">low</span> <span class="p">]:</span> <span class="n">returnVal</span> <span class="o">=</span> <span class="n">low</span>
+    <span class="k">elif</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="n">knots</span><span class="p">[</span><span class="n">high</span><span class="p">]:</span> <span class="n">returnVal</span> <span class="o">=</span> <span class="n">high</span><span class="o">-</span><span class="mi">1</span>
+    <span class="k">else</span><span class="p">:</span>
+        <span class="c1"># Perform binary search</span>
+        <span class="n">span</span> <span class="o">=</span> <span class="p">(</span><span class="n">low</span><span class="o">+</span><span class="n">high</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span>
+        <span class="k">while</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="n">knots</span><span class="p">[</span><span class="n">span</span><span class="p">]</span> <span class="ow">or</span> <span class="n">x</span> <span class="o">&gt;=</span> <span class="n">knots</span><span class="p">[</span><span class="n">span</span><span class="o">+</span><span class="mi">1</span><span class="p">]:</span>
+            <span class="k">if</span> <span class="n">x</span> <span class="o">&lt;</span> <span class="n">knots</span><span class="p">[</span><span class="n">span</span><span class="p">]:</span>
+                <span class="n">high</span> <span class="o">=</span> <span class="n">span</span>
+            <span class="k">else</span><span class="p">:</span>
+                <span class="n">low</span>  <span class="o">=</span> <span class="n">span</span>
+            <span class="n">span</span> <span class="o">=</span> <span class="p">(</span><span class="n">low</span><span class="o">+</span><span class="n">high</span><span class="p">)</span><span class="o">//</span><span class="mi">2</span>
+        <span class="n">returnVal</span> <span class="o">=</span> <span class="n">span</span>
+
+    <span class="k">return</span> <span class="n">returnVal</span>
+</pre></div>
+</div>
+<p>The second point is implemented by the following function, that returns all non-vanishing B-Splines at <span class="math notranslate nohighlight">\(x\)</span></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">all_bsplines</span><span class="p">(</span> <span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">span</span> <span class="p">):</span>
+    <span class="n">left</span>   <span class="o">=</span> <span class="n">empty</span><span class="p">(</span> <span class="n">degree</span>  <span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">float</span> <span class="p">)</span>
+    <span class="n">right</span>  <span class="o">=</span> <span class="n">empty</span><span class="p">(</span> <span class="n">degree</span>  <span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">float</span> <span class="p">)</span>
+    <span class="n">values</span> <span class="o">=</span> <span class="n">empty</span><span class="p">(</span> <span class="n">degree</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">float</span> <span class="p">)</span>
+
+    <span class="n">values</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">1.0</span>
+    <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">degree</span><span class="p">):</span>
+        <span class="n">left</span> <span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="n">knots</span><span class="p">[</span><span class="n">span</span><span class="o">-</span><span class="n">j</span><span class="p">]</span>
+        <span class="n">right</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="o">=</span> <span class="n">knots</span><span class="p">[</span><span class="n">span</span><span class="o">+</span><span class="mi">1</span><span class="o">+</span><span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">x</span>
+        <span class="n">saved</span>    <span class="o">=</span> <span class="mf">0.0</span>
+        <span class="k">for</span> <span class="n">r</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+            <span class="n">temp</span>      <span class="o">=</span> <span class="n">values</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">/</span> <span class="p">(</span><span class="n">right</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">+</span> <span class="n">left</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="n">r</span><span class="p">])</span>
+            <span class="n">values</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">=</span> <span class="n">saved</span> <span class="o">+</span> <span class="n">right</span><span class="p">[</span><span class="n">r</span><span class="p">]</span> <span class="o">*</span> <span class="n">temp</span>
+            <span class="n">saved</span>     <span class="o">=</span> <span class="n">left</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="n">r</span><span class="p">]</span> <span class="o">*</span> <span class="n">temp</span>
+        <span class="n">values</span><span class="p">[</span><span class="n">j</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">saved</span>
+
+    <span class="k">return</span> <span class="n">values</span>
+</pre></div>
+</div>
+<p>The following function plots all B-Splines given a knot vector and a polynomial degree.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">plot_splines</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">100</span><span class="p">):</span>
+    <span class="n">xmin</span> <span class="o">=</span> <span class="n">knots</span><span class="p">[</span><span class="n">degree</span><span class="p">]</span>
+    <span class="n">xmax</span> <span class="o">=</span> <span class="n">knots</span><span class="p">[</span><span class="o">-</span><span class="n">degree</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+    
+    <span class="c1"># grid points for evaluation</span>
+    <span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">xmin</span><span class="p">,</span><span class="n">xmax</span><span class="p">,</span><span class="n">nx</span><span class="p">)</span>
+
+    <span class="c1"># this is the number of the BSplines in the Schoenberg space</span>
+    <span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">degree</span> <span class="o">-</span> <span class="mi">1</span>
+
+    <span class="n">ys</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">N</span><span class="p">,</span><span class="n">nx</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">double</span><span class="p">)</span>
+    <span class="k">for</span> <span class="n">ix</span><span class="p">,</span><span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">xs</span><span class="p">):</span>
+        <span class="n">span</span> <span class="o">=</span> <span class="n">find_span</span><span class="p">(</span> <span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">x</span> <span class="p">)</span>    
+        <span class="n">b</span>    <span class="o">=</span> <span class="n">all_bsplines</span><span class="p">(</span> <span class="n">knots</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">span</span> <span class="p">)</span>  
+        <span class="n">ys</span><span class="p">[</span><span class="n">span</span><span class="o">-</span><span class="n">degree</span><span class="p">:</span><span class="n">span</span><span class="o">+</span><span class="mi">1</span><span class="p">,</span> <span class="n">ix</span><span class="p">]</span> <span class="o">=</span> <span class="n">b</span><span class="p">[:]</span>
+        
+    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">N</span><span class="p">):</span>
+        <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">xs</span><span class="p">,</span><span class="n">ys</span><span class="p">[</span><span class="n">i</span><span class="p">,:],</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$N_</span><span class="si">{}</span><span class="s1">$&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">9</span><span class="p">,</span> <span class="n">ncol</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="knots-vector-families">
+<h2>Knots vector families<a class="headerlink" href="#knots-vector-families" title="Link to this heading">#</a></h2>
+<p>There are two kind of <strong>knots vectors</strong>, called <strong>clamped</strong> and <strong>unclamped</strong>. Both families contains <strong>uniform</strong> and <strong>non-uniform</strong> sequences.</p>
+<p>The following are examples of such knots vectors</p>
+<section id="clamped-knots-open-knots-vector">
+<h3>Clamped knots (open knots vector)<a class="headerlink" href="#clamped-knots-open-knots-vector" title="Link to this heading">#</a></h3>
+<section id="uniform">
+<h4>uniform<a class="headerlink" href="#uniform" title="Link to this heading">#</a></h4>
+<div class="proof example admonition" id="bsplines-ex-10">
+<p class="admonition-title"><span class="caption-number">Example 17 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">T</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
+
+<span class="n">plot_splines</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_30_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-11">
+<p class="admonition-title"><span class="caption-number">Example 18 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">T</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">]</span>
+
+<span class="n">plot_splines</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_31_0.png" /></p>
+</section>
+</div></section>
+<section id="non-uniform">
+<h4>non-uniform<a class="headerlink" href="#non-uniform" title="Link to this heading">#</a></h4>
+<div class="proof example admonition" id="bsplines-ex-12">
+<p class="admonition-title"><span class="caption-number">Example 19 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">T</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
+
+<span class="n">plot_splines</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_33_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-13">
+<p class="admonition-title"><span class="caption-number">Example 20 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">T</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">]</span>
+
+<span class="n">plot_splines</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_34_0.png" /></p>
+</section>
+</div></section>
+</section>
+<section id="unclamped-knots">
+<h3>Unclamped knots<a class="headerlink" href="#unclamped-knots" title="Link to this heading">#</a></h3>
+<section id="id1">
+<h4>uniform<a class="headerlink" href="#id1" title="Link to this heading">#</a></h4>
+<div class="proof example admonition" id="bsplines-ex-14">
+<p class="admonition-title"><span class="caption-number">Example 21 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">T</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]</span>
+
+<span class="n">plot_splines</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_36_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-15">
+<p class="admonition-title"><span class="caption-number">Example 22 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">T</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">]</span>
+
+<span class="n">plot_splines</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_37_0.png" /></p>
+</section>
+</div></section>
+<section id="id2">
+<h4>non-uniform<a class="headerlink" href="#id2" title="Link to this heading">#</a></h4>
+<div class="proof example admonition" id="bsplines-ex-16">
+<p class="admonition-title"><span class="caption-number">Example 23 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">T</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">9</span><span class="p">]</span>
+
+<span class="n">plot_splines</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_39_0.png" /></p>
+</section>
+</div><div class="proof example admonition" id="bsplines-ex-17">
+<p class="admonition-title"><span class="caption-number">Example 24 </span></p>
+<section class="example-content" id="proof-content">
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">T</span> <span class="o">=</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">]</span>
+
+<span class="n">plot_splines</span><span class="p">(</span><span class="n">T</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_40_0.png" /></p>
+</section>
+</div></section>
+</section>
+</section>
+<div class="toctree-wrapper compound">
+</div>
+</section>
+
+    <script type="text/x-thebe-config">
+    {
+        requestKernel: true,
+        binderOptions: {
+            repo: "binder-examples/jupyter-stacks-datascience",
+            ref: "master",
+        },
+        codeMirrorConfig: {
+            theme: "abcdef",
+            mode: "python"
+        },
+        kernelOptions: {
+            name: "python3",
+            path: "./chapter0"
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+    <!-- Table of contents -->
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+<li class="toc-h1 nav-item toc-entry"><a class="reference internal nav-link" href="#">Data Structure</a></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">2D Case</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-profile">Matrix profile</a></li>
+</ul>
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+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="data-structure">
+<h1>Data Structure<a class="headerlink" href="#data-structure" title="Link to this heading">#</a></h1>
+<p>In the sequel, we shall use <strong>StencilMatrix</strong> and <strong>StencilVector</strong> from the <strong>psydac</strong> library.</p>
+<p>For the moment, we are only interested about having an appropriate data structure to store our computations.
+Once the assembly is done, we will convert the matrix into a scipy sparse matrix, while the vector will be converted to numpy array.</p>
+<p>We will see later how one can impose boundary conditions.</p>
+</section>
+<section class="tex2jax_ignore mathjax_ignore" id="d-case">
+<h1>1D Case<a class="headerlink" href="#d-case" title="Link to this heading">#</a></h1>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="kn">from</span> <span class="nn">psydac.fem.splines</span> <span class="kn">import</span> <span class="n">SplineSpace</span>
+<span class="kn">from</span> <span class="nn">psydac.fem.tensor</span>  <span class="kn">import</span> <span class="n">TensorFemSpace</span>
+<span class="kn">from</span> <span class="nn">psydac.linalg.stencil</span> <span class="kn">import</span> <span class="n">StencilMatrix</span>
+<span class="kn">from</span> <span class="nn">psydac.linalg.stencil</span> <span class="kn">import</span> <span class="n">StencilVector</span>
+<span class="kn">from</span> <span class="nn">psydac.ddm.cart</span> <span class="kn">import</span> <span class="n">DomainDecomposition</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">gallery_section_03</span> <span class="kn">import</span> <span class="n">assemble_stiffness_1d</span>
+<span class="kn">from</span> <span class="nn">gallery_section_03</span> <span class="kn">import</span> <span class="n">assemble_vector_1d</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">xmin</span> <span class="o">=</span> <span class="mf">0.</span> <span class="p">;</span> <span class="n">xmax</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">nelements</span> <span class="o">=</span> <span class="mi">8</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">grid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">nelements</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
+<span class="n">V</span> <span class="o">=</span> <span class="n">SplineSpace</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid</span><span class="p">)</span>
+<span class="n">dd</span> <span class="o">=</span> <span class="n">DomainDecomposition</span><span class="p">(</span><span class="n">ncells</span><span class="o">=</span><span class="p">[</span><span class="n">V</span><span class="o">.</span><span class="n">ncells</span><span class="p">],</span> <span class="n">periods</span><span class="o">=</span><span class="p">[</span><span class="kc">False</span><span class="p">])</span>
+<span class="n">V</span> <span class="o">=</span> <span class="n">TensorFemSpace</span><span class="p">(</span><span class="n">dd</span><span class="p">,</span> <span class="n">V</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">stiffness</span> <span class="o">=</span> <span class="n">StencilMatrix</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">,</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">stiffness</span> <span class="o">=</span> <span class="n">assemble_stiffness_1d</span><span class="p">(</span> <span class="n">V</span><span class="p">,</span> <span class="n">matrix</span><span class="o">=</span><span class="n">stiffness</span> <span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">stiffness</span> <span class="o">=</span> <span class="n">stiffness</span><span class="o">.</span><span class="n">tosparse</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mf">2.</span>    
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">rhs</span> <span class="o">=</span> <span class="n">StencilVector</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">rhs</span> <span class="o">=</span> <span class="n">assemble_vector_1d</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">rhs</span><span class="o">=</span><span class="n">rhs</span> <span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">rhs</span> <span class="o">=</span> <span class="n">rhs</span><span class="o">.</span><span class="n">toarray</span><span class="p">()</span>
+</pre></div>
+</div>
+<section id="id1">
+<h2>2D Case<a class="headerlink" href="#id1" title="Link to this heading">#</a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">gallery_section_03</span> <span class="kn">import</span> <span class="n">assemble_stiffness_2d</span>
+<span class="kn">from</span> <span class="nn">gallery_section_03</span> <span class="kn">import</span> <span class="n">assemble_vector_2d</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">x1min</span> <span class="o">=</span> <span class="mf">0.</span> <span class="p">;</span> <span class="n">x1max</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">x2min</span> <span class="o">=</span> <span class="mf">0.</span> <span class="p">;</span> <span class="n">x2max</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">nelements1</span> <span class="o">=</span> <span class="mi">8</span>
+<span class="n">nelements2</span> <span class="o">=</span> <span class="mi">8</span>
+<span class="n">degree1</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">degree2</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">grid1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">x1min</span><span class="p">,</span> <span class="n">x1max</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">nelements1</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
+<span class="n">grid2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">x2min</span><span class="p">,</span> <span class="n">x2max</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">nelements2</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
+
+<span class="n">V1</span> <span class="o">=</span> <span class="n">SplineSpace</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree1</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid1</span><span class="p">)</span>
+<span class="n">V2</span> <span class="o">=</span> <span class="n">SplineSpace</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree2</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid2</span><span class="p">)</span>
+
+<span class="n">dd</span> <span class="o">=</span> <span class="n">DomainDecomposition</span><span class="p">(</span><span class="n">ncells</span><span class="o">=</span><span class="p">[</span><span class="n">V1</span><span class="o">.</span><span class="n">ncells</span><span class="p">,</span> <span class="n">V2</span><span class="o">.</span><span class="n">ncells</span><span class="p">],</span> <span class="n">periods</span><span class="o">=</span><span class="p">[</span><span class="kc">False</span><span class="p">,</span> <span class="kc">False</span><span class="p">])</span>
+<span class="n">V</span> <span class="o">=</span> <span class="n">TensorFemSpace</span><span class="p">(</span><span class="n">dd</span><span class="p">,</span> <span class="n">V1</span><span class="p">,</span> <span class="n">V2</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">stiffness</span> <span class="o">=</span> <span class="n">StencilMatrix</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">,</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">stiffness</span> <span class="o">=</span> <span class="n">assemble_stiffness_2d</span><span class="p">(</span> <span class="n">V</span><span class="p">,</span> <span class="n">matrix</span><span class="o">=</span><span class="n">stiffness</span> <span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">stiffness</span> <span class="o">=</span> <span class="n">stiffness</span><span class="o">.</span><span class="n">tosparse</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># convert the sparse matrix to dense</span>
+<span class="n">stiffness</span> <span class="o">=</span> <span class="n">stiffness</span><span class="o">.</span><span class="n">toarray</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="mf">2.</span>    
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">rhs</span> <span class="o">=</span> <span class="n">StencilVector</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">rhs</span> <span class="o">=</span> <span class="n">assemble_vector_2d</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">rhs</span><span class="o">=</span><span class="n">rhs</span> <span class="p">)</span>
+</pre></div>
+</div>
+<section id="matrix-profile">
+<h3>Matrix profile<a class="headerlink" href="#matrix-profile" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span> 
+
+<span class="n">plt</span><span class="o">.</span><span class="n">spy</span><span class="p">(</span><span class="n">stiffness</span><span class="p">)</span>
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+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.analytical_mapping.html">sympde.topology.analytical_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.basic.html">sympde.topology.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.callable_mapping.html">sympde.topology.callable_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.datatype.html">sympde.topology.datatype</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.domain.html">sympde.topology.domain</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_field_2d.html">sympde.topology.tests.test_field_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.html">sympde.expr</a><input class="toctree-checkbox" id="toctree-checkbox-18" name="toctree-checkbox-18" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-18"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.equation.html">sympde.expr.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_equation_2d.html">sympde.expr.tests.test_equation_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
+</li>
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+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.form.html">sympde.exterior.form</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_compiler.html">sympde.exterior.tests.test_compiler</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
+</ul>
+</li>
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+</ul>
+</li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.expr.html">psydac.api.ast.expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.glt.html">psydac.api.ast.glt</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.utilities.html">psydac.api.ast.utilities</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.discretization.html">psydac.api.discretization</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.equation.html">psydac.api.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.essential_bc.html">psydac.api.essential_bc</a></li>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization-of-the-poisson-problem">Discretization of the Poisson problem</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#expressions">Expressions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#notations">Notations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#assembly-procedures">Assembly procedures</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#pre-process">Pre-process</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-a-discrete-field-on-a-given-element">Evaluation of a discrete field on a given element</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#kernel-for-an-expression-with-arity-1">Kernel for an expression with arity = 1</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-contributions-of-test-basis-functions-1d">Computing contributions of test basis functions (1D)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-contributions-of-test-basis-functions-2d">Computing contributions of test basis functions (2D)</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#kernel-for-an-expression-with-arity-2">Kernel for an expression with arity = 2</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-contributions-between-test-and-trial-basis-functions-1d">Computing contributions between test and trial basis functions (1D)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-contributions-between-test-and-trial-basis-functions-2d">Computing contributions between test and trial basis functions (2D)</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="introduction-to-b-splines-fem">
+<h1>Introduction to B-Splines FEM<a class="headerlink" href="#introduction-to-b-splines-fem" title="Link to this heading">#</a></h1>
+<p>Let <span class="math notranslate nohighlight">\(\Omega \subset \mathbb{R}^d\)</span> be a computational domain that is the image of a logical domain <span class="math notranslate nohighlight">\(\mathcal{P}\)</span>, <em>i.e.</em> a unit line (in <em>1d</em>), square (in <em>2d</em>) or a cube (in <em>3d</em>) with a <strong>mapping</strong> function</p>
+<div class="math notranslate nohighlight">
+\[
+\Omega = \mathcal{G} (\mathcal{P}) 
+\]</div>
+<p>We consider the Poisson problem with Homogeneous Dirichlet boundary conditions:</p>
+<div class="math notranslate nohighlight">
+\[
+  \mbox{Find} ~  u \in H^1_0(\Omega), ~ \mbox{such that}
+\]</div>
+<div class="math notranslate nohighlight">
+\[
+  - \nabla^2 u = f, ~~ \Omega
+\]</div>
+<p>Using a Galerkin-Rietz method, we introduce a discrete finite elements space <span class="math notranslate nohighlight">\(\mathcal{V}_h = \mathbf{span}\{ \phi_j, 1 \leq j \leq n_V \}\)</span> for trial and test functions. Here the index <span class="math notranslate nohighlight">\(j\)</span> is a multi-index. For example, in <span class="math notranslate nohighlight">\(2D\)</span>, we have <span class="math notranslate nohighlight">\(j = (j_1, j_2)\)</span></p>
+<p>Let <span class="math notranslate nohighlight">\(u_h \in \mathcal{V}_h\)</span> such that <span class="math notranslate nohighlight">\(u_h = \sum_{j=1}^{n_V} u_j \phi_j\)</span>. Then the weak formulation for the Poisson equation writes</p>
+<div class="math notranslate nohighlight">
+\[
+  \sum_{j=1}^{n} u_j \int_{\Omega} \nabla \phi_j \cdot \nabla \phi_i = \int_{\Omega} f \phi_i, \quad \forall  1 \leq i \leq n 
+\]</div>
+<p>which can be written in a matrix form</p>
+<div class="math notranslate nohighlight">
+\[
+  M U = F
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(U\)</span> denotes the coefficients <span class="math notranslate nohighlight">\((u_j, ~ 1 \leq j \leq n)\)</span> and <span class="math notranslate nohighlight">\(F\)</span> is a vector consisting of the terms <span class="math notranslate nohighlight">\(\int_{\Omega} f \phi_i\)</span> for <span class="math notranslate nohighlight">\(1 \leq i \leq n\)</span>. Finally, the matrix <span class="math notranslate nohighlight">\(M\)</span> is called the <strong>stiffness</strong> matrix and its entries are defined as</p>
+<div class="math notranslate nohighlight">
+\[
+  M_{ij} = \int_{\Omega} \nabla \phi_j \cdot \nabla \phi_i
+\]</div>
+<p>We will denote our basis function by <span class="math notranslate nohighlight">\(b_i\)</span> and <span class="math notranslate nohighlight">\(b_j\)</span> rather than <span class="math notranslate nohighlight">\(\phi_i\)</span> and <span class="math notranslate nohighlight">\(\phi_j\)</span>. In this case, in order to solve the Poisson problem, one needs to</p>
+<ol class="arabic simple">
+<li><p>Assemble the matrix <span class="math notranslate nohighlight">\(M\)</span> of entries
+$<span class="math notranslate nohighlight">\(
+M_{ij} = \int_{\Omega} \nabla b_j \cdot \nabla b_i
+\)</span>$</p></li>
+<li><p>Assemble the right hand side <span class="math notranslate nohighlight">\(F\)</span> of entries
+$<span class="math notranslate nohighlight">\(
+F_{i} = \int_{\Omega} f b_i
+\)</span>$</p></li>
+<li><p>Solve the linear system
+$<span class="math notranslate nohighlight">\(
+M U = F
+\)</span>$</p></li>
+</ol>
+<p>Since at the end, we need to compute the <span class="math notranslate nohighlight">\(L^2\)</span>-norm of the error <span class="math notranslate nohighlight">\(u-u_h\)</span>, we need also to evaluate (discretize) the following (scalar) expression:</p>
+<div class="math notranslate nohighlight">
+\[
+\| e_h \|_{L^2}^2 := \| u-u_h \|_{L^2}^2 = \int_{\Omega} (u-u_h)^2
+\]</div>
+<section id="discretization-of-the-poisson-problem">
+<h2>Discretization of the Poisson problem<a class="headerlink" href="#discretization-of-the-poisson-problem" title="Link to this heading">#</a></h2>
+<p>Now let’s go back to a matrix entry <span class="math notranslate nohighlight">\(M_{ij}\)</span>, we have
+$<span class="math notranslate nohighlight">\(
+M_{ij} = \sum_e \int_{Q_e} \nabla b_i \cdot \nabla b_j
+\)</span><span class="math notranslate nohighlight">\(
+We know that every basis function \)</span>b_i<span class="math notranslate nohighlight">\( is a polynomial over the element \)</span>Q_i$. We can then use one of the Gauss quadrature formulae.</p>
+<p><strong>Note</strong>:
+In the case of <em>B-splines</em> we will avoid having to evaluate on the boundary of our element (B-Splines are only defined on the right, if we don’t do so, evaluating derivatives may lead to wrong results). Hence we will use the Gauss-Legendre formula.</p>
+<p><strong>Note</strong>:
+In general, the quadrature formulae are given on the interval <span class="math notranslate nohighlight">\([-1, 1]\)</span>. We therefor need to remap these points for every element, in out partition.</p>
+<p>Let’s assume now that our quadrature points have been remapped, and let’s call them <span class="math notranslate nohighlight">\(\xi_k\)</span>, <span class="math notranslate nohighlight">\(\omega_k\)</span> will denote the associated weight to <span class="math notranslate nohighlight">\(\xi_k\)</span>.
+In this case <span class="math notranslate nohighlight">\(M_{ij}\)</span> can be written as</p>
+<div class="math notranslate nohighlight">
+\[
+M_{ij} = \sum_e \sum_k \omega_k \nabla b_i(\xi_k) \cdot \nabla b_j( \xi_k)
+\]</div>
+<p>Now let’s go back to a rhs entry <span class="math notranslate nohighlight">\(F_{i}\)</span>, we have
+$<span class="math notranslate nohighlight">\(
+F_{i} = \sum_e \int_{Q_e} f b_i 
+\)</span><span class="math notranslate nohighlight">\(
+We know that every basis function \)</span>b_i<span class="math notranslate nohighlight">\( is a polynomial over the element \)</span>Q_i$. We can then use one of the Gauss quadrature formulae.</p>
+<p>Since our quadrature points have been remapped ( points : <span class="math notranslate nohighlight">\(\xi_k\)</span>, weights : <span class="math notranslate nohighlight">\(\omega_k\)</span> ).</p>
+<p>In this case <span class="math notranslate nohighlight">\(F_{i}\)</span> can be written as</p>
+<div class="math notranslate nohighlight">
+\[
+F_{i} = \sum_e \sum_k \omega_k b_i(\xi_k) f( \xi_k)
+\]</div>
+<p>Now let’s go back to a norm <span class="math notranslate nohighlight">\(e_h\)</span>, we have
+$<span class="math notranslate nohighlight">\(
+\| e_h \|_{L^2}^2 = \sum_e \int_{Q_e} (u-u_h)^2 
+\)</span>$</p>
+<p>which leads to</p>
+<div class="math notranslate nohighlight">
+\[
+\| e_h \|_{L^2}^2 = \sum_e \sum_k \omega_k (u(\xi_k) - u_h( \xi_k))^2
+\]</div>
+<p>The missing part here, is how to evaluate the discrete field <span class="math notranslate nohighlight">\(u_h\)</span>. This will be covered in the next part.</p>
+</section>
+<section id="expressions">
+<h2>Expressions<a class="headerlink" href="#expressions" title="Link to this heading">#</a></h2>
+<p>When using a Finite Elements method, one needs to compute the following kind of expressions:</p>
+<ul class="simple">
+<li><p><strong>bilinear form</strong>. Example : <span class="math notranslate nohighlight">\(\int_{\Omega} \phi_i \phi_j ~ d\Omega\)</span></p></li>
+<li><p><strong>linear form</strong>. Example : <span class="math notranslate nohighlight">\(\int_{\Omega} f \phi_i ~ d\Omega\)</span></p></li>
+<li><p><strong>functional</strong>, i.e. integral of a scalar (vector) expression. Example : <span class="math notranslate nohighlight">\(\int_{\Omega} (u-u_h)^2\)</span></p></li>
+</ul>
+<p>A general formal definition of a variational form <span class="math notranslate nohighlight">\(a\)</span> is given by Eq. \ref{eq:var-form-def}, as real-valued \textit{multi-linear} map on the product of a sequence of function spaces  <span class="math notranslate nohighlight">\(\{V_j\}_{j=1}^\rho\)</span> :</p>
+<p>\begin{align}
+a =
+\begin{cases}
+V_\rho \times \ldots \times V_2 \times V_1 \times \mathcal{P} \rightarrow \mathbb{R}
+\
+a(v_\rho, \dots, v_1; w_1, \dots, w_\nu, c_1, \dots, c_\eta)
+\end{cases}
+\label{eq:var-form-def}
+\end{align}</p>
+<p>where <span class="math notranslate nohighlight">\(\rho\)</span> is the arity of the variational form, and the special cases <span class="math notranslate nohighlight">\(\rho=0,1,2\)</span> yield to the notions of \emph{functional}, \emph{linear form} and \emph{bilinear form}, respectively.
+Each of the spaces <span class="math notranslate nohighlight">\(V_j\)</span> may be a scalar/vector function space, or a product of other function spaces.
+<span class="math notranslate nohighlight">\(\mathcal{P}\)</span> is the space of all parameters (which can be empty), containing either functions or constants (with respect to differential operators).
+A function parameter is referred to as a \emph{field} and is an element of a function space, therefore it may be represented by its coefficients over a given basis.
+The general definition of the space of parameters is given as <span class="math notranslate nohighlight">\(\mathcal{P} := \left( W_1 \times \ldots \times W_\nu \right) \times \mathbb{R}^\eta\)</span> where <span class="math notranslate nohighlight">\(\left( W_i \right)_{1 \le i \le \nu}\)</span> can be scalar or vector function spaces and <span class="math notranslate nohighlight">\(\eta\)</span> is the number of (real) constants.
+We shall refer to the parameters as \textit{free variables}, as they will not appear in the signature of the variational form, and are defined as global variables (or local to the current scope).</p>
+<p>In the sequel, we shall see how the different expressions can be evaluated. We will developp the different needed algorithms, that we will use in the next chapters.</p>
+<p>Since, we will end up by computing an integral over the domain, we will need the following procedures</p>
+<ul class="simple">
+<li><p>Assembly over the domain: this will loop over the elements of the domain. In each element, we shall call a kernel procedure that performs the computation over one single element.</p></li>
+<li><p>depending on the multi-linear map (expression) to evaluate, we shall have three kinds of kernels:</p>
+<ul>
+<li><p>a kernel for computing expressions with arity of 2</p></li>
+<li><p>a kernel for computing expressions with arity of 1</p></li>
+<li><p>a kernel for computing expressions with arity of 0</p></li>
+</ul>
+</li>
+<li><p>we shall not discuss for the moment the data structure used to store the results of the kernels</p></li>
+</ul>
+</section>
+<section id="notations">
+<h2>Notations<a class="headerlink" href="#notations" title="Link to this heading">#</a></h2>
+<p>In the sequel, we shall use the following arguments</p>
+<ul class="simple">
+<li><p>n_elements_d (d=1,2,3) number of elements of the grid in each direction</p></li>
+<li><p>an element will always refer to a grid element, which considered as the a tuple</p></li>
+<li><p>k_d (d=1,2,3) will denote the number of quadrature points in each direction, with respect to current element</p></li>
+<li><p>The operator <span class="math notranslate nohighlight">\(\texttt{coeff}\)</span> will denote the local coefficients of a field over the current element</p></li>
+<li><p>The operator <span class="math notranslate nohighlight">\(\texttt{value}\)</span> will denote the values of a field over the current element</p></li>
+</ul>
+</section>
+<section id="assembly-procedures">
+<h2>Assembly procedures<a class="headerlink" href="#assembly-procedures" title="Link to this heading">#</a></h2>
+<p>The generic form of the assembly procedure is described next. In general, it is splitted in 3 parts:</p>
+<ul class="simple">
+<li><p>Pre-process: select the needed data, related to the current element.</p></li>
+<li><p>Process: call the kernel for computations on the given element</p></li>
+<li><p>Post-process:</p></li>
+</ul>
+<section id="pre-process">
+<h3>Pre-process<a class="headerlink" href="#pre-process" title="Link to this heading">#</a></h3>
+<p>For each function space (both <span class="math notranslate nohighlight">\(V_i\)</span> and <span class="math notranslate nohighlight">\(W_i\)</span>) involved in the expression, we need to get the basis functions over the current element, as well as their spans.</p>
+<p>The following examples are not general, since we consider only one function space.</p>
+</section>
+</section>
+<section id="evaluation-of-a-discrete-field-on-a-given-element">
+<h2>Evaluation of a discrete field on a given element<a class="headerlink" href="#evaluation-of-a-discrete-field-on-a-given-element" title="Link to this heading">#</a></h2>
+<p>When a free-variable is used in the expression, one needs to evaluate it on the quadrature points. In what follows, we describe the evaluation of a discrete field, with respect to a given element</p>
+<p><img alt="title" src="../_images/eval_field_1d.png" />
+<img alt="title" src="../_images/eval_field_2d.png" /></p>
+</section>
+<section id="kernel-for-an-expression-with-arity-1">
+<h2>Kernel for an expression with arity = 1<a class="headerlink" href="#kernel-for-an-expression-with-arity-1" title="Link to this heading">#</a></h2>
+<p><img alt="title" src="../_images/kernel_arity1_1d.png" />
+<img alt="title" src="../_images/kernel_arity1_2d.png" /></p>
+<section id="computing-contributions-of-test-basis-functions-1d">
+<h3>Computing contributions of test basis functions (1D)<a class="headerlink" href="#computing-contributions-of-test-basis-functions-1d" title="Link to this heading">#</a></h3>
+<p><img alt="title" src="../_images/contribution_arity1_1d_ex1.png" />
+<img alt="title" src="../_images/contribution_arity1_1d_ex2.png" /></p>
+</section>
+<section id="computing-contributions-of-test-basis-functions-2d">
+<h3>Computing contributions of test basis functions (2D)<a class="headerlink" href="#computing-contributions-of-test-basis-functions-2d" title="Link to this heading">#</a></h3>
+<p><img alt="title" src="../_images/contribution_arity1_2d_ex1.png" />
+<img alt="title" src="../_images/contribution_arity1_2d_ex2.png" /></p>
+</section>
+</section>
+<section id="kernel-for-an-expression-with-arity-2">
+<h2>Kernel for an expression with arity = 2<a class="headerlink" href="#kernel-for-an-expression-with-arity-2" title="Link to this heading">#</a></h2>
+<p><img alt="title" src="../_images/kernel_arity2_1d.png" />
+<img alt="title" src="../_images/kernel_arity2_2d.png" /></p>
+<section id="computing-contributions-between-test-and-trial-basis-functions-1d">
+<h3>Computing contributions between test and trial basis functions (1D)<a class="headerlink" href="#computing-contributions-between-test-and-trial-basis-functions-1d" title="Link to this heading">#</a></h3>
+<p><img alt="title" src="../_images/contribution_arity2_1d_ex1.png" />
+<img alt="title" src="../_images/contribution_arity2_1d_ex2.png" /></p>
+</section>
+<section id="computing-contributions-between-test-and-trial-basis-functions-2d">
+<h3>Computing contributions between test and trial basis functions (2D)<a class="headerlink" href="#computing-contributions-between-test-and-trial-basis-functions-2d" title="Link to this heading">#</a></h3>
+<p><img alt="title" src="../_images/contribution_arity2_2d_ex1.png" />
+<img alt="title" src="../_images/contribution_arity2_2d_ex2.png" /></p>
+</section>
+</section>
+</section>
+
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization-of-the-poisson-problem">Discretization of the Poisson problem</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#expressions">Expressions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#notations">Notations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#assembly-procedures">Assembly procedures</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#pre-process">Pre-process</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-a-discrete-field-on-a-given-element">Evaluation of a discrete field on a given element</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#kernel-for-an-expression-with-arity-1">Kernel for an expression with arity = 1</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-contributions-of-test-basis-functions-1d">Computing contributions of test basis functions (1D)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-contributions-of-test-basis-functions-2d">Computing contributions of test basis functions (2D)</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#kernel-for-an-expression-with-arity-2">Kernel for an expression with arity = 2</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-contributions-between-test-and-trial-basis-functions-1d">Computing contributions between test and trial basis functions (1D)</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-contributions-between-test-and-trial-basis-functions-2d">Computing contributions between test and trial basis functions (2D)</a></li>
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+                    IGA-Python
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Introduction</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">What to expect from IGA-Python</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="cad.html">Computer Aided Design</a><input class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-1"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2 has-children"><a class="reference internal" href="bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="bezier-curves.html">Bézier curves</a></li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="bsplines.html">B-Splines</a><input class="toctree-checkbox" id="toctree-checkbox-3" name="toctree-checkbox-3" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-3"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="bsplines-curves.html">B-Splines curves</a></li>
+<li class="toctree-l3"><a class="reference internal" href="bsplines-surfaces.html">B-Splines surfaces</a></li>
+<li class="toctree-l3"><a class="reference internal" href="bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
+</li>
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+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="iga.html">Isogeometric Analysis</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="data-structure.html">Data Structure</a></li>
+
+<li class="toctree-l2"><a class="reference internal" href="poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/install.html">Installation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter1/sympde.html">SymPDE concepts and their mathematical meaning</a><input class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-5"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/topology.html">Topology concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/space.html">Function Space concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/sympde.html">SymPDE API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-13" name="toctree-checkbox-13" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-13"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/install.html">Installation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter1/sympde.html">SymPDE concepts and their mathematical meaning</a><input class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-5"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/topology.html">Topology concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/space.html">Function Space concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/sympde.html">SymPDE API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-13" name="toctree-checkbox-13" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-13"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.algebra.html">sympde.core.algebra</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.basic.html">sympde.core.basic</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.core.tests.test_utils.html">sympde.core.tests.test_utils</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.utils.html">sympde.core.utils</a></li>
+</ul>
+</li>
+<li class="toctree-l2"><a class="reference internal" href="../documentation/_autosummary/sympde.calculus.html">sympde.calculus</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.analytical_mapping.html">sympde.topology.analytical_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.basic.html">sympde.topology.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.callable_mapping.html">sympde.topology.callable_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.datatype.html">sympde.topology.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.derivatives.html">sympde.topology.derivatives</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.domain.html">sympde.topology.domain</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.html">sympde.topology.tests</a><input class="toctree-checkbox" id="toctree-checkbox-17" name="toctree-checkbox-17" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-17"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_callable_mapping.html">sympde.topology.tests.test_callable_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_derivatives.html">sympde.topology.tests.test_derivatives</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_field_2d.html">sympde.topology.tests.test_field_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.html">sympde.expr</a><input class="toctree-checkbox" id="toctree-checkbox-18" name="toctree-checkbox-18" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-18"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.basic.html">sympde.expr.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.equation.html">sympde.expr.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_equation_2d.html">sympde.expr.tests.test_equation_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.html">sympde.printing</a><input class="toctree-checkbox" id="toctree-checkbox-20" name="toctree-checkbox-20" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-20"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.latex.html">sympde.printing.latex</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.tests.test_latex.html">sympde.printing.tests.test_latex</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.html">sympde.exterior</a><input class="toctree-checkbox" id="toctree-checkbox-22" name="toctree-checkbox-22" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-22"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.calculus.html">sympde.exterior.calculus</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.form.html">sympde.exterior.form</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_compiler.html">sympde.exterior.tests.test_compiler</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.utilities.html">sympde.utilities</a><input class="toctree-checkbox" id="toctree-checkbox-24" name="toctree-checkbox-24" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-24"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.utilities.utils.html">sympde.utilities.utils</a></li>
+</ul>
+</li>
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+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Performance and Acceleration</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#d-case">1D Case</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#pure-python">Pure Python</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#timing-using-pure-python">Timing using pure Python</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#using-pyccel">Using Pyccel</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#timing">Timing</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">2D case</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Pure Python</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">Timing</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id4">Using Pyccel</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id5">Timing</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercises">Exercises</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="performance-and-acceleration">
+<h1>Performance and Acceleration<a class="headerlink" href="#performance-and-acceleration" title="Link to this heading">#</a></h1>
+<p>In this section, we shall see how to accelerate our Python code assembly and get native speed.
+We will be using <a class="reference external" href="https://numba.pydata.org/">Numba</a> and <a class="reference external" href="https://github.com/pyccel/pyccel">Pyccel</a>.</p>
+<p>The <strong>StencilMatrix</strong> format are based on a negative indexing, which we provide through a syntactic-sugar approach by overiding the <strong><strong>getitem</strong></strong> and <strong><strong>setitem</strong></strong> methods. The entries are in fact stored in the the private attribute <strong>_data</strong> which is a <em>numpy.NdArray</em>. We therefor use a shift by adding <strong>p</strong> as described in the <strong>StencilMatrix</strong> method</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">StencilMatrix</span><span class="p">(</span> <span class="nb">object</span> <span class="p">):</span>
+    <span class="o">...</span>
+    <span class="k">def</span> <span class="nf">_shift_index</span><span class="p">(</span> <span class="n">index</span><span class="p">,</span> <span class="n">shift</span> <span class="p">):</span>
+        <span class="k">if</span> <span class="nb">isinstance</span><span class="p">(</span> <span class="n">index</span><span class="p">,</span> <span class="nb">slice</span> <span class="p">):</span>
+            <span class="n">start</span> <span class="o">=</span> <span class="kc">None</span> <span class="k">if</span> <span class="n">index</span><span class="o">.</span><span class="n">start</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">index</span><span class="o">.</span><span class="n">start</span> <span class="o">+</span> <span class="n">shift</span>
+            <span class="n">stop</span>  <span class="o">=</span> <span class="kc">None</span> <span class="k">if</span> <span class="n">index</span><span class="o">.</span><span class="n">stop</span>  <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">index</span><span class="o">.</span><span class="n">stop</span>  <span class="o">+</span> <span class="n">shift</span>
+            <span class="k">return</span> <span class="nb">slice</span><span class="p">(</span><span class="n">start</span><span class="p">,</span> <span class="n">stop</span><span class="p">,</span> <span class="n">index</span><span class="o">.</span><span class="n">step</span><span class="p">)</span>
+        <span class="k">else</span><span class="p">:</span>
+            <span class="k">return</span> <span class="n">index</span> <span class="o">+</span> <span class="n">shift</span>
+</pre></div>
+</div>
+<p>In order to use a Python accelerator, we will first need to pass the <em>NdArray</em> as argument to the assembly function and not the <strong>StencilMatrix</strong> object.</p>
+<p>Since this will be <em>ugly</em> it is highly recommanded that one creates an <em>interface</em> function that will be calling the <em>core</em> assembly function. It is the later one that we will accelerate using Numba or Pyccel.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psydac.fem.splines</span> <span class="kn">import</span> <span class="n">SplineSpace</span>
+<span class="kn">from</span> <span class="nn">psydac.fem.tensor</span>  <span class="kn">import</span> <span class="n">TensorFemSpace</span>
+<span class="kn">from</span> <span class="nn">psydac.linalg.stencil</span> <span class="kn">import</span> <span class="n">StencilMatrix</span>
+<span class="kn">from</span> <span class="nn">psydac.linalg.stencil</span> <span class="kn">import</span> <span class="n">StencilVector</span>
+<span class="kn">from</span> <span class="nn">psydac.ddm.cart</span> <span class="kn">import</span> <span class="n">DomainDecomposition</span>
+</pre></div>
+</div>
+<section id="d-case">
+<h2>1D Case<a class="headerlink" href="#d-case" title="Link to this heading">#</a></h2>
+<section id="pure-python">
+<h3>Pure Python<a class="headerlink" href="#pure-python" title="Link to this heading">#</a></h3>
+<p>The novelties here are</p>
+<ul class="simple">
+<li><p>we add Python Typing Syntax to our function</p></li>
+<li><p>we use the matrix as <strong>inout</strong> argument, in the spirit of Fortran. This means that we shall have a procedure and not a function.</p></li>
+<li><p>we shift the access to memory when storing the Matrix entries</p></li>
+</ul>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">assemble_stiffness_1d_pure</span><span class="p">(</span><span class="n">nk1</span><span class="p">:</span> <span class="s1">&#39;int&#39;</span><span class="p">,</span> 
+                               <span class="n">p1</span><span class="p">:</span> <span class="s1">&#39;int&#39;</span><span class="p">,</span>
+                               <span class="n">nq1</span><span class="p">:</span> <span class="s1">&#39;int&#39;</span><span class="p">,</span> 
+                               <span class="n">spans_1</span><span class="p">:</span> <span class="s1">&#39;int[:]&#39;</span><span class="p">,</span> 
+                               <span class="n">basis_1</span><span class="p">:</span> <span class="s1">&#39;double[:,:,:,:]&#39;</span><span class="p">,</span> 
+                               <span class="n">weights_1</span><span class="p">:</span> <span class="s1">&#39;double[:,:]&#39;</span><span class="p">,</span> 
+                               <span class="n">points_1</span><span class="p">:</span> <span class="s1">&#39;double[:,:]&#39;</span><span class="p">,</span> 
+                               <span class="n">matrix</span><span class="p">:</span> <span class="s1">&#39;double[:,:]&#39;</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    assembling the stiffness matrix using stencil forms</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># ... build matrices</span>
+    <span class="k">for</span> <span class="n">ie1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">nk1</span><span class="p">):</span>
+        <span class="n">i_span_1</span> <span class="o">=</span> <span class="n">spans_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">]</span>
+        <span class="k">for</span> <span class="n">il_1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">p1</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+            <span class="k">for</span> <span class="n">jl_1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">p1</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+                <span class="n">i1</span> <span class="o">=</span> <span class="n">i_span_1</span> <span class="o">-</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">il_1</span>
+                <span class="n">j1</span> <span class="o">=</span> <span class="n">i_span_1</span> <span class="o">-</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">jl_1</span>
+
+                <span class="n">v</span> <span class="o">=</span> <span class="mf">0.0</span>
+                <span class="k">for</span> <span class="n">g1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">nq1</span><span class="p">):</span>
+                    <span class="n">bi_0</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">il_1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>
+                    <span class="n">bi_x</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">il_1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>
+
+                    <span class="n">bj_0</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">jl_1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>
+                    <span class="n">bj_x</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">jl_1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>
+
+                    <span class="n">wvol</span> <span class="o">=</span> <span class="n">weights_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>
+
+                    <span class="n">v</span> <span class="o">+=</span> <span class="p">(</span><span class="n">bi_x</span> <span class="o">*</span> <span class="n">bj_x</span><span class="p">)</span> <span class="o">*</span> <span class="n">wvol</span>
+
+                <span class="c1"># we shift the test index by p1</span>
+                <span class="n">matrix</span><span class="p">[</span><span class="n">i1</span><span class="p">,</span> <span class="n">p1</span><span class="o">+</span><span class="n">j1</span><span class="o">-</span><span class="n">i1</span><span class="p">]</span>  <span class="o">+=</span> <span class="n">v</span>
+    <span class="c1"># ...</span>
+
+    <span class="c1"># NOTE: we will not return the matrix. </span>
+    <span class="c1">#       explainations will come later</span>
+    <span class="c1">#return matrix</span>
+</pre></div>
+</div>
+<section id="timing-using-pure-python">
+<h4>Timing using pure Python<a class="headerlink" href="#timing-using-pure-python" title="Link to this heading">#</a></h4>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="n">xmin</span> <span class="o">=</span> <span class="mf">0.</span> <span class="p">;</span> <span class="n">xmax</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">nelements</span> <span class="o">=</span> <span class="mi">400</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">grid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">nelements</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
+<span class="n">V</span> <span class="o">=</span> <span class="n">SplineSpace</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid</span><span class="p">)</span>
+<span class="n">dd</span> <span class="o">=</span> <span class="n">DomainDecomposition</span><span class="p">(</span><span class="n">ncells</span><span class="o">=</span><span class="p">[</span><span class="n">V</span><span class="o">.</span><span class="n">ncells</span><span class="p">],</span> <span class="n">periods</span><span class="o">=</span><span class="p">[</span><span class="kc">False</span><span class="p">])</span>
+<span class="n">V</span> <span class="o">=</span> <span class="n">TensorFemSpace</span><span class="p">(</span><span class="n">dd</span><span class="p">,</span> <span class="n">V</span><span class="p">)</span>
+
+<span class="n">M</span> <span class="o">=</span> <span class="n">StencilMatrix</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">,</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">)</span>
+
+<span class="c1"># Sizes</span>
+<span class="p">[</span><span class="n">s1</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">starts</span>
+<span class="p">[</span><span class="n">e1</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">ends</span>
+<span class="p">[</span><span class="n">p1</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">pads</span>
+
+<span class="c1"># Quadrature data</span>
+<span class="n">nk1</span>       <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">num_elements</span>
+<span class="n">nq1</span>       <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">num_quad_pts</span>
+<span class="n">spans_1</span>   <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">spans</span>
+<span class="n">basis_1</span>   <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">basis</span>
+<span class="n">points_1</span>  <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">points</span>
+<span class="n">weights_1</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">weights</span>
+
+<span class="o">%</span><span class="n">timeit</span> <span class="n">assemble_stiffness_1d_pure</span><span class="p">(</span> <span class="n">nk1</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">nq1</span><span class="p">,</span> <span class="n">spans_1</span><span class="p">,</span> <span class="n">basis_1</span><span class="p">,</span> <span class="n">weights_1</span><span class="p">,</span> <span class="n">points_1</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">_data</span> <span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>13.2 ms ± 126 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
+</pre></div>
+</div>
+</section>
+</section>
+<section id="using-pyccel">
+<h3>Using Pyccel<a class="headerlink" href="#using-pyccel" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">pyccel.epyccel</span> <span class="kn">import</span> <span class="n">epyccel</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">assemble_stiffness_1d_pyccel</span> <span class="o">=</span> <span class="n">epyccel</span><span class="p">(</span><span class="n">assemble_stiffness_1d_pure</span><span class="p">)</span>
+</pre></div>
+</div>
+<section id="timing">
+<h4>Timing<a class="headerlink" href="#timing" title="Link to this heading">#</a></h4>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="n">xmin</span> <span class="o">=</span> <span class="mf">0.</span> <span class="p">;</span> <span class="n">xmax</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">nelements</span> <span class="o">=</span> <span class="mi">400</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">grid</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">xmin</span><span class="p">,</span> <span class="n">xmax</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">nelements</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
+<span class="n">V</span> <span class="o">=</span> <span class="n">SplineSpace</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid</span><span class="p">)</span>
+<span class="n">dd</span> <span class="o">=</span> <span class="n">DomainDecomposition</span><span class="p">(</span><span class="n">ncells</span><span class="o">=</span><span class="p">[</span><span class="n">V</span><span class="o">.</span><span class="n">ncells</span><span class="p">],</span> <span class="n">periods</span><span class="o">=</span><span class="p">[</span><span class="kc">False</span><span class="p">])</span>
+<span class="n">V</span> <span class="o">=</span> <span class="n">TensorFemSpace</span><span class="p">(</span><span class="n">dd</span><span class="p">,</span> <span class="n">V</span><span class="p">)</span>
+
+<span class="n">M</span> <span class="o">=</span> <span class="n">StencilMatrix</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">,</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">)</span>
+
+<span class="c1"># Sizes</span>
+<span class="p">[</span><span class="n">s1</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">starts</span>
+<span class="p">[</span><span class="n">e1</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">ends</span>
+<span class="p">[</span><span class="n">p1</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">pads</span>
+
+<span class="c1"># Quadrature data</span>
+<span class="n">nk1</span>       <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">num_elements</span>
+<span class="n">nq1</span>       <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">num_quad_pts</span>
+<span class="n">spans_1</span>   <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">spans</span>
+<span class="n">basis_1</span>   <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">basis</span>
+<span class="n">points_1</span>  <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">points</span>
+<span class="n">weights_1</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">weights</span>
+
+<span class="o">%</span><span class="n">timeit</span> <span class="n">assemble_stiffness_1d_pyccel</span><span class="p">(</span> <span class="n">nk1</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">nq1</span><span class="p">,</span> <span class="n">spans_1</span><span class="p">,</span> <span class="n">basis_1</span><span class="p">,</span> <span class="n">weights_1</span><span class="p">,</span> <span class="n">points_1</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">_data</span> <span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>12.2 µs ± 139 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
+</pre></div>
+</div>
+</section>
+</section>
+</section>
+<section id="id1">
+<h2>2D case<a class="headerlink" href="#id1" title="Link to this heading">#</a></h2>
+<section id="id2">
+<h3>Pure Python<a class="headerlink" href="#id2" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">assemble_stiffness_2d_pure</span><span class="p">(</span><span class="n">nk1</span><span class="p">:</span> <span class="s1">&#39;int&#39;</span><span class="p">,</span> <span class="n">nk2</span><span class="p">:</span> <span class="s1">&#39;int&#39;</span><span class="p">,</span> 
+                               <span class="n">p1</span><span class="p">:</span>  <span class="s1">&#39;int&#39;</span><span class="p">,</span> <span class="n">p2</span><span class="p">:</span>  <span class="s1">&#39;int&#39;</span><span class="p">,</span> 
+                               <span class="n">nq1</span><span class="p">:</span>  <span class="s1">&#39;int&#39;</span><span class="p">,</span> <span class="n">nq2</span><span class="p">:</span>  <span class="s1">&#39;int&#39;</span><span class="p">,</span> 
+                               <span class="n">spans_1</span><span class="p">:</span>          <span class="s1">&#39;int[:]&#39;</span><span class="p">,</span> <span class="n">spans_2</span><span class="p">:</span>          <span class="s1">&#39;int[:]&#39;</span><span class="p">,</span> 
+                               <span class="n">basis_1</span><span class="p">:</span> <span class="s1">&#39;double[:,:,:,:]&#39;</span><span class="p">,</span> <span class="n">basis_2</span><span class="p">:</span> <span class="s1">&#39;double[:,:,:,:]&#39;</span><span class="p">,</span> 
+                               <span class="n">weights_1</span><span class="p">:</span>   <span class="s1">&#39;double[:,:]&#39;</span><span class="p">,</span> <span class="n">weights_2</span><span class="p">:</span>   <span class="s1">&#39;double[:,:]&#39;</span><span class="p">,</span> 
+                               <span class="n">points_1</span><span class="p">:</span>    <span class="s1">&#39;double[:,:]&#39;</span><span class="p">,</span> <span class="n">points_2</span><span class="p">:</span>    <span class="s1">&#39;double[:,:]&#39;</span><span class="p">,</span> 
+                               <span class="n">matrix</span><span class="p">:</span> <span class="s1">&#39;double[:,:,:,:]&#39;</span><span class="p">):</span>
+<span class="w">    </span><span class="sd">&quot;&quot;&quot;</span>
+<span class="sd">    assembling the stiffness matrix using stencil forms</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+
+    <span class="c1"># ... build matrices</span>
+    <span class="k">for</span> <span class="n">ie1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">nk1</span><span class="p">):</span>
+        <span class="n">i_span_1</span> <span class="o">=</span> <span class="n">spans_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">]</span>
+        <span class="k">for</span> <span class="n">ie2</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">nk2</span><span class="p">):</span>
+            <span class="n">i_span_2</span> <span class="o">=</span> <span class="n">spans_2</span><span class="p">[</span><span class="n">ie2</span><span class="p">]</span>
+            <span class="c1"># evaluation dependant uniquement de l&#39;element</span>
+
+            <span class="k">for</span> <span class="n">il_1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">p1</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+                <span class="k">for</span> <span class="n">il_2</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">p2</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+                    <span class="k">for</span> <span class="n">jl_1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">p1</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+                        <span class="k">for</span> <span class="n">jl_2</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">p2</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+                            <span class="n">i1</span> <span class="o">=</span> <span class="n">i_span_1</span> <span class="o">-</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">il_1</span>
+                            <span class="n">j1</span> <span class="o">=</span> <span class="n">i_span_1</span> <span class="o">-</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">jl_1</span>
+
+                            <span class="n">i2</span> <span class="o">=</span> <span class="n">i_span_2</span> <span class="o">-</span> <span class="n">p2</span> <span class="o">+</span> <span class="n">il_2</span>
+                            <span class="n">j2</span> <span class="o">=</span> <span class="n">i_span_2</span> <span class="o">-</span> <span class="n">p2</span> <span class="o">+</span> <span class="n">jl_2</span>
+
+                            <span class="n">v</span> <span class="o">=</span> <span class="mf">0.0</span>
+                            <span class="k">for</span> <span class="n">g1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">nq1</span><span class="p">):</span>
+                                <span class="k">for</span> <span class="n">g2</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">nq2</span><span class="p">):</span>
+                                    <span class="n">bi_0</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">il_1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span> <span class="o">*</span> <span class="n">basis_2</span><span class="p">[</span><span class="n">ie2</span><span class="p">,</span> <span class="n">il_2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g2</span><span class="p">]</span>
+                                    <span class="n">bi_x</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">il_1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span> <span class="o">*</span> <span class="n">basis_2</span><span class="p">[</span><span class="n">ie2</span><span class="p">,</span> <span class="n">il_2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g2</span><span class="p">]</span>
+                                    <span class="n">bi_y</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">il_1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span> <span class="o">*</span> <span class="n">basis_2</span><span class="p">[</span><span class="n">ie2</span><span class="p">,</span> <span class="n">il_2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">g2</span><span class="p">]</span>
+
+                                    <span class="n">bj_0</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">jl_1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span> <span class="o">*</span> <span class="n">basis_2</span><span class="p">[</span><span class="n">ie2</span><span class="p">,</span> <span class="n">jl_2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g2</span><span class="p">]</span>
+                                    <span class="n">bj_x</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">jl_1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span> <span class="o">*</span> <span class="n">basis_2</span><span class="p">[</span><span class="n">ie2</span><span class="p">,</span> <span class="n">jl_2</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g2</span><span class="p">]</span>
+                                    <span class="n">bj_y</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">jl_1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span> <span class="o">*</span> <span class="n">basis_2</span><span class="p">[</span><span class="n">ie2</span><span class="p">,</span> <span class="n">jl_2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">g2</span><span class="p">]</span>
+
+                                    <span class="n">wvol</span> <span class="o">=</span> <span class="n">weights_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span> <span class="o">*</span> <span class="n">weights_2</span><span class="p">[</span><span class="n">ie2</span><span class="p">,</span> <span class="n">g2</span><span class="p">]</span>
+
+                                    <span class="n">v</span> <span class="o">+=</span> <span class="p">(</span><span class="n">bi_x</span> <span class="o">*</span> <span class="n">bj_x</span> <span class="o">+</span> <span class="n">bi_y</span> <span class="o">*</span> <span class="n">bj_y</span><span class="p">)</span> <span class="o">*</span> <span class="n">wvol</span>
+
+                            <span class="n">matrix</span><span class="p">[</span><span class="n">i1</span><span class="p">,</span> <span class="n">i2</span><span class="p">,</span> <span class="n">p1</span><span class="o">+</span><span class="n">j1</span><span class="o">-</span><span class="n">i1</span><span class="p">,</span> <span class="n">p2</span><span class="o">+</span><span class="n">j2</span><span class="o">-</span><span class="n">i2</span><span class="p">]</span>  <span class="o">+=</span> <span class="n">v</span>
+    <span class="c1"># ...</span>
+</pre></div>
+</div>
+<section id="id3">
+<h4>Timing<a class="headerlink" href="#id3" title="Link to this heading">#</a></h4>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the spline space for each direction</span>
+<span class="n">x1min</span> <span class="o">=</span> <span class="mf">0.</span> <span class="p">;</span> <span class="n">x1max</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">nelements1</span> <span class="o">=</span> <span class="mi">32</span>
+<span class="n">degree1</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">grid1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">x1min</span><span class="p">,</span> <span class="n">x1max</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">nelements1</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
+<span class="n">V1</span> <span class="o">=</span> <span class="n">SplineSpace</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree1</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid1</span><span class="p">)</span>
+
+<span class="n">x2min</span> <span class="o">=</span> <span class="mf">0.</span> <span class="p">;</span> <span class="n">x2max</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">nelements2</span> <span class="o">=</span> <span class="mi">32</span>
+<span class="n">degree2</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">grid2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">x2min</span><span class="p">,</span> <span class="n">x2max</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">nelements2</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
+<span class="n">V2</span> <span class="o">=</span> <span class="n">SplineSpace</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree2</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid2</span><span class="p">)</span>
+
+<span class="c1"># create the tensor space</span>
+<span class="n">dd</span> <span class="o">=</span> <span class="n">DomainDecomposition</span><span class="p">(</span><span class="n">ncells</span><span class="o">=</span><span class="p">[</span><span class="n">V1</span><span class="o">.</span><span class="n">ncells</span><span class="p">,</span> <span class="n">V2</span><span class="o">.</span><span class="n">ncells</span><span class="p">],</span> <span class="n">periods</span><span class="o">=</span><span class="p">[</span><span class="kc">False</span><span class="p">,</span> <span class="kc">False</span><span class="p">])</span>
+<span class="n">V</span> <span class="o">=</span> <span class="n">TensorFemSpace</span><span class="p">(</span><span class="n">dd</span><span class="p">,</span> <span class="n">V1</span><span class="p">,</span> <span class="n">V2</span><span class="p">)</span>
+
+<span class="n">M</span> <span class="o">=</span> <span class="n">StencilMatrix</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">,</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">)</span>
+
+<span class="c1"># Sizes</span>
+<span class="p">[</span><span class="n">s1</span><span class="p">,</span> <span class="n">s2</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">starts</span>
+<span class="p">[</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">ends</span>
+<span class="p">[</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">pads</span>
+
+<span class="c1"># Quadrature data</span>
+<span class="p">[</span>      <span class="n">nk1</span><span class="p">,</span>       <span class="n">nk2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">num_elements</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+<span class="p">[</span>      <span class="n">nq1</span><span class="p">,</span>       <span class="n">nq2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">num_quad_pts</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+<span class="p">[</span>  <span class="n">spans_1</span><span class="p">,</span>   <span class="n">spans_2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">spans</span>        <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+<span class="p">[</span>  <span class="n">basis_1</span><span class="p">,</span>   <span class="n">basis_2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">basis</span>        <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+<span class="p">[</span> <span class="n">points_1</span><span class="p">,</span>  <span class="n">points_2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">points</span>       <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+<span class="p">[</span><span class="n">weights_1</span><span class="p">,</span> <span class="n">weights_2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">weights</span>      <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+
+<span class="o">%</span><span class="n">timeit</span> <span class="n">assemble_stiffness_2d_pure</span><span class="p">(</span> <span class="n">nk1</span><span class="p">,</span> <span class="n">nk2</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">nq1</span><span class="p">,</span> <span class="n">nq2</span><span class="p">,</span> <span class="n">spans_1</span><span class="p">,</span> <span class="n">spans_2</span><span class="p">,</span> <span class="n">basis_1</span><span class="p">,</span> <span class="n">basis_2</span><span class="p">,</span> <span class="n">weights_1</span><span class="p">,</span> <span class="n">weights_2</span><span class="p">,</span> <span class="n">points_1</span><span class="p">,</span> <span class="n">points_2</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">_data</span> <span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>5.97 s ± 491 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
+</pre></div>
+</div>
+</section>
+</section>
+<section id="id4">
+<h3>Using Pyccel<a class="headerlink" href="#id4" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">assemble_stiffness_2d_pyccel</span> <span class="o">=</span> <span class="n">epyccel</span><span class="p">(</span><span class="n">assemble_stiffness_2d_pure</span><span class="p">)</span>
+</pre></div>
+</div>
+<section id="id5">
+<h4>Timing<a class="headerlink" href="#id5" title="Link to this heading">#</a></h4>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the spline space for each direction</span>
+<span class="n">x1min</span> <span class="o">=</span> <span class="mf">0.</span> <span class="p">;</span> <span class="n">x1max</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">nelements1</span> <span class="o">=</span> <span class="mi">32</span>
+<span class="n">degree1</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">grid1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">x1min</span><span class="p">,</span> <span class="n">x1max</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">nelements1</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
+<span class="n">V1</span> <span class="o">=</span> <span class="n">SplineSpace</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree1</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid1</span><span class="p">)</span>
+
+<span class="n">x2min</span> <span class="o">=</span> <span class="mf">0.</span> <span class="p">;</span> <span class="n">x2max</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">nelements2</span> <span class="o">=</span> <span class="mi">32</span>
+<span class="n">degree2</span> <span class="o">=</span> <span class="mi">3</span>
+<span class="n">grid2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">x2min</span><span class="p">,</span> <span class="n">x2max</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">nelements2</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
+<span class="n">V2</span> <span class="o">=</span> <span class="n">SplineSpace</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree2</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid2</span><span class="p">)</span>
+
+<span class="c1"># create the tensor space</span>
+<span class="n">dd</span> <span class="o">=</span> <span class="n">DomainDecomposition</span><span class="p">(</span><span class="n">ncells</span><span class="o">=</span><span class="p">[</span><span class="n">V1</span><span class="o">.</span><span class="n">ncells</span><span class="p">,</span> <span class="n">V2</span><span class="o">.</span><span class="n">ncells</span><span class="p">],</span> <span class="n">periods</span><span class="o">=</span><span class="p">[</span><span class="kc">False</span><span class="p">,</span> <span class="kc">False</span><span class="p">])</span>
+<span class="n">V</span> <span class="o">=</span> <span class="n">TensorFemSpace</span><span class="p">(</span><span class="n">dd</span><span class="p">,</span> <span class="n">V1</span><span class="p">,</span> <span class="n">V2</span><span class="p">)</span>
+
+<span class="n">M</span> <span class="o">=</span> <span class="n">StencilMatrix</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">,</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">)</span>
+
+<span class="c1"># Sizes</span>
+<span class="p">[</span><span class="n">s1</span><span class="p">,</span> <span class="n">s2</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">starts</span>
+<span class="p">[</span><span class="n">e1</span><span class="p">,</span> <span class="n">e2</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">ends</span>
+<span class="p">[</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">]</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">pads</span>
+
+<span class="c1"># Quadrature data</span>
+<span class="p">[</span>      <span class="n">nk1</span><span class="p">,</span>       <span class="n">nk2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">num_elements</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+<span class="p">[</span>      <span class="n">nq1</span><span class="p">,</span>       <span class="n">nq2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">num_quad_pts</span> <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+<span class="p">[</span>  <span class="n">spans_1</span><span class="p">,</span>   <span class="n">spans_2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">spans</span>        <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+<span class="p">[</span>  <span class="n">basis_1</span><span class="p">,</span>   <span class="n">basis_2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">basis</span>        <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+<span class="p">[</span> <span class="n">points_1</span><span class="p">,</span>  <span class="n">points_2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">points</span>       <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+<span class="p">[</span><span class="n">weights_1</span><span class="p">,</span> <span class="n">weights_2</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">g</span><span class="o">.</span><span class="n">weights</span>      <span class="k">for</span> <span class="n">g</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">quad_grids</span><span class="p">()]</span>
+
+<span class="o">%</span><span class="n">timeit</span> <span class="n">assemble_stiffness_2d_pyccel</span><span class="p">(</span> <span class="n">nk1</span><span class="p">,</span> <span class="n">nk2</span><span class="p">,</span> <span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">nq1</span><span class="p">,</span> <span class="n">nq2</span><span class="p">,</span> <span class="n">spans_1</span><span class="p">,</span> <span class="n">spans_2</span><span class="p">,</span> <span class="n">basis_1</span><span class="p">,</span> <span class="n">basis_2</span><span class="p">,</span> <span class="n">weights_1</span><span class="p">,</span> <span class="n">weights_2</span><span class="p">,</span> <span class="n">points_1</span><span class="p">,</span> <span class="n">points_2</span><span class="p">,</span> <span class="n">M</span><span class="o">.</span><span class="n">_data</span> <span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>3.56 ms ± 127 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
+</pre></div>
+</div>
+</section>
+</section>
+</section>
+<section id="exercises">
+<h2>Exercises<a class="headerlink" href="#exercises" title="Link to this heading">#</a></h2>
+<ol class="arabic simple">
+<li><p>Write the accelerated version of the rhs assembly produce using Pyccel and Numba and compute their timing.</p></li>
+<li><p>Perform a benchmark between Pyccel and Numba while varying the B-Spline degree.</p></li>
+</ol>
+</section>
+</section>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#pure-python">Pure Python</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#timing-using-pure-python">Timing using pure Python</a></li>
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+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id4">Using Pyccel</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id5">Timing</a></li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+    <h1>B-splines FEM solver for Poisson equation (1D)</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analytical-solution">Analytical solution</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-spline-space-class">The Spline Space class</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-profile">Matrix profile</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#next-step">Next step</a></li>
+</ul>
+</li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-1">Exercise 1.</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-2">Exercise 2.</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-3">Exercise 3.</a></li>
+</ul>
+</li>
+</ul>
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+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="b-splines-fem-solver-for-poisson-equation-1d">
+<h1>B-splines FEM solver for Poisson equation (1D)<a class="headerlink" href="#b-splines-fem-solver-for-poisson-equation-1d" title="Link to this heading">#</a></h1>
+<p>Following the previous <a class="reference external" href="http://nbviewer.jupyter.org/github/ratnania/IGA-Python/blob/main/lessons/Chapter1/01_introduction_fem.ipynb">section</a>, we implement here a B-Splines FEM for the Poisson problem in 1D, with homogeneous boundary conditions.</p>
+<p>Appropriate data-structures for matrices and vectors, involved when assembling a bilinear and linear form, will be discussed in the next section.</p>
+<p>We consider the following Poisson problem in 1D:
+$<span class="math notranslate nohighlight">\(
+-u^{\prime \prime} = f, \quad (0,1)
+\\
+u(0) = 0  ~~ \mbox{and} ~~ u(1) = 0
+\)</span>$</p>
+<p>Using the algorithms developped in section 1., for assembly a bilinear form and a linear form, we see that we need to assemble</p>
+<ul class="simple">
+<li><p>The Stiffness matrix</p></li>
+<li><p>The right hand side (rhs)</p></li>
+</ul>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># needed imports</span>
+<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">ones</span><span class="p">,</span> <span class="n">linspace</span><span class="p">,</span> <span class="n">zeros_like</span><span class="p">,</span> <span class="n">asarray</span>
+<span class="kn">from</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">import</span> <span class="n">plot</span><span class="p">,</span> <span class="n">show</span>
+<span class="o">%</span><span class="n">matplotlib</span> <span class="n">inline</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># ... assembling the stiffness matrix using stencil forms</span>
+<span class="k">def</span> <span class="nf">assemble_stiffness</span><span class="p">(</span><span class="n">nelements</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">spans</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="n">weights</span><span class="p">,</span> <span class="n">points</span><span class="p">,</span> <span class="n">matrix</span><span class="p">):</span>
+
+    <span class="c1"># ... sizes</span>
+    <span class="n">ne1</span>       <span class="o">=</span> <span class="n">nelements</span>
+    <span class="n">p1</span>        <span class="o">=</span> <span class="n">degree</span>
+    <span class="n">spans_1</span>   <span class="o">=</span> <span class="n">spans</span>
+    <span class="n">basis_1</span>   <span class="o">=</span> <span class="n">basis</span>
+    <span class="n">weights_1</span> <span class="o">=</span> <span class="n">weights</span>
+    <span class="n">points_1</span>  <span class="o">=</span> <span class="n">points</span>
+    
+    <span class="n">k1</span> <span class="o">=</span> <span class="n">weights</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+    <span class="c1"># ...</span>
+
+    <span class="c1"># ... build matrices</span>
+    <span class="k">for</span> <span class="n">ie1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">ne1</span><span class="p">):</span>
+        <span class="n">i_span_1</span> <span class="o">=</span> <span class="n">spans_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">]</span>
+        <span class="k">for</span> <span class="n">il_1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">p1</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+            <span class="k">for</span> <span class="n">jl_1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">p1</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+                <span class="n">i1</span> <span class="o">=</span> <span class="n">i_span_1</span> <span class="o">-</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">il_1</span>
+                <span class="n">j1</span> <span class="o">=</span> <span class="n">i_span_1</span> <span class="o">-</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">jl_1</span>
+
+                <span class="n">v</span> <span class="o">=</span> <span class="mf">0.0</span>
+                <span class="k">for</span> <span class="n">g1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">k1</span><span class="p">):</span>
+                    <span class="n">bi_0</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">il_1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>
+                    <span class="n">bi_x</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">il_1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>                    
+
+                    <span class="n">bj_0</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">jl_1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>
+                    <span class="n">bj_x</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">jl_1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>                    
+
+                    <span class="n">wvol</span> <span class="o">=</span> <span class="n">weights_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>
+
+                    <span class="n">v</span> <span class="o">+=</span> <span class="p">(</span><span class="n">bi_x</span> <span class="o">*</span> <span class="n">bj_x</span><span class="p">)</span> <span class="o">*</span> <span class="n">wvol</span>
+
+                <span class="n">matrix</span><span class="p">[</span><span class="n">i1</span><span class="p">,</span> <span class="n">j1</span><span class="p">]</span>  <span class="o">+=</span> <span class="n">v</span>
+    <span class="c1"># ...</span>
+
+    <span class="k">return</span> <span class="n">matrix</span>    
+<span class="c1"># ...</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># ... Assembly procedure for the rhs</span>
+<span class="k">def</span> <span class="nf">assemble_vector</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">nelements</span><span class="p">,</span> <span class="n">degree</span><span class="p">,</span> <span class="n">spans</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="n">weights</span><span class="p">,</span> <span class="n">points</span><span class="p">,</span> <span class="n">rhs</span><span class="p">):</span>
+
+    <span class="c1"># ... sizes</span>
+    <span class="n">ne1</span>       <span class="o">=</span> <span class="n">nelements</span>
+    <span class="n">p1</span>        <span class="o">=</span> <span class="n">degree</span>
+    <span class="n">spans_1</span>   <span class="o">=</span> <span class="n">spans</span>
+    <span class="n">basis_1</span>   <span class="o">=</span> <span class="n">basis</span>
+    <span class="n">weights_1</span> <span class="o">=</span> <span class="n">weights</span>
+    <span class="n">points_1</span>  <span class="o">=</span> <span class="n">points</span>
+    
+    <span class="n">k1</span> <span class="o">=</span> <span class="n">weights</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
+    <span class="c1"># ...</span>
+
+    <span class="c1"># ... build rhs</span>
+    <span class="k">for</span> <span class="n">ie1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">ne1</span><span class="p">):</span>
+        <span class="n">i_span_1</span> <span class="o">=</span> <span class="n">spans_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">]</span>
+        <span class="k">for</span> <span class="n">il_1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">p1</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
+            <span class="n">i1</span> <span class="o">=</span> <span class="n">i_span_1</span> <span class="o">-</span> <span class="n">p1</span> <span class="o">+</span> <span class="n">il_1</span>
+
+            <span class="n">v</span> <span class="o">=</span> <span class="mf">0.0</span>
+            <span class="k">for</span> <span class="n">g1</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">k1</span><span class="p">):</span>
+                <span class="n">bi_0</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">il_1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>
+                <span class="n">bi_x</span> <span class="o">=</span> <span class="n">basis_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">il_1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>  
+
+                <span class="n">x1</span>    <span class="o">=</span> <span class="n">points_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>
+                <span class="n">wvol</span>  <span class="o">=</span> <span class="n">weights_1</span><span class="p">[</span><span class="n">ie1</span><span class="p">,</span> <span class="n">g1</span><span class="p">]</span>
+
+                <span class="n">v</span> <span class="o">+=</span> <span class="n">bi_0</span> <span class="o">*</span> <span class="n">f</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span> <span class="o">*</span> <span class="n">wvol</span>
+
+            <span class="n">rhs</span><span class="p">[</span><span class="n">i1</span><span class="p">]</span> <span class="o">+=</span> <span class="n">v</span>
+    <span class="c1"># ...</span>
+
+    <span class="c1"># ...</span>
+    <span class="k">return</span> <span class="n">rhs</span>
+    <span class="c1"># ...</span>
+<span class="c1"># ...</span>
+</pre></div>
+</div>
+<hr class="docutils" />
+<p><strong>Note</strong>:
+We added the function <strong>f</strong> as an argument of the <em>assemble_rhs</em> function. In general, we will avoid calling a function over the quadrature points for performance reasons.</p>
+<hr class="docutils" />
+<p>We will need the following imports from the <strong>psydac</strong> library.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># nedded imports</span>
+<span class="kn">from</span> <span class="nn">psydac.core.bsplines</span> <span class="kn">import</span> <span class="n">elements_spans</span>  <span class="c1"># computes the span for each element</span>
+<span class="kn">from</span> <span class="nn">psydac.core.bsplines</span> <span class="kn">import</span> <span class="n">make_knots</span>      <span class="c1"># create a knot sequence from a grid</span>
+<span class="kn">from</span> <span class="nn">psydac.core.bsplines</span> <span class="kn">import</span> <span class="n">quadrature_grid</span> <span class="c1"># create a quadrature rule over the whole 1d grid</span>
+<span class="kn">from</span> <span class="nn">psydac.core.bsplines</span> <span class="kn">import</span> <span class="n">basis_ders_on_quad_grid</span> <span class="c1"># evaluates all bsplines and their derivatives on the quad grid</span>
+<span class="kn">from</span> <span class="nn">psydac.utilities.quadratures</span> <span class="kn">import</span> <span class="n">gauss_legendre</span> <span class="c1"># computes gauss legendre quadrature points &amp; weights</span>
+<span class="kn">from</span> <span class="nn">utilities.plot</span> <span class="kn">import</span> <span class="n">plot_field_1d</span> <span class="c1"># plot a solution for 1d problems</span>
+</pre></div>
+</div>
+<p>Now, let’s create a B-spline Finite Element Space <span class="math notranslate nohighlight">\(V\)</span> over a grid of <span class="math notranslate nohighlight">\(32\)</span> elements, and using <strong>cubic</strong> <em>B-splines</em>.</p>
+<p>A B-Spline space is defined here through a grid, we then call the function <strong>make_knots</strong> to create an open knot sequence. We finally, compute the spans for every element, to avoid doing it for each quadrature point on the current element, in the assembly procedure.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">p</span>  <span class="o">=</span> <span class="mi">3</span>    <span class="c1"># spline degree</span>
+<span class="n">ne</span> <span class="o">=</span> <span class="mi">32</span>   <span class="c1"># number of elements</span>
+
+<span class="n">grid</span>  <span class="o">=</span> <span class="n">linspace</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="n">ne</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
+<span class="n">knots</span> <span class="o">=</span> <span class="n">make_knots</span><span class="p">(</span><span class="n">grid</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">periodic</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
+<span class="n">spans</span> <span class="o">=</span> <span class="n">elements_spans</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>  
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">nelements</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">grid</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
+<span class="n">nbasis</span>    <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">knots</span><span class="p">)</span> <span class="o">-</span> <span class="n">p</span> <span class="o">-</span> <span class="mi">1</span>
+
+<span class="c1"># we need the value a B-Spline and its first derivative</span>
+<span class="n">nderiv</span> <span class="o">=</span> <span class="mi">1</span>
+
+<span class="c1"># create the gauss-legendre rule, on [-1, 1]</span>
+<span class="n">u</span><span class="p">,</span> <span class="n">w</span> <span class="o">=</span> <span class="n">gauss_legendre</span><span class="p">(</span> <span class="n">p</span> <span class="p">)</span>
+
+<span class="c1"># for each element on the grid, we create a local quadrature grid</span>
+<span class="n">points</span><span class="p">,</span> <span class="n">weights</span> <span class="o">=</span> <span class="n">quadrature_grid</span><span class="p">(</span> <span class="n">grid</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">w</span> <span class="p">)</span>
+
+<span class="c1"># for each element and a quadrature points, </span>
+<span class="c1"># we compute the non-vanishing B-Splines</span>
+<span class="n">basis</span> <span class="o">=</span> <span class="n">basis_ders_on_quad_grid</span><span class="p">(</span> <span class="n">knots</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">points</span><span class="p">,</span> <span class="n">nderiv</span><span class="p">,</span> <span class="n">normalization</span><span class="o">=</span><span class="kc">False</span> <span class="p">)</span>
+</pre></div>
+</div>
+<p>Assembling the <strong>Stiffness</strong> matrix is then done using</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">stiffness</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">((</span><span class="n">nbasis</span><span class="p">,</span> <span class="n">nbasis</span><span class="p">))</span>
+<span class="n">stiffness</span> <span class="o">=</span> <span class="n">assemble_stiffness</span><span class="p">(</span><span class="n">nelements</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">spans</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="n">weights</span><span class="p">,</span> <span class="n">points</span><span class="p">,</span> <span class="n">matrix</span><span class="o">=</span><span class="n">stiffness</span><span class="p">)</span>
+</pre></div>
+</div>
+<section id="analytical-solution">
+<h2>Analytical solution<a class="headerlink" href="#analytical-solution" title="Link to this heading">#</a></h2>
+<p>In finite elements, one of the first tests we do is to check polynomial solutions of with a degree that smaller or equal to the one we chose for our discretization.
+Here, we take the analytical solution <span class="math notranslate nohighlight">\(u(x) = x(1-x)\)</span> which leads to the rhs <span class="math notranslate nohighlight">\(f(x) = - \partial_{xx} u(x) = 2\)</span>.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="mf">2.</span>    
+<span class="n">rhs</span> <span class="o">=</span> <span class="n">zeros</span><span class="p">(</span><span class="n">nbasis</span><span class="p">)</span>
+<span class="n">rhs</span> <span class="o">=</span> <span class="n">assemble_vector</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">nelements</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">spans</span><span class="p">,</span> <span class="n">basis</span><span class="p">,</span> <span class="n">weights</span><span class="p">,</span> <span class="n">points</span><span class="p">,</span> <span class="n">rhs</span><span class="o">=</span><span class="n">rhs</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>In order to apply the homogeneous Dirichlet boundary conditions, we only have to remove the first and the last splines, since their values at the boundary is <span class="math notranslate nohighlight">\(1\)</span>. Due to the partition of unity, the remaining splines are <span class="math notranslate nohighlight">\(0\)</span> on the boundary.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># apply homogeneous dirichlet boundary conditions</span>
+<span class="n">rhs</span> <span class="o">=</span> <span class="n">rhs</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+<span class="n">stiffness</span> <span class="o">=</span> <span class="n">stiffness</span><span class="p">[</span><span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
+</pre></div>
+</div>
+<p>We use a <strong>cg</strong> solver from <strong>scipy</strong> to solve the linear system.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">scipy.sparse.linalg</span> <span class="kn">import</span> <span class="n">cg</span>
+<span class="n">u</span><span class="p">,</span> <span class="n">info</span> <span class="o">=</span> <span class="n">cg</span><span class="p">(</span> <span class="n">stiffness</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="mf">1e-6</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="mi">5000</span> <span class="p">)</span>
+</pre></div>
+</div>
+<p>Now, we need to add the first and last control points, which are equal to <strong>0</strong>, then proceed to the plot.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">u</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.</span><span class="p">]</span> <span class="o">+</span> <span class="nb">list</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">+</span> <span class="p">[</span><span class="mf">0.</span><span class="p">]</span>
+<span class="n">u</span> <span class="o">=</span> <span class="n">asarray</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+<span class="n">plot_field_1d</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">nx</span><span class="o">=</span><span class="mi">401</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_20_01.png" /></p>
+</section>
+<section id="the-spline-space-class">
+<h2>The Spline Space class<a class="headerlink" href="#the-spline-space-class" title="Link to this heading">#</a></h2>
+<p>Since we will be calling over and over the block-lines [5] and [6], we created a class called <strong>SplineSpace</strong> that you can import from <strong>psydac</strong> library.</p>
+<section id="matrix-profile">
+<h3>Matrix profile<a class="headerlink" href="#matrix-profile" title="Link to this heading">#</a></h3>
+<p>Let’s check the profile of our matrix.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span> 
+
+<span class="n">plt</span><span class="o">.</span><span class="n">spy</span><span class="p">(</span><span class="n">stiffness</span><span class="p">)</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_23_1.png" /></p>
+<p>As expected, it is a sparse matrix and it has a typical profile (band one).</p>
+</section>
+<section id="next-step">
+<h3>Next step<a class="headerlink" href="#next-step" title="Link to this heading">#</a></h3>
+<p>In fact, we are not done yet!
+The problem now, is that the data structure we are using for the matrix (here as a dense matrix) will consume too much memory, and is not taken into account the locality!</p>
+<p>Since on each element, there are exactly <span class="math notranslate nohighlight">\(p+1\)</span> consecutive non-vanishing <em>B-splines</em>, we know that on each element there are at most <span class="math notranslate nohighlight">\(2p+1\)</span> non zeros entries (in every direction). This means that we will need to set entries as the following, in <span class="math notranslate nohighlight">\(1D\)</span>:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">M</span><span class="p">[</span><span class="n">i1</span><span class="p">,</span> <span class="n">j1</span> <span class="o">-</span> <span class="n">i1</span><span class="p">]</span>  <span class="o">+=</span> <span class="n">v_s</span>
+</pre></div>
+</div>
+<p>in <span class="math notranslate nohighlight">\(2D\)</span>:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">M</span><span class="p">[</span><span class="n">i1</span><span class="p">,</span> <span class="n">i2</span><span class="p">,</span> <span class="n">j1</span> <span class="o">-</span> <span class="n">i1</span><span class="p">,</span> <span class="n">j2</span> <span class="o">-</span> <span class="n">i2</span><span class="p">]</span>  <span class="o">+=</span> <span class="n">v_s</span>
+</pre></div>
+</div>
+<p>and in <span class="math notranslate nohighlight">\(3D\)</span>:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">M</span><span class="p">[</span><span class="n">i1</span><span class="p">,</span> <span class="n">i2</span><span class="p">,</span> <span class="n">i3</span><span class="p">,</span> <span class="n">j1</span> <span class="o">-</span> <span class="n">i1</span><span class="p">,</span> <span class="n">j2</span> <span class="o">-</span> <span class="n">i2</span><span class="p">,</span> <span class="n">j3</span> <span class="o">-</span> <span class="n">i3</span><span class="p">]</span>  <span class="o">+=</span> <span class="n">v_s</span>
+</pre></div>
+</div>
+<p>Unfortunatly, this can not be represented as a <strong>numpy.ndarray</strong>, because of the negative indexing!
+Moreover, if our aim is to write a parallel code, other consideration must be taken into account.</p>
+<p>More details will be found in the next <a class="reference external" href="http://nbviewer.jupyter.org/github/ratnania/IGA-Python/blob/main/lessons/Chapter1/03_data_structure.ipynb">section</a>.</p>
+</section>
+</section>
+<section id="exercises">
+<h2>Exercises<a class="headerlink" href="#exercises" title="Link to this heading">#</a></h2>
+<section id="exercise-1">
+<h3>Exercise 1.<a class="headerlink" href="#exercise-1" title="Link to this heading">#</a></h3>
+<ol class="arabic simple">
+<li><p>Implement a 1D Poisson solver using non homogeneous Dirichlet boundary conditions.</p></li>
+<li><p>Implement a 2D Poisson solver using homogeneous and non homogeneous Dirichlet boundary conditions.</p></li>
+</ol>
+</section>
+<section id="exercise-2">
+<h3>Exercise 2.<a class="headerlink" href="#exercise-2" title="Link to this heading">#</a></h3>
+<p>We consider the Heat equation in 1D</p>
+<div class="math notranslate nohighlight">
+\[
+\partial_t u = \nabla^2 u
+\]</div>
+<p>with homogeneous boundary conditions.</p>
+<ol class="arabic simple">
+<li><p>Write an explicit FEM solver. What do you notice about the dependency of the CFL with respect to the degree? what about the regularity?</p></li>
+<li><p>Write an implicit FEM solver.</p></li>
+</ol>
+</section>
+<section id="exercise-3">
+<h3>Exercise 3.<a class="headerlink" href="#exercise-3" title="Link to this heading">#</a></h3>
+<p>We consider a unidirectional flow in the velocity/pressure formulation</p>
+<div class="math notranslate nohighlight">
+\[
+\partial u_x = - \frac{1}{\rho} \partial_x p + \nu \partial_{yy}^2 u_x + g_x
+\]</div>
+<p>where the unknown <span class="math notranslate nohighlight">\(u_x\)</span> is a function of <span class="math notranslate nohighlight">\(y\)</span> only.</p>
+<p>We will consider <span class="math notranslate nohighlight">\(\partial_x p = -2\)</span> and <span class="math notranslate nohighlight">\(g_x = 0.4\)</span>. The boundary conditions are given by <span class="math notranslate nohighlight">\(u_x(y=0) = u_x(y=h) = 0\)</span>.</p>
+<ol class="arabic simple">
+<li><p>Write an explicit and implicit time schemes and solvers.</p></li>
+<li><p>Find numerical CFL in the case of explicit time scheme.</p></li>
+<li><p>Find the steady state.</p></li>
+</ol>
+</section>
+</section>
+</section>
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+    <i class="fa-solid fa-list"></i> Contents
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+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analytical-solution">Analytical solution</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-spline-space-class">The Spline Space class</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-profile">Matrix profile</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#next-step">Next step</a></li>
+</ul>
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+    <h1>B-splines FEM solver for Poisson equation (2D)</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#create-the-finite-elements-space">Create the Finite Elements Space</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#assemble-the-stiffness-matrix">Assemble the Stiffness Matrix</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#assemble-the-rhs">Assemble the rhs</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#imposing-boundary-conditions">Imposing boundary conditions</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="b-splines-fem-solver-for-poisson-equation-2d">
+<h1>B-splines FEM solver for Poisson equation (2D)<a class="headerlink" href="#b-splines-fem-solver-for-poisson-equation-2d" title="Link to this heading">#</a></h1>
+<p>In this section, we show hoa to use <strong>simplines</strong> to solve a 2D Poisson problem with homogeneous boundary conditions
+$$</p>
+<ul class="simple">
+<li><p>\nabla^2 u = f, \Omega,
+\
+u = 0, \partial \Omega
+$<span class="math notranslate nohighlight">\(
+where the computation domain \)</span>\Omega$ is the unit square.</p></li>
+</ul>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># needed imports</span>
+<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">zeros</span><span class="p">,</span> <span class="n">ones</span><span class="p">,</span> <span class="n">linspace</span><span class="p">,</span> <span class="n">zeros_like</span><span class="p">,</span> <span class="n">asarray</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">from</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">import</span> <span class="n">plot</span><span class="p">,</span> <span class="n">show</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="o">%</span><span class="n">matplotlib</span> <span class="n">inline</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psydac.fem.splines</span> <span class="kn">import</span> <span class="n">SplineSpace</span>
+<span class="kn">from</span> <span class="nn">psydac.fem.tensor</span>  <span class="kn">import</span> <span class="n">TensorFemSpace</span>
+<span class="kn">from</span> <span class="nn">psydac.linalg.stencil</span> <span class="kn">import</span> <span class="n">StencilMatrix</span>
+<span class="kn">from</span> <span class="nn">psydac.linalg.stencil</span> <span class="kn">import</span> <span class="n">StencilVector</span>
+<span class="kn">from</span> <span class="nn">psydac.ddm.cart</span> <span class="kn">import</span> <span class="n">DomainDecomposition</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">gallery_section_04</span> <span class="kn">import</span> <span class="n">assemble_stiffness_2d</span>
+<span class="kn">from</span> <span class="nn">gallery_section_04</span> <span class="kn">import</span> <span class="n">assemble_vector_2d</span>
+</pre></div>
+</div>
+<section id="create-the-finite-elements-space">
+<h2>Create the Finite Elements Space<a class="headerlink" href="#create-the-finite-elements-space" title="Link to this heading">#</a></h2>
+<p>In 2D, our Spline function space is defined as</p>
+<div class="math notranslate nohighlight">
+\[
+\mathcal{V}_h := \texttt{span}\{ B_{i_1}^{p_1} B_{i_2}^{p_1}, ~ 1 \le i_1 \le n_1, ~ 1 \le i_2 \le n_2\}
+\]</div>
+<p>which is basicaly
+$<span class="math notranslate nohighlight">\(
+\mathcal{V}_h = \mathcal{V}_h^1 \otimes \mathcal{V}_h^2 
+\)</span><span class="math notranslate nohighlight">\(
+where
+\)</span><span class="math notranslate nohighlight">\(
+\mathcal{V}_h^1 := \texttt{span}\{ B_{i_1}^{p_1}, ~ 1 \le i_1 \le n_1\}
+\)</span><span class="math notranslate nohighlight">\(
+and
+\)</span><span class="math notranslate nohighlight">\(
+\mathcal{V}_h^2 := \texttt{span}\{ B_{i_2}^{p_2}, ~ 1 \le i_2 \le n_2\}
+\)</span>$</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the spline space for each direction</span>
+<span class="n">x1min</span> <span class="o">=</span> <span class="mf">0.</span> <span class="p">;</span> <span class="n">x1max</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">nelements1</span> <span class="o">=</span> <span class="mi">1</span>
+<span class="n">degree1</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">grid1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">x1min</span><span class="p">,</span> <span class="n">x1max</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">nelements1</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
+<span class="n">V1</span> <span class="o">=</span> <span class="n">SplineSpace</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree1</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid1</span><span class="p">)</span>
+
+<span class="n">x2min</span> <span class="o">=</span> <span class="mf">0.</span> <span class="p">;</span> <span class="n">x2max</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">nelements2</span> <span class="o">=</span> <span class="mi">1</span>
+<span class="n">degree2</span> <span class="o">=</span> <span class="mi">2</span>
+<span class="n">grid2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span> <span class="n">x2min</span><span class="p">,</span> <span class="n">x2max</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="n">nelements2</span><span class="o">+</span><span class="mi">1</span> <span class="p">)</span>
+<span class="n">V2</span> <span class="o">=</span> <span class="n">SplineSpace</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="n">degree2</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid2</span><span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">dd</span> <span class="o">=</span> <span class="n">DomainDecomposition</span><span class="p">(</span><span class="n">ncells</span><span class="o">=</span><span class="p">[</span><span class="n">V1</span><span class="o">.</span><span class="n">ncells</span><span class="p">,</span> <span class="n">V2</span><span class="o">.</span><span class="n">ncells</span><span class="p">],</span> <span class="n">periods</span><span class="o">=</span><span class="p">[</span><span class="kc">False</span><span class="p">,</span> <span class="kc">False</span><span class="p">])</span>
+<span class="c1"># create the tensor space</span>
+<span class="n">V</span> <span class="o">=</span> <span class="n">TensorFemSpace</span><span class="p">(</span><span class="n">dd</span><span class="p">,</span> <span class="n">V1</span><span class="p">,</span> <span class="n">V2</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="assemble-the-stiffness-matrix">
+<h2>Assemble the Stiffness Matrix<a class="headerlink" href="#assemble-the-stiffness-matrix" title="Link to this heading">#</a></h2>
+<p>The stiffness matrix entries are defined as</p>
+<div class="math notranslate nohighlight">
+\[
+M_{\textbf{i}, \textbf{j}} := \int_{\Omega} \nabla B_{\textbf{i}} \cdot \nabla B_{\textbf{j}}
+\]</div>
+<p>where
+$<span class="math notranslate nohighlight">\(
+B_{\textbf{i}}(x_1,x_2) := B_{i_1}(x_1)B_{i_2}(x_2), \quad \textbf{i} := (i_1,i_2) 
+\)</span><span class="math notranslate nohighlight">\(
+and
+\)</span><span class="math notranslate nohighlight">\(
+B_{\textbf{j}}(x_1,x_2) := B_{j_1}(x_1)B_{j_2}(x_2), \quad \textbf{j} := (j_1,j_2)
+\)</span>$</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">stiffness</span> <span class="o">=</span> <span class="n">StencilMatrix</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">,</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">)</span>
+<span class="n">stiffness</span> <span class="o">=</span> <span class="n">assemble_stiffness_2d</span><span class="p">(</span> <span class="n">V</span><span class="p">,</span> <span class="n">matrix</span><span class="o">=</span><span class="n">stiffness</span> <span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="assemble-the-rhs">
+<h2>Assemble the rhs<a class="headerlink" href="#assemble-the-rhs" title="Link to this heading">#</a></h2>
+<p>The right hand side entries are defined as</p>
+<div class="math notranslate nohighlight">
+\[
+F_{\textbf{i}} := \int_{\Omega} f B_{\textbf{i}} ~d\Omega
+\]</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">rhs</span> <span class="o">=</span> <span class="n">StencilVector</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="p">)</span>
+
+<span class="n">f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">:</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">y</span><span class="p">)</span> 
+<span class="n">rhs</span> <span class="o">=</span> <span class="n">assemble_vector_2d</span><span class="p">(</span> <span class="n">f</span><span class="p">,</span> <span class="n">V</span><span class="p">,</span> <span class="n">rhs</span><span class="o">=</span><span class="n">rhs</span> <span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="imposing-boundary-conditions">
+<h2>Imposing boundary conditions<a class="headerlink" href="#imposing-boundary-conditions" title="Link to this heading">#</a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">s1</span><span class="p">,</span> <span class="n">s2</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">starts</span>
+<span class="n">e1</span><span class="p">,</span> <span class="n">e2</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">vector_space</span><span class="o">.</span><span class="n">ends</span>
+
+<span class="c1"># ... needed for iterative solvers</span>
+<span class="c1"># left  bc at x=0.</span>
+<span class="n">stiffness</span><span class="p">[</span><span class="n">s1</span><span class="p">,:,:,:]</span> <span class="o">=</span> <span class="mf">0.</span>
+<span class="n">rhs</span><span class="p">[</span><span class="n">s1</span><span class="p">,:]</span>           <span class="o">=</span> <span class="mf">0.</span>
+<span class="c1"># right bc at x=1.</span>
+<span class="n">stiffness</span><span class="p">[</span><span class="n">e1</span><span class="p">,:,:,:]</span> <span class="o">=</span> <span class="mf">0.</span>
+<span class="n">rhs</span><span class="p">[</span><span class="n">e1</span><span class="p">,:]</span>           <span class="o">=</span> <span class="mf">0.</span>
+<span class="c1"># lower bc at y=0.</span>
+<span class="n">stiffness</span><span class="p">[:,</span><span class="n">s2</span><span class="p">,:,:]</span> <span class="o">=</span> <span class="mf">0.</span>
+<span class="n">rhs</span><span class="p">[:,</span><span class="n">s2</span><span class="p">]</span>           <span class="o">=</span> <span class="mf">0.</span>
+<span class="c1"># upper bc at y=1.</span>
+<span class="n">stiffness</span><span class="p">[:,</span><span class="n">e2</span><span class="p">,:,:]</span> <span class="o">=</span> <span class="mf">0.</span>
+<span class="n">rhs</span><span class="p">[:,</span><span class="n">e2</span><span class="p">]</span>           <span class="o">=</span> <span class="mf">0.</span>
+<span class="c1"># ...</span>
+
+<span class="c1"># ... needed for direct solvers</span>
+<span class="c1"># boundary x = 0</span>
+<span class="c1">#stiffness[s1,:,0,:] = 1.</span>
+<span class="c1"># boundary x = 1</span>
+<span class="c1">#stiffness[e1,:,0,:] = 1.</span>
+<span class="c1"># boundary y = 0</span>
+<span class="c1">#stiffness[:,s2,:,0] = 1.</span>
+<span class="c1"># boundary y = 1</span>
+<span class="c1">#stiffness[:,e2,:,0] = 1.    </span>
+<span class="c1"># ...</span>
+</pre></div>
+</div>
+<p>From now on, you can use the function <strong>apply_dirichlet</strong> to set the dirichlet boundary conditions for both the matrix and rhs.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># convert the stencil matrix to scipy sparse</span>
+<span class="n">stiffness</span> <span class="o">=</span> <span class="n">stiffness</span><span class="o">.</span><span class="n">tosparse</span><span class="p">()</span> 
+
+
+<span class="err">```</span><span class="n">python</span>
+<span class="c1"># convert the stencil vector to a nd_array</span>
+<span class="n">rhs</span> <span class="o">=</span> <span class="n">rhs</span><span class="o">.</span><span class="n">toarray</span><span class="p">()</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1">#from scipy.sparse import csc_matrix, linalg as sla</span>
+
+<span class="c1">#lu = sla.splu(csc_matrix(stiffness))</span>
+<span class="c1">#x = lu.solve(rhs)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">scipy.sparse.linalg</span> <span class="kn">import</span> <span class="n">cg</span>
+<span class="n">x</span><span class="p">,</span> <span class="n">info</span> <span class="o">=</span> <span class="n">cg</span><span class="p">(</span> <span class="n">stiffness</span><span class="p">,</span> <span class="n">rhs</span><span class="p">,</span> <span class="n">rtol</span><span class="o">=</span><span class="mf">1e-7</span><span class="p">,</span> <span class="n">maxiter</span><span class="o">=</span><span class="mi">100</span> <span class="p">)</span>
+</pre></div>
+</div>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">utilities.plot</span> <span class="kn">import</span> <span class="n">plot_field_2d</span>
+<span class="n">nbasis</span>  <span class="o">=</span> <span class="p">[</span><span class="n">W</span><span class="o">.</span><span class="n">nbasis</span> <span class="k">for</span> <span class="n">W</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">spaces</span><span class="p">]</span>
+<span class="n">knots</span>   <span class="o">=</span> <span class="p">[</span><span class="n">W</span><span class="o">.</span><span class="n">knots</span> <span class="k">for</span> <span class="n">W</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">spaces</span><span class="p">]</span>
+<span class="n">degrees</span> <span class="o">=</span> <span class="p">[</span><span class="n">W</span><span class="o">.</span><span class="n">degree</span> <span class="k">for</span> <span class="n">W</span> <span class="ow">in</span> <span class="n">V</span><span class="o">.</span><span class="n">spaces</span><span class="p">]</span>
+<span class="n">u</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">nbasis</span><span class="p">)</span>
+<span class="n">plot_field_2d</span><span class="p">(</span><span class="n">knots</span><span class="p">,</span> <span class="n">degrees</span><span class="p">,</span> <span class="n">u</span><span class="p">)</span> <span class="p">;</span> <span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">()</span>
+</pre></div>
+</div>
+<p><img alt="png" src="../_images/output_20_1.png" /></p>
+</section>
+</section>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#assemble-the-stiffness-matrix">Assemble the Stiffness Matrix</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#assemble-the-rhs">Assemble the rhs</a></li>
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+<h1>Installation<a class="headerlink" href="#installation" title="Link to this heading">#</a></h1>
+<p>It is recommanded to use a virtual environement.</p>
+<div class="highlight-shell notranslate"><div class="highlight"><pre><span></span><span class="c1"># create a virtual environement</span>
+python3<span class="w"> </span>-m<span class="w"> </span>venv<span class="w"> </span>.iga-python
+
+<span class="c1"># activate this environement using </span>
+<span class="nb">source</span><span class="w"> </span>.iga-python/bin/activate
+</pre></div>
+</div>
+<p>Then install dependencies using</p>
+<div class="highlight-shell notranslate"><div class="highlight"><pre><span></span>pip3<span class="w"> </span>install<span class="w"> </span>wheel
+pip3<span class="w"> </span>install<span class="w"> </span>numpy
+pip3<span class="w"> </span>install<span class="w"> </span>pyccel<span class="w"> </span>
+pip3<span class="w"> </span>install<span class="w"> </span>ipykernel
+pip3<span class="w"> </span>install<span class="w"> </span>git+https://github.com/pyccel/sympde.git
+pip3<span class="w"> </span>install<span class="w"> </span>git+https://github.com/pyccel/gelato.git
+pip3<span class="w"> </span>install<span class="w"> </span>git+https://github.com/pyccel/psydac.git
+pip3<span class="w"> </span>install<span class="w"> </span>git+https://github.com/girving/igakit.git
+</pre></div>
+</div>
+<p>Adding the virtual environement to Jupyter notebook, can be done using</p>
+<div class="highlight-shell notranslate"><div class="highlight"><pre><span></span>python3<span class="w"> </span>-m<span class="w"> </span>ipykernel<span class="w"> </span>install<span class="w"> </span>--user<span class="w"> </span>--name<span class="o">=</span>.iga-python
+</pre></div>
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+</section>
+
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diff --git a/chapter1/poisson.html b/chapter1/poisson.html
new file mode 100644
index 0000000..7b6b9e8
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@@ -0,0 +1,1242 @@
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+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter0/bsplines.html">B-Splines</a><input class="toctree-checkbox" id="toctree-checkbox-3" name="toctree-checkbox-3" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-3"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter0/bsplines-curves.html">B-Splines curves</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter0/bsplines-surfaces.html">B-Splines surfaces</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter0/iga.html">Isogeometric Analysis</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/data-structure.html">Data Structure</a></li>
+
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="install.html">Installation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="sympde.html">SymPDE concepts and their mathematical meaning</a><input class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-5"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="topology.html">Topology concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="space.html">Function Space concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/sympde.html">SymPDE API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-13" name="toctree-checkbox-13" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-13"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.algebra.html">sympde.core.algebra</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.basic.html">sympde.core.basic</a></li>
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+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.utils.html">sympde.core.utils</a></li>
+</ul>
+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.analytical_mapping.html">sympde.topology.analytical_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.basic.html">sympde.topology.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.callable_mapping.html">sympde.topology.callable_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.datatype.html">sympde.topology.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.derivatives.html">sympde.topology.derivatives</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.domain.html">sympde.topology.domain</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_derivatives.html">sympde.topology.tests.test_derivatives</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_field_2d.html">sympde.topology.tests.test_field_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.html">sympde.expr</a><input class="toctree-checkbox" id="toctree-checkbox-18" name="toctree-checkbox-18" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-18"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.equation.html">sympde.expr.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_equation_2d.html">sympde.expr.tests.test_equation_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.html">sympde.printing</a><input class="toctree-checkbox" id="toctree-checkbox-20" name="toctree-checkbox-20" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-20"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.latex.html">sympde.printing.latex</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.tests.test_latex.html">sympde.printing.tests.test_latex</a></li>
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+</li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.calculus.html">sympde.exterior.calculus</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.form.html">sympde.exterior.form</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_compiler.html">sympde.exterior.tests.test_compiler</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
+</ul>
+</li>
+</ul>
+</li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Your first code using SymPDE & PsyDAC</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#variational-formulation">Variational Formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="your-first-code-using-sympde-psydac">
+<h1>Your first code using SymPDE &amp; PsyDAC<a class="headerlink" href="#your-first-code-using-sympde-psydac" title="Link to this heading">#</a></h1>
+<p>We first start by writing our first example using SymPDE.
+Let <span class="math notranslate nohighlight">\(\Omega := (0,1)^2\)</span>. We consider the Poisson problem with homogeneous Dirichlet boundary conditions.</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  - \nabla^2 u = f \quad \text{in $\Omega$}, \quad \quad 
+  u = 0            \quad \text{on $\partial \Omega$}. 
+\end{align}
+\]</div>
+<section id="variational-formulation">
+<h2>Variational Formulation<a class="headerlink" href="#variational-formulation" title="Link to this heading">#</a></h2>
+<p>An <span class="math notranslate nohighlight">\(H^1\)</span>-conforming variational formulation of the previous problem reads</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \text{find $u \in V$ such that} \quad a(u,v) = l(v) \quad \forall v \in V,
+\end{align}
+\]</div>
+<p>where</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(V \subset H^1_0(\Omega)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(a(u,v) := \int_{\Omega} \nabla u \cdot \nabla v ~ d\Omega\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(l(v) := \int_{\Omega} f v ~ d\Omega\)</span>.</p></li>
+</ul>
+</section>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+
+<span class="n">V</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span> <span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))))</span>
+
+<span class="c1"># linear form</span>
+<span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">f</span><span class="o">*</span><span class="n">v</span><span class="p">))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>This very simple Python code reflects well the abstract mathematical framework needed for variational formulations.
+The structure of the code is as follows,</p>
+<ol class="arabic simple">
+<li><p>Create a domain.</p></li>
+<li><p>Create a space of <em>scalar</em> functions over the domain.</p></li>
+<li><p>Create elements from this function space. These elements will denote the test and trial functions.</p></li>
+<li><p>Create the Bilinear and Linear forms, <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(l\)</span> respectively.</p></li>
+<li><p>Create Essential Boundary Conditions.</p></li>
+<li><p>Create the variational problem.</p></li>
+</ol>
+<p>Most of the time, you will need to follow the same steps, with some minor variants depending on the problem you’re considering.</p>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<p>We shall need the <strong>discretize</strong> function from <strong>PsyDAC</strong>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">16</span><span class="p">,</span><span class="mi">16</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">Vh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">uh</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<p>When the analytical solution is available, you might be interested in computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm or <span class="math notranslate nohighlight">\(H^1_0\)</span> semi-norm.
+SymPDE allows you to do so, by creating the <strong>Norm</strong> object.
+In this example, the analytical solution is given by</p>
+<div class="math notranslate nohighlight">
+\[
+u_e = \sin(\pi x) \sin(\pi y)
+\]</div>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ue</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+
+<span class="n">u</span> <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+
+<span class="c1"># create the formal Norm object</span>
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">l2norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">l2_error</span> <span class="o">=</span> <span class="n">l2norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">l2_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>9.49756716724664e-07
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-semi-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm<a class="headerlink" href="#computing-the-h-1-semi-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">SemiNorm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">h1norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">h1_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>9.768702410721585e-05
+</pre></div>
+</div>
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm<a class="headerlink" href="#computing-the-h-1-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">h1norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">h1_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>9.769164089397408e-05
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
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+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.datatype.html">sympde.topology.datatype</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.equation.html">sympde.expr.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_equation_2d.html">sympde.expr.tests.test_equation_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
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+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.form.html">sympde.exterior.form</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_compiler.html">sympde.exterior.tests.test_compiler</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
+</ul>
+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.utilities.utils.html">sympde.utilities.utils</a></li>
+</ul>
+</li>
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+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.expr.html">psydac.api.ast.expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.glt.html">psydac.api.ast.glt</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.utilities.html">psydac.api.ast.utilities</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.discretization.html">psydac.api.discretization</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.equation.html">psydac.api.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.essential_bc.html">psydac.api.essential_bc</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.expr.html">psydac.api.expr</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.feec.html">psydac.api.feec</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.fem.html">psydac.api.fem</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.glt.html">psydac.api.glt</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.grid.html">psydac.api.grid</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.postprocessing.html">psydac.api.postprocessing</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/psydac.api.printing.html">psydac.api.printing</a><input class="toctree-checkbox" id="toctree-checkbox-29" name="toctree-checkbox-29" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-29"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+    <h1>Algebraic and differential operators evalution rules</h1>
+    <!-- Table of contents -->
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-the-mathrm-grad-operator">Evaluation of the <span class="math notranslate nohighlight">\(\mathrm{grad}\)</span> operator</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-mathrm-curl-and-mathrm-rot-operators-in-2d">Evaluation of <span class="math notranslate nohighlight">\(\mathrm{curl}\)</span> and <span class="math notranslate nohighlight">\(\mathrm{rot}\)</span> operators in 2D</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-mathrm-curl-operator-in-3d">Evaluation of <span class="math notranslate nohighlight">\(\mathrm{curl}\)</span> operator in 3D</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-mathrm-div-operator">Evaluation of <span class="math notranslate nohighlight">\(\mathrm{div}\)</span> operator</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-mathrm-laplace-operator">Evaluation of <span class="math notranslate nohighlight">\(\mathrm{laplace}\)</span> operator</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-specific-combination-of-operators">Evaluation of specific combination of operators</a></li>
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+  <section class="tex2jax_ignore mathjax_ignore" id="algebraic-and-differential-operators-evalution-rules">
+<h1>Algebraic and differential operators evalution rules<a class="headerlink" href="#algebraic-and-differential-operators-evalution-rules" title="Link to this heading">#</a></h1>
+<section id="evaluation-of-the-mathrm-grad-operator">
+<h2>Evaluation of the <span class="math notranslate nohighlight">\(\mathrm{grad}\)</span> operator<a class="headerlink" href="#evaluation-of-the-mathrm-grad-operator" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Rule ID</p></th>
+<th class="head"><p>symbolic expression</p></th>
+<th class="head"><p>evaluation</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OG1]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{grad}(f+g)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{grad}(f) + \mathrm{grad}(g)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OG2]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{grad}(\alpha ~ f)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\alpha~\mathrm{grad}(f)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OG3]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{grad}(f ~ g)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(f~\mathrm{grad}(g) + g~\mathrm{grad}(f)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OG4]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{grad}(\frac{f}{g})\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(-\frac{f}{g^2}\mathrm{grad}(g) + \frac{1}{g}\mathrm{grad}(f)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OG5]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{grad}(F+G)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{grad}(F) + \mathrm{grad}(G)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OG6]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{grad}(\alpha ~ F)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\alpha~\mathrm{grad}(F)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OG7]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{grad}(\mathrm{dot}(F, G))\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{convect}(F, G) + \mathrm{convect}(G, F) + \mathrm{cross}(F, \mathrm{curl}(G)) - \mathrm{cross}(\mathrm{curl}(F), G)\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="evaluation-of-mathrm-curl-and-mathrm-rot-operators-in-2d">
+<h2>Evaluation of <span class="math notranslate nohighlight">\(\mathrm{curl}\)</span> and <span class="math notranslate nohighlight">\(\mathrm{rot}\)</span> operators in 2D<a class="headerlink" href="#evaluation-of-mathrm-curl-and-mathrm-rot-operators-in-2d" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Rule ID</p></th>
+<th class="head"><p>symbolic expression</p></th>
+<th class="head"><p>evaluation</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OC1a]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{curl}(f+g)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{curl}(f) + \mathrm{curl}(g)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OC2a]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{curl}(\alpha ~ f)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\alpha~\mathrm{curl}(f)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OC1b]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{rot}(F+G)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{rot}(F) + \mathrm{rot}(G)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OC2b]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{rot}(\alpha ~ F)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\alpha~\mathrm{rot}(F)\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="evaluation-of-mathrm-curl-operator-in-3d">
+<h2>Evaluation of <span class="math notranslate nohighlight">\(\mathrm{curl}\)</span> operator in 3D<a class="headerlink" href="#evaluation-of-mathrm-curl-operator-in-3d" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Rule ID</p></th>
+<th class="head"><p>symbolic expression</p></th>
+<th class="head"><p>evaluation</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OC1]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{curl}(F+G)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{curl}(F) + \mathrm{curl}(G)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OC2]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{curl}(\alpha ~ F)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\alpha~\mathrm{curl}(F)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OC3]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{curl}(f  ~ F)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(f~\mathrm{curl}(F) + \mathrm{cross}(\mathrm{grad}(f), F)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OC4]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{curl}(\mathrm{cross}(F, G))\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{div}(G)~F - \mathrm{div}(F)~G - \mathrm{convect}(F, G) +\mathrm{convect}(G, F)\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="evaluation-of-mathrm-div-operator">
+<h2>Evaluation of <span class="math notranslate nohighlight">\(\mathrm{div}\)</span> operator<a class="headerlink" href="#evaluation-of-mathrm-div-operator" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Rule ID</p></th>
+<th class="head"><p>symbolic expression</p></th>
+<th class="head"><p>evaluation</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OD1]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{div}(F+G)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{div}(F) + \mathrm{div}(G)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OD2]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{div}(\alpha ~ F)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\alpha~\mathrm{div}(F)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OD3]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{div}(f ~ G)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(f~\mathrm{div}(G) + \mathrm{dot}(G, \mathrm{grad}(f))\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OD4]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{div}(\mathrm{cross}(F, G))\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(-\mathrm{dot}(F, \mathrm{curl}(G)) + \mathrm{dot}(G, \mathrm{curl}(F))\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="evaluation-of-mathrm-laplace-operator">
+<h2>Evaluation of <span class="math notranslate nohighlight">\(\mathrm{laplace}\)</span> operator<a class="headerlink" href="#evaluation-of-mathrm-laplace-operator" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Rule ID</p></th>
+<th class="head"><p>symbolic expression</p></th>
+<th class="head"><p>evaluation</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OL1]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{laplace}(f+g)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{laplace}(f) + \mathrm{laplace}(g)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OL2]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{laplace}(\alpha ~ f)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\alpha~\mathrm{laplace}(f)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OL3]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{laplace}(f~g)\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(f ~ \mathrm{laplace}(g) + g ~ \mathrm{laplace}(f) + 2 \mathrm{dot}(\mathrm{grad}(f), \mathrm{grad}(g))\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="evaluation-of-specific-combination-of-operators">
+<h2>Evaluation of specific combination of operators<a class="headerlink" href="#evaluation-of-specific-combination-of-operators" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Rule ID</p></th>
+<th class="head"><p>symbolic expression</p></th>
+<th class="head"><p>evaluation</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OS1]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{curl}(\mathrm{grad}(f))\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OS2]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{div}(\mathrm{curl}(F))\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OS3]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{div}(\mathrm{cross}(\mathrm{grad}(F), \mathrm{grad}(G)))\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">[OS4]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{curl}(\mathrm{curl}(F))\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{grad}(\mathrm{div}(F)) - \mathrm{laplace}(F)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">[OS5]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{curl}(f~\mathrm{grad}(g))\)</span></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathrm{cross}(\mathrm{grad}(f), \mathrm{grad}(g))\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+</section>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-the-mathrm-grad-operator">Evaluation of the <span class="math notranslate nohighlight">\(\mathrm{grad}\)</span> operator</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-mathrm-curl-and-mathrm-rot-operators-in-2d">Evaluation of <span class="math notranslate nohighlight">\(\mathrm{curl}\)</span> and <span class="math notranslate nohighlight">\(\mathrm{rot}\)</span> operators in 2D</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-mathrm-curl-operator-in-3d">Evaluation of <span class="math notranslate nohighlight">\(\mathrm{curl}\)</span> operator in 3D</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-mathrm-div-operator">Evaluation of <span class="math notranslate nohighlight">\(\mathrm{div}\)</span> operator</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-mathrm-laplace-operator">Evaluation of <span class="math notranslate nohighlight">\(\mathrm{laplace}\)</span> operator</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#evaluation-of-specific-combination-of-operators">Evaluation of specific combination of operators</a></li>
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+    <h1>Function Space concepts</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#basicfunctionspace">BasicFunctionSpace</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#scalarfunctionspace">ScalarFunctionSpace</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#vectorfunctionspace">VectorFunctionSpace</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#typed-spaces">Typed Spaces</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#productspace">ProductSpace</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#derham">Derham</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#functions">Functions</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#scalarfunction">ScalarFunction</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#indexedvectorfunction">IndexedVectorFunction</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#vectorfunction">VectorFunction</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#trace">Trace</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#trace-0">trace_0</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#trace-1">trace_1</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#utilities">Utilities</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#element-of-and-elements-of">element_of and elements_of</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#projection-experimental">Projection (Experimental)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#projector-experimental">Projector (Experimental)</a></li>
+</ul>
+            </nav>
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+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="function-space-concepts">
+<h1>Function Space concepts<a class="headerlink" href="#function-space-concepts" title="Link to this heading">#</a></h1>
+<p>SymPDE provides two Python classes to describe scalar and vector function spaces, respectively. There is no notion of a discrete representation; for example, we do not need to mention that a space is a Brezzi-Douglas-Marini (BDM) space. In fact, these kind of function spaces can be seen as parametric types having as a basic type <strong>ScalarFunctionSpace</strong> or <strong>VectorFunctionSpace</strong>. SymPDE only needs to know if an element of the space can be indexed or not. A BDM space would then be identified by an annotation added by a third party library to uniquely define a function space at the discrete level.</p>
+<section id="basicfunctionspace">
+<h2>BasicFunctionSpace<a class="headerlink" href="#basicfunctionspace" title="Link to this heading">#</a></h2>
+<p>This represents the base class of all our function spaces. It does not reflect a continuous space, but it is more a <strong>type</strong> at the formal/abstract level.</p>
+</section>
+<section id="scalarfunctionspace">
+<h2>ScalarFunctionSpace<a class="headerlink" href="#scalarfunctionspace" title="Link to this heading">#</a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">domain</span> <span class="o">=</span> <span class="n">Domain</span><span class="p">(</span><span class="s1">&#39;Omega&#39;</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">V1</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V1&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="vectorfunctionspace">
+<h2>VectorFunctionSpace<a class="headerlink" href="#vectorfunctionspace" title="Link to this heading">#</a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">domain</span> <span class="o">=</span> <span class="n">Domain</span><span class="p">(</span><span class="s1">&#39;Omega&#39;</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">W1</span> <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;W1&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="typed-spaces">
+<h2>Typed Spaces<a class="headerlink" href="#typed-spaces" title="Link to this heading">#</a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">domain</span> <span class="o">=</span> <span class="n">Domain</span><span class="p">(</span><span class="s1">&#39;Omega&#39;</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">H1</span>    <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V0&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;H1&#39;</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Hcurl</span> <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V1&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Hcurl&#39;</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Hdiv</span>  <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V2&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Hdiv&#39;</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">L2</span>    <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V3&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;L2&#39;</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">H1</span><span class="o">.</span><span class="n">kind</span>    <span class="o">==</span> <span class="n">H1Space</span><span class="p">)</span>                
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">Hcurl</span><span class="o">.</span><span class="n">kind</span> <span class="o">==</span> <span class="n">HcurlSpace</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">Hdiv</span><span class="o">.</span><span class="n">kind</span>  <span class="o">==</span> <span class="n">HdivSpace</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">L2</span><span class="o">.</span><span class="n">kind</span>    <span class="o">==</span> <span class="n">L2Space</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">V</span><span class="o">.</span><span class="n">kind</span>     <span class="o">==</span> <span class="n">UndefinedSpace</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">W</span><span class="o">.</span><span class="n">kind</span>     <span class="o">==</span> <span class="n">UndefinedSpace</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">H1</span><span class="o">.</span><span class="n">regularity</span>    <span class="o">&gt;</span> <span class="n">L2</span><span class="o">.</span><span class="n">regularity</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">H1</span><span class="o">.</span><span class="n">regularity</span>    <span class="o">&gt;</span> <span class="n">Hcurl</span><span class="o">.</span><span class="n">regularity</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">Hcurl</span><span class="o">.</span><span class="n">regularity</span> <span class="o">&gt;</span> <span class="n">L2</span><span class="o">.</span><span class="n">regularity</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="productspace">
+<h2>ProductSpace<a class="headerlink" href="#productspace" title="Link to this heading">#</a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">domain</span> <span class="o">=</span> <span class="n">Domain</span><span class="p">(</span><span class="s1">&#39;Omega&#39;</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">V1</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V1&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">V2</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V2&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">W1</span> <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;W1&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">V1</span> <span class="o">*</span> <span class="n">V2</span> <span class="o">*</span> <span class="n">W1</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="c1"># or </span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">X</span> <span class="o">=</span> <span class="n">ProductSpace</span><span class="p">(</span><span class="n">V1</span><span class="p">,</span> <span class="n">V2</span><span class="p">,</span> <span class="n">W1</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="derham">
+<h2>Derham<a class="headerlink" href="#derham" title="Link to this heading">#</a></h2>
+</section>
+<section id="functions">
+<h2>Functions<a class="headerlink" href="#functions" title="Link to this heading">#</a></h2>
+<section id="scalarfunction">
+<h3>ScalarFunction<a class="headerlink" href="#scalarfunction" title="Link to this heading">#</a></h3>
+<p>A scalar function is necessary an element of the <strong>ScalarFunctionSpace</strong>.</p>
+</section>
+<section id="indexedvectorfunction">
+<h3>IndexedVectorFunction<a class="headerlink" href="#indexedvectorfunction" title="Link to this heading">#</a></h3>
+</section>
+<section id="vectorfunction">
+<h3>VectorFunction<a class="headerlink" href="#vectorfunction" title="Link to this heading">#</a></h3>
+<p>A vector function is necessary an element of the <strong>VectorFunctionSpace</strong>. It is an extension of the <strong>IndexedVectorFunction</strong>, which means that one can access to its components.</p>
+</section>
+</section>
+<section id="trace">
+<h2>Trace<a class="headerlink" href="#trace" title="Link to this heading">#</a></h2>
+<section id="trace-0">
+<h3>trace_0<a class="headerlink" href="#trace-0" title="Link to this heading">#</a></h3>
+</section>
+<section id="trace-1">
+<h3>trace_1<a class="headerlink" href="#trace-1" title="Link to this heading">#</a></h3>
+</section>
+</section>
+<section id="utilities">
+<h2>Utilities<a class="headerlink" href="#utilities" title="Link to this heading">#</a></h2>
+<section id="element-of-and-elements-of">
+<h3>element_of and elements_of<a class="headerlink" href="#element-of-and-elements-of" title="Link to this heading">#</a></h3>
+</section>
+</section>
+<section id="projection-experimental">
+<h2>Projection (Experimental)<a class="headerlink" href="#projection-experimental" title="Link to this heading">#</a></h2>
+</section>
+<section id="projector-experimental">
+<h2>Projector (Experimental)<a class="headerlink" href="#projector-experimental" title="Link to this heading">#</a></h2>
+</section>
+</section>
+
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.html">sympde.printing</a><input class="toctree-checkbox" id="toctree-checkbox-20" name="toctree-checkbox-20" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-20"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.form.html">sympde.exterior.form</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
+</ul>
+</li>
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+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.utilities.utils.html">sympde.utilities.utils</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.utilities.html">psydac.api.ast.utilities</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.discretization.html">psydac.api.discretization</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.equation.html">psydac.api.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.essential_bc.html">psydac.api.essential_bc</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.expr.html">psydac.api.expr</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.feec.html">psydac.api.feec</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.fem.html">psydac.api.fem</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.glt.html">psydac.api.glt</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.grid.html">psydac.api.grid</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.postprocessing.html">psydac.api.postprocessing</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/psydac.api.printing.html">psydac.api.printing</a><input class="toctree-checkbox" id="toctree-checkbox-29" name="toctree-checkbox-29" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-29"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.printing.pycode.html">psydac.api.printing.pycode</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.settings.html">psydac.api.settings</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/psydac.api.tests.html">psydac.api.tests</a><input class="toctree-checkbox" id="toctree-checkbox-30" name="toctree-checkbox-30" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-30"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.tests.build_domain.html">psydac.api.tests.build_domain</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.tests.test_2d_biharmonic.html">psydac.api.tests.test_2d_biharmonic</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.tests.test_2d_complex.html">psydac.api.tests.test_2d_complex</a></li>
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+    <h1>SymPDE concepts and their mathematical meaning</h1>
+    <!-- Table of contents -->
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+            <div>
+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#domain">Domain</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mapping">Mapping</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#function-spaces">Function spaces</a><ul class="nav section-nav flex-column">
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#typed-function-spaces">Typed function spaces</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#projections">Projections</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#atomic-variables">Atomic variables</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#algebraic-operators">Algebraic operators</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#differential-operators">Differential operators</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#additional-operators">Additional operators</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#integral-operators">Integral operators</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="sympde-concepts-and-their-mathematical-meaning">
+<h1>SymPDE concepts and their mathematical meaning<a class="headerlink" href="#sympde-concepts-and-their-mathematical-meaning" title="Link to this heading">#</a></h1>
+<section id="domain">
+<h2>Domain<a class="headerlink" href="#domain" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>sympde notation</p></th>
+<th class="head"><p>Mathematical notion</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">Domain('Omega',</span> <span class="pre">dim=n)</span></code></p></td>
+<td><p>an abstract domain <span class="math notranslate nohighlight">\(\Omega \subset \mathbb{R}^n\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">Domain('Omega',</span> <span class="pre">interiors,</span> <span class="pre">boundaries,</span> <span class="pre">connectivity)</span></code></p></td>
+<td><p>an abstract domain <span class="math notranslate nohighlight">\(\Omega \subset \mathbb{R}^n\)</span>, represented by its interiors, boundaries and connectivity</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">Interval('I',</span> <span class="pre">coordinate=x)</span></code></p></td>
+<td><p>an abstract (interior) interval <span class="math notranslate nohighlight">\(I \subset \mathbb{R}\)</span>, with the coordinate <span class="math notranslate nohighlight">\(x\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">Union(A,</span> <span class="pre">B,</span> <span class="pre">C)</span></code></p></td>
+<td><p>union of domains <span class="math notranslate nohighlight">\(A \cup B \cup C\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">ProductDomain(A,</span> <span class="pre">B)</span></code></p></td>
+<td><p>product of two domains <span class="math notranslate nohighlight">\(A \times B\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">Indicator(A)</span></code></p></td>
+<td><p>Indicator/characteristic function on <span class="math notranslate nohighlight">\(A\)</span>, <em>i.e.</em> <span class="math notranslate nohighlight">\(\mathbf{1}\_A\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">Line('Omega')</span></code></p></td>
+<td><p>an abstract line segment, <span class="math notranslate nohighlight">\(\Omega \subset \mathbb{R}\)</span>, having one interval as interior domain</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">Square('Omega')</span></code></p></td>
+<td><p>an abstract square, <span class="math notranslate nohighlight">\(\Omega \subset \mathbb{R}^2\)</span>, having one interior domain as product of two intervals</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">Cube('Omega')</span></code></p></td>
+<td><p>an abstract cube, <span class="math notranslate nohighlight">\(\Omega \subset \mathbb{R}^3\)</span>, having one interior domain as product of three intervals</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">NormalVector('n')</span></code></p></td>
+<td><p>Normal vector <span class="math notranslate nohighlight">\(\mathbf{n}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">TangentVector('t')</span></code></p></td>
+<td><p>Tangent vector <span class="math notranslate nohighlight">\(\mathbf{t}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">e</span> <span class="pre">=</span> <span class="pre">ElementDomain(Omega)</span></code></p></td>
+<td><p>an element of an abstract domain <span class="math notranslate nohighlight">\(e \in \mathcal{T}(\Omega)\)</span> or <span class="math notranslate nohighlight">\(e = d\Omega\)</span>, where <span class="math notranslate nohighlight">\(\mathcal{T}(\Omega)\)</span> is a given tessellation of <span class="math notranslate nohighlight">\(\Omega\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">DomainArea(Omega)</span></code></p></td>
+<td><p>Area of an abstract domain <span class="math notranslate nohighlight">\(\Omega\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">ElementArea(Omega)</span></code></p></td>
+<td><p>Area of an abstract element of a domain <span class="math notranslate nohighlight">\(\Omega\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">Area(A)</span></code></p></td>
+<td><p>Area of an expression of topological domain notions</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="mapping">
+<h2>Mapping<a class="headerlink" href="#mapping" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>sympde notation</p></th>
+<th class="head"><p>Mathematical notion</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">Mapping('F',</span> <span class="pre">n)</span></code></p></td>
+<td><p>a mapping functor <span class="math notranslate nohighlight">\(\mathbf{F}\_{\Omega} := \left( \mathbf{F}, \Omega \right): \Omega \rightarrow \mathbb{R}^n\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">F[i]</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(i^{th}\)</span> physical coordinate from <span class="math notranslate nohighlight">\(\{x,y,z\}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">F.jacobian</span></code></p></td>
+<td><p>the jacobian matrix <span class="math notranslate nohighlight">\(\mathcal{D}\_\mathbf{F}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">F.det_jacobian</span></code></p></td>
+<td><p>determinant of the jacobian matrix, <span class="math notranslate nohighlight">\(J_\mathbf{F}:= \mathrm{det} ~\mathcal{D}\_\mathbf{F}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">F.covariant</span></code></p></td>
+<td><p>the covariant matrix <span class="math notranslate nohighlight">\(\left( \mathcal{D}\_\mathbf{F} \right)^{-T}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">F.contravariant</span></code></p></td>
+<td><p>the contravariant matrix <span class="math notranslate nohighlight">\(\frac{1}{J_\mathbf{F}} \mathcal{D}\_\mathbf{F}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">F.hessian</span></code></p></td>
+<td><p>the hessian matrix <span class="math notranslate nohighlight">\(\mathcal{H}\_\mathbf{F}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">Covariant(F,</span> <span class="pre">v)</span></code></p></td>
+<td><p>action of the covariant matrix of <span class="math notranslate nohighlight">\(\mathbf{F}\)</span> on <span class="math notranslate nohighlight">\(\mathbf{v}\)</span>, <em>i.e.</em> <span class="math notranslate nohighlight">\(\left( \mathcal{D}\_\mathbf{F} \right)^{-T} \mathbf{v}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">Contravariant(F,</span> <span class="pre">v)</span></code></p></td>
+<td><p>action of the contravariant matrix of <span class="math notranslate nohighlight">\(\mathbf{F}\)</span> on <span class="math notranslate nohighlight">\(\mathbf{v}\)</span>, <em>i.e.</em> <span class="math notranslate nohighlight">\(\frac{1}{J_\mathbf{F}} \mathcal{D}\_\mathbf{F} \mathbf{v}\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="function-spaces">
+<h2>Function spaces<a class="headerlink" href="#function-spaces" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>sympde notation</p></th>
+<th class="head"><p>Mathematical notion</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">ScalarFunctionSpace('V',</span> <span class="pre">Omega)</span></code></p></td>
+<td><p>scalar function space <span class="math notranslate nohighlight">\(\mathcal{V}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">VectorFunctionSpace('W',</span> <span class="pre">Omega)</span></code></p></td>
+<td><p>vector function space <span class="math notranslate nohighlight">\(\mathbf{\mathcal{W}}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">ProductSpace(V,</span> <span class="pre">W)</span></code> or <code class="docutils literal notranslate"><span class="pre">V*W</span></code></p></td>
+<td><p>product of spaces, <em>i.e.</em> <span class="math notranslate nohighlight">\(\mathcal{V} \times \mathcal{W}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">element_of_space(V,</span> <span class="pre">'v')</span></code></p></td>
+<td><p>scalar function <span class="math notranslate nohighlight">\(v \in \mathcal{V}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">element_of_space(W,</span> <span class="pre">'w')</span></code></p></td>
+<td><p>vector function <span class="math notranslate nohighlight">\(\mathbf{w} \in \mathcal{W}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">element_of_space(V*W,</span> <span class="pre">['v',</span> <span class="pre">'w'])</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\left(v,\mathbf{w}\right) \in \mathcal{V} \times \mathcal{W}\)</span></p></td>
+</tr>
+</tbody>
+</table>
+<section id="function-space-types">
+<h3>Function space types<a class="headerlink" href="#function-space-types" title="Link to this heading">#</a></h3>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>sympde notation</p></th>
+<th class="head"><p>Mathematical notion</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">H1SpaceType</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(H^1\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">HcurlSpaceType</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\({H}{\mbox{curl}}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">HdivSpaceType</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\({H}{\mbox{div}}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">L2SpaceType</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(L^2\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="typed-function-spaces">
+<h3>Typed function spaces<a class="headerlink" href="#typed-function-spaces" title="Link to this heading">#</a></h3>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>sympde notation</p></th>
+<th class="head"><p>Mathematical notion</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">ScalarFunctionSpace('V0',</span> <span class="pre">Omega,</span> <span class="pre">kind='H1')</span></code></p></td>
+<td><p>scalar function space <span class="math notranslate nohighlight">\(\mathcal{V}\_0 \subseteq H^1(\Omega)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">VectorFunctionSpace('V1',</span> <span class="pre">Omega,</span> <span class="pre">kind='Hcurl')</span></code></p></td>
+<td><p>vector function space <span class="math notranslate nohighlight">\(\mathcal{V}\_1 \subseteq H(\mbox{curl}, \Omega)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">VectorFunctionSpace('V2',</span> <span class="pre">Omega,</span> <span class="pre">kind='Hdiv')</span> </code></p></td>
+<td><p>vector function space <span class="math notranslate nohighlight">\(\mathcal{V}\_2 \subseteq H(\mbox{div}, \Omega)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">ScalarFunctionSpace('V3',</span> <span class="pre">Omega,</span> <span class="pre">kind='L2')</span></code></p></td>
+<td><p>scalar function space <span class="math notranslate nohighlight">\(\mathcal{V}\_3 \subseteq L^2(\Omega)\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+</section>
+<section id="projections">
+<h2>Projections<a class="headerlink" href="#projections" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>sympde notation</p></th>
+<th class="head"><p>Mathematical notion</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">P_V</span> <span class="pre">:=</span> <span class="pre">Projector(V)</span></code></p></td>
+<td><p>Projector onto the function space <span class="math notranslate nohighlight">\(\mathcal{V}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">P_V(expr)</span></code></p></td>
+<td><p>Projection of the expression <code class="docutils literal notranslate"><span class="pre">expr</span></code> onto the function space <span class="math notranslate nohighlight">\(\mathcal{V}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">Pi_V</span> <span class="pre">:=</span> <span class="pre">Projector(V,</span> <span class="pre">'commuting')</span></code></p></td>
+<td><p>Commuting projector onto the typed function space <span class="math notranslate nohighlight">\(\mathcal{V}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">Pi_V(expr)</span></code></p></td>
+<td><p>Commuting projection of the expression <code class="docutils literal notranslate"><span class="pre">expr</span></code> onto the typed function space <span class="math notranslate nohighlight">\(\mathcal{V}\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="atomic-variables">
+<h2>Atomic variables<a class="headerlink" href="#atomic-variables" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>sympde notation</p></th>
+<th class="head"><p>Mathematical notion</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">ScalarFunction(V,</span> <span class="pre">'u')</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(u \in \mathcal{V}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">VectorFunction(W,</span> <span class="pre">'w')</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathbf{w} \in \mathcal{W}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">Constant('mu',</span> <span class="pre">real=True)</span></code></p></td>
+<td><p>constant number <span class="math notranslate nohighlight">\(\mu \in \mathbb{R}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">int</span></code></p></td>
+<td><p>integer number</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">float</span></code></p></td>
+<td><p>floating-point number</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="algebraic-operators">
+<h2>Algebraic operators<a class="headerlink" href="#algebraic-operators" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>sympde notation</p></th>
+<th class="head"><p>Mathematical notion</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">dot(u,</span> <span class="pre">v)</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathbf{u} \cdot \mathbf{v}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">inner(u,</span> <span class="pre">v)</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathbf{u} : \mathbf{v}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">cross(u,</span> <span class="pre">v)</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathbf{u} \times \mathbf{v}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">outer(u,</span> <span class="pre">v)</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\mathbf{u} \otimes \mathbf{v}\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="differential-operators">
+<h2>Differential operators<a class="headerlink" href="#differential-operators" title="Link to this heading">#</a></h2>
+<p>boldface font is used for vector functions/expressions</p>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>sympde notation</p></th>
+<th class="head"><p>Mathematical notion</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">dx(u)</span></code>, <code class="docutils literal notranslate"><span class="pre">dy(u)</span></code> or <code class="docutils literal notranslate"><span class="pre">dz(u)</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\partial_x u\)</span>, <span class="math notranslate nohighlight">\(\partial_y u\)</span> or <span class="math notranslate nohighlight">\(\partial_z u\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">grad(u)</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\nabla u\)</span> or <span class="math notranslate nohighlight">\(\nabla \mathbf{u}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">div(u)</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\nabla \cdot \mathbf{u}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">curl(u)</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\nabla \times \mathbf{u}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">rot(u)</span> </code></p></td>
+<td><p>2D rotational <span class="math notranslate nohighlight">\(\nabla \times u\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">convect(a,</span> <span class="pre">u)</span></code></p></td>
+<td><p><span class="math notranslate nohighlight">\(\left( \mathbf{a} \cdot \nabla \right) \mathbf{u}\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">trace(u)</span></code></p></td>
+<td><p>trace <span class="math notranslate nohighlight">\(\gamma(u)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">Dn(u)</span></code></p></td>
+<td><p>normal derivative <span class="math notranslate nohighlight">\(\partial_{\mathbf{n}} u\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">D(u)</span></code></p></td>
+<td><p>Strain tensor <span class="math notranslate nohighlight">\(\frac{1}{2}\left(\nabla \mathbf{u} + \nabla \mathbf{u}^T \right) \)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">laplace(u)</span></code></p></td>
+<td><p>Laplace <span class="math notranslate nohighlight">\(\Delta u\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">hessian(u)</span></code></p></td>
+<td><p>Hessian <span class="math notranslate nohighlight">\(H(u)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">bracket(u,v)</span></code></p></td>
+<td><p>Poisson Bracket <span class="math notranslate nohighlight">\([u,v]\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="additional-operators">
+<h2>Additional operators<a class="headerlink" href="#additional-operators" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>sympde notation</p></th>
+<th class="head"><p>Mathematical notion</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">jump(u)</span></code></p></td>
+<td><p>jump of <span class="math notranslate nohighlight">\(u\)</span>, <em>i.e.</em> <span class="math notranslate nohighlight">\([u]\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">avg(u)</span></code></p></td>
+<td><p>average of <span class="math notranslate nohighlight">\(u\)</span>, <em>i.e.</em> <span class="math notranslate nohighlight">\(\langle u \rangle\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">conv(K,u)</span></code></p></td>
+<td><p>convolution of <span class="math notranslate nohighlight">\(u\)</span> with a kernel <span class="math notranslate nohighlight">\(K\)</span>, <em>i.e.</em> <span class="math notranslate nohighlight">\(K * u\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+<section id="integral-operators">
+<h2>Integral operators<a class="headerlink" href="#integral-operators" title="Link to this heading">#</a></h2>
+<table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>sympde notation</p></th>
+<th class="head"><p>Mathematical notion</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">DomainIntegral(f)</span></code></p></td>
+<td><p>integral over a domain, <em>i.e.</em> <span class="math notranslate nohighlight">\((f, \Omega) \mapsto \int_{\Omega} f ~d\Omega\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">BoundaryIntegral(f)</span></code></p></td>
+<td><p>integral over a boundary, <em>i.e.</em> <span class="math notranslate nohighlight">\((f, \Gamma) \mapsto \int_{\Gamma} f ~d\partial\Omega\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="docutils literal notranslate"><span class="pre">PathIntegral(F)</span></code></p></td>
+<td><p>integral over an oriented path, <em>i.e.</em> <span class="math notranslate nohighlight">\((\mathbf{F}, C) \mapsto \int_{C} \mathbf{F} \cdot d\mathbf{s}\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="docutils literal notranslate"><span class="pre">SurfaceIntegral(F)</span></code></p></td>
+<td><p>integral over an oriented surface, <em>i.e.</em> <span class="math notranslate nohighlight">\((\mathbf{F}, S) \mapsto \int_{S} \mathbf{F} \cdot d\mathbf{S}\)</span></p></td>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
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+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.utilities.utils.html">sympde.utilities.utils</a></li>
+</ul>
+</li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.expr.html">psydac.api.ast.expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.glt.html">psydac.api.ast.glt</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.utilities.html">psydac.api.ast.utilities</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.discretization.html">psydac.api.discretization</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.equation.html">psydac.api.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.essential_bc.html">psydac.api.essential_bc</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.expr.html">psydac.api.expr</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.feec.html">psydac.api.feec</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.fem.html">psydac.api.fem</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.glt.html">psydac.api.glt</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.grid.html">psydac.api.grid</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.postprocessing.html">psydac.api.postprocessing</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.printing.pycode.html">psydac.api.printing.pycode</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.settings.html">psydac.api.settings</a></li>
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+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ncubeinterior">NCubeInterior</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id7">Properties</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#additional-geometry-objects">Additional geometry objects</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#vectors">Vectors</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#boundaryvector">BoundaryVector</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#normalvector">NormalVector</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#minusnormalvector">MinusNormalVector</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id8">MinusNormalVector</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tangentvector">TangentVector</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#elementdomain">ElementDomain</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#basicarea">BasicArea</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#domainarea">DomainArea</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#elementarea">ElementArea</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#geometry-operations">Geometry Operations</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#basicgeometryoperator">BasicGeometryOperator</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#area">Area</a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="topology-concepts">
+<h1>Topology concepts<a class="headerlink" href="#topology-concepts" title="Link to this heading">#</a></h1>
+<section id="domain">
+<h2>Domain<a class="headerlink" href="#domain" title="Link to this heading">#</a></h2>
+<p>Mathematicaly speaking, a domain <span class="math notranslate nohighlight">\(\Omega\)</span> is defined as an interior <span class="math notranslate nohighlight">\(\a{\Omega}\)</span> and a boundary <span class="math notranslate nohighlight">\(\partial \Omega\)</span>.
+A domain is defined by its <strong>interior</strong> and its <strong>boundaries</strong>. For instance, a domain without a boundary is periodic, since, <strong>sympde</strong> does not consider infinite domains.</p>
+<p>In addition to these two attributes, a <strong>Domain</strong> can also be defined as the image of a <em>logical domain</em> using a <em>mapping</em>.</p>
+<p>Finally, a <strong>Domain</strong> can have subdomains, in which case, a <strong>connectivity</strong> must be provided as well.</p>
+<p>The simplest <strong>Domain</strong> is the <strong>NCube</strong> object. For instance, <strong>Line</strong>, <strong>Square</strong> and <strong>Cube</strong> are specific cases of <strong>NCube</strong>.</p>
+<p>Let’s take a look at the <strong>Line</strong> object.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympde</span> <span class="kn">import</span> <span class="n">topology</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">domain</span> <span class="o">=</span> <span class="n">topology</span><span class="o">.</span><span class="n">Line</span><span class="p">()</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">domain</span><span class="p">))</span>
+<span class="go">sympde.topology.domain.Line</span>
+</pre></div>
+</div>
+<p>The parent class of <strong>Line</strong> is <strong>NCube</strong> as we can see in our example</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">topology</span><span class="o">.</span><span class="n">Line</span><span class="o">.</span><span class="vm">__bases__</span><span class="p">))</span>
+<span class="go">sympde.topology.domain.NCube</span>
+</pre></div>
+</div>
+<p>In fact, the objects <strong>Line</strong>, <strong>Square</strong> and <strong>Cube</strong> are subclasses of <strong>NCube</strong> which is a subclasse of <strong>Domain</strong> that is an extension of the <strong>BasicDomain</strong> object.</p>
+<section id="properties">
+<h3>Properties<a class="headerlink" href="#properties" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p><strong>name</strong>: returns the domain’s name</p></li>
+<li><p><strong>interior</strong>: returns the interior of the domain</p></li>
+<li><p><strong>boundary</strong>: returns the boundary of the domain</p></li>
+<li><p><strong>mapping</strong>: returns the mapping, if associated to the domain</p></li>
+<li><p><strong>logical_domain</strong>: returns the logical domain if a mapping is used</p></li>
+<li><p><strong>connectivity</strong>: returns information about the interfaces of the subdomains</p></li>
+<li><p><strong>dim</strong>: dimension of the space</p></li>
+<li><p><strong>dtype</strong>: a dictionary that contains information about the domain</p></li>
+<li><p><strong>interfaces</strong>: returns the interfaces of the subdomains</p></li>
+<li><p><strong>corners</strong>: returns the corners of the domain</p></li>
+</ul>
+<p><strong>TODO</strong> add inherited properties</p>
+</section>
+<section id="examples">
+<h3>Examples<a class="headerlink" href="#examples" title="Link to this heading">#</a></h3>
+<section id="example-1">
+<h4>Example 1<a class="headerlink" href="#example-1" title="Link to this heading">#</a></h4>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">InteriorDomain</span><span class="p">,</span> <span class="n">Boundary</span><span class="p">,</span> <span class="n">Domain</span>
+
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Omega_1</span> <span class="o">=</span> <span class="n">InteriorDomain</span><span class="p">(</span><span class="s1">&#39;Omega_1&#39;</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Omega_2</span> <span class="o">=</span> <span class="n">InteriorDomain</span><span class="p">(</span><span class="s1">&#39;Omega_2&#39;</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Gamma_1</span> <span class="o">=</span> <span class="n">Boundary</span><span class="p">(</span><span class="s1">&#39;Gamma_1&#39;</span><span class="p">,</span> <span class="n">Omega_1</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Gamma_2</span> <span class="o">=</span> <span class="n">Boundary</span><span class="p">(</span><span class="s1">&#39;Gamma_2&#39;</span><span class="p">,</span> <span class="n">Omega_2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Gamma_3</span> <span class="o">=</span> <span class="n">Boundary</span><span class="p">(</span><span class="s1">&#39;Gamma_3&#39;</span><span class="p">,</span> <span class="n">Omega_2</span><span class="p">)</span>
+
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Omega</span> <span class="o">=</span> <span class="n">Domain</span><span class="p">(</span><span class="s1">&#39;Omega&#39;</span><span class="p">,</span>
+<span class="go">                   interiors=[Omega_1, Omega_2],</span>
+<span class="go">                   boundaries=[Gamma_1, Gamma_2, Gamma_3])</span>
+
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span> <span class="n">Omega</span><span class="o">.</span><span class="n">dim</span> <span class="p">)</span>
+<span class="go">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span> <span class="nb">len</span><span class="p">(</span><span class="n">Omega</span><span class="o">.</span><span class="n">interior</span><span class="p">)</span> <span class="p">)</span>
+<span class="go">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span> <span class="nb">len</span><span class="p">(</span><span class="n">Omega</span><span class="o">.</span><span class="n">boundary</span><span class="p">)</span> <span class="p">)</span>
+<span class="go">3</span>
+</pre></div>
+</div>
+</section>
+<section id="example-2">
+<h4>Example 2<a class="headerlink" href="#example-2" title="Link to this heading">#</a></h4>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">InteriorDomain</span><span class="p">,</span> <span class="n">Boundary</span><span class="p">,</span> <span class="n">Domain</span>
+
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Omega_1</span> <span class="o">=</span> <span class="n">InteriorDomain</span><span class="p">(</span><span class="s1">&#39;Omega_1&#39;</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Gamma_1</span> <span class="o">=</span> <span class="n">Boundary</span><span class="p">(</span><span class="s1">&#39;Gamma_1&#39;</span><span class="p">,</span> <span class="n">Omega_1</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Gamma_2</span> <span class="o">=</span> <span class="n">Boundary</span><span class="p">(</span><span class="s1">&#39;Gamma_2&#39;</span><span class="p">,</span> <span class="n">Omega_1</span><span class="p">)</span>
+
+<span class="gp">&gt;&gt;&gt; </span><span class="n">Omega</span> <span class="o">=</span> <span class="n">Domain</span><span class="p">(</span><span class="s1">&#39;Omega&#39;</span><span class="p">,</span>
+<span class="go">                   interiors=[Omega_1],</span>
+<span class="go">                   boundaries=[Gamma_1, Gamma_2])</span>
+
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">Omega</span><span class="o">.</span><span class="n">boundary</span> <span class="o">==</span> <span class="n">Union</span><span class="p">(</span><span class="n">Gamma_1</span><span class="p">,</span> <span class="n">Gamma_2</span><span class="p">))</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">Omega</span><span class="o">.</span><span class="n">boundary</span><span class="o">.</span><span class="n">complement</span><span class="p">(</span><span class="n">Gamma_1</span><span class="p">)</span> <span class="o">==</span> <span class="n">Gamma_2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span><span class="n">Omega</span><span class="o">.</span><span class="n">boundary</span> <span class="o">-</span> <span class="n">Gamma_1</span> <span class="o">==</span> <span class="n">Gamma_2</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+</section>
+</section>
+<section id="periodicdomain">
+<h2>PeriodicDomain<a class="headerlink" href="#periodicdomain" title="Link to this heading">#</a></h2>
+<p>Similar to the <strong>Domain</strong> object, a <strong>PeriodicDomain</strong> is defined by giving a <strong>Domain</strong> and the associated <strong>periods</strong>.</p>
+<section id="id1">
+<h3>Properties<a class="headerlink" href="#id1" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p><strong>domain</strong>: returns the domain</p></li>
+<li><p><strong>periods</strong>: returns the flags for each axis</p></li>
+<li><p><strong>boundary</strong>: return the boundary of the domain</p></li>
+<li><p><strong>dim</strong> returns the dimension of the domain</p></li>
+<li><p><strong>coordinates</strong> return the coordinates symbols</p></li>
+</ul>
+</section>
+</section>
+<section id="ncube">
+<h2>NCube<a class="headerlink" href="#ncube" title="Link to this heading">#</a></h2>
+<p>An <strong>NCube</strong> is a <strong>Domain</strong> that models hypercubes in <span class="math notranslate nohighlight">\(\mathbb{R}^n\)</span>. It is defined by specifying its <strong>name</strong>, <strong>dim</strong>, <strong>min_coords</strong> and <strong>max_coords</strong>.</p>
+<p>The interior of a <strong>NCube</strong> is defined as <strong>NCubeInterior</strong>.
+To make things easier, we provide specific classes in the case where <span class="math notranslate nohighlight">\(n \in \{ 1, 2, 3 \}\)</span>.</p>
+<section id="line">
+<h3>Line<a class="headerlink" href="#line" title="Link to this heading">#</a></h3>
+<p>This corresponds to the case where <span class="math notranslate nohighlight">\(n=1\)</span>, meaning it is <span class="math notranslate nohighlight">\((x_{min}, x_{max})\)</span>.</p>
+<section id="id2">
+<h4>Properties<a class="headerlink" href="#id2" title="Link to this heading">#</a></h4>
+<ul class="simple">
+<li><p><strong>bounds</strong>: the bounds for the line, i.e. <span class="math notranslate nohighlight">\(x_{min}\)</span> and <span class="math notranslate nohighlight">\(x_{max}\)</span>.</p></li>
+</ul>
+</section>
+</section>
+<section id="square">
+<h3>Square<a class="headerlink" href="#square" title="Link to this heading">#</a></h3>
+<p>This corresponds to the case where <span class="math notranslate nohighlight">\(n=2\)</span>, meaning it is <span class="math notranslate nohighlight">\((x_{min}, x_{max}) \times (y_{min}, y_{max})\)</span>.</p>
+<section id="id3">
+<h4>Properties<a class="headerlink" href="#id3" title="Link to this heading">#</a></h4>
+<ul class="simple">
+<li><p><strong>bounds1</strong>: the bounds in the first axis, i.e. <span class="math notranslate nohighlight">\(x_{min}\)</span> and <span class="math notranslate nohighlight">\(x_{max}\)</span>.</p></li>
+<li><p><strong>bounds2</strong>: the bounds in the first axis, i.e. <span class="math notranslate nohighlight">\(y_{min}\)</span> and <span class="math notranslate nohighlight">\(y_{max}\)</span>.</p></li>
+</ul>
+</section>
+</section>
+<section id="cube">
+<h3>Cube<a class="headerlink" href="#cube" title="Link to this heading">#</a></h3>
+<p>This corresponds to the case where <span class="math notranslate nohighlight">\(n=3\)</span>, meaning it is <span class="math notranslate nohighlight">\((x_{min}, x_{max}) \times (y_{min}, y_{max}) \times (z_{min}, z_{max})\)</span>.</p>
+<section id="id4">
+<h4>Properties<a class="headerlink" href="#id4" title="Link to this heading">#</a></h4>
+<ul class="simple">
+<li><p><strong>bounds1</strong>: the bounds in the first axis, i.e. <span class="math notranslate nohighlight">\(x_{min}\)</span> and <span class="math notranslate nohighlight">\(x_{max}\)</span>.</p></li>
+<li><p><strong>bounds2</strong>: the bounds in the first axis, i.e. <span class="math notranslate nohighlight">\(y_{min}\)</span> and <span class="math notranslate nohighlight">\(y_{max}\)</span>.</p></li>
+<li><p><strong>bounds3</strong>: the bounds in the first axis, i.e. <span class="math notranslate nohighlight">\(z_{min}\)</span> and <span class="math notranslate nohighlight">\(z_{max}\)</span>.</p></li>
+</ul>
+</section>
+</section>
+</section>
+<section id="interiordomain">
+<h2>InteriorDomain<a class="headerlink" href="#interiordomain" title="Link to this heading">#</a></h2>
+<p><strong>TODO</strong></p>
+<section id="id5">
+<h3>Examples<a class="headerlink" href="#id5" title="Link to this heading">#</a></h3>
+<section id="id6">
+<h4>Example 1<a class="headerlink" href="#id6" title="Link to this heading">#</a></h4>
+<p>In this example, we create two interior domains and then we build their union.</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">InteriorDomain</span><span class="p">,</span> <span class="n">Union</span>
+
+<span class="gp">&gt;&gt;&gt; </span><span class="n">D1</span> <span class="o">=</span> <span class="n">InteriorDomain</span><span class="p">(</span><span class="s1">&#39;D1&#39;</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="n">D2</span> <span class="o">=</span> <span class="n">InteriorDomain</span><span class="p">(</span><span class="s1">&#39;D2&#39;</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+
+<span class="gp">&gt;&gt;&gt; </span><span class="c1"># Union is commutative</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="k">assert</span><span class="p">(</span> <span class="n">Union</span><span class="p">(</span><span class="n">D2</span><span class="p">,</span> <span class="n">D1</span><span class="p">)</span> <span class="o">==</span> <span class="n">Union</span><span class="p">(</span><span class="n">D1</span><span class="p">,</span> <span class="n">D2</span><span class="p">)</span> <span class="p">)</span>
+
+<span class="gp">&gt;&gt;&gt; </span><span class="n">D</span> <span class="o">=</span> <span class="n">Union</span><span class="p">(</span><span class="n">D1</span><span class="p">,</span> <span class="n">D2</span><span class="p">)</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="c1"># the union will be of dimension 2, and has a length of 2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="n">D</span><span class="o">.</span><span class="n">dim</span><span class="p">)</span>
+<span class="go">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">D</span><span class="p">))</span>
+<span class="go">2</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="c1"># we see it clearly when converting it to a dictionary</span>
+<span class="gp">&gt;&gt;&gt; </span><span class="nb">print</span><span class="p">(</span><span class="n">D</span><span class="o">.</span><span class="n">todict</span><span class="p">())</span>
+<span class="go">[{&#39;name&#39;: &#39;D1&#39;, &#39;mapping&#39;: &#39;None&#39;}, {&#39;name&#39;: &#39;D2&#39;, &#39;mapping&#39;: &#39;None&#39;}]</span>
+</pre></div>
+</div>
+</section>
+</section>
+</section>
+<section id="ncubeinterior">
+<h2>NCubeInterior<a class="headerlink" href="#ncubeinterior" title="Link to this heading">#</a></h2>
+<p>This class defines the interior of a domain which is <strong>NCube</strong>.</p>
+<section id="id7">
+<h3>Properties<a class="headerlink" href="#id7" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p><strong>min_coords</strong></p></li>
+<li><p><strong>max_coords</strong></p></li>
+<li><p><strong>boundary</strong></p></li>
+</ul>
+</section>
+</section>
+<section id="additional-geometry-objects">
+<h2>Additional geometry objects<a class="headerlink" href="#additional-geometry-objects" title="Link to this heading">#</a></h2>
+<section id="vectors">
+<h3>Vectors<a class="headerlink" href="#vectors" title="Link to this heading">#</a></h3>
+<section id="boundaryvector">
+<h4>BoundaryVector<a class="headerlink" href="#boundaryvector" title="Link to this heading">#</a></h4>
+<p>is an extension of the <strong>IndexedBase</strong> object from <strong>sympy</strong>.</p>
+</section>
+<section id="normalvector">
+<h4>NormalVector<a class="headerlink" href="#normalvector" title="Link to this heading">#</a></h4>
+<p>defines the normal vector to boundaries. It is an extension of the <strong>BoundaryVector</strong> object.</p>
+</section>
+<section id="minusnormalvector">
+<h4>MinusNormalVector<a class="headerlink" href="#minusnormalvector" title="Link to this heading">#</a></h4>
+<p>defines the inward normal vector to a boundary. It is an extension of the <strong>NormalVector</strong> object.</p>
+</section>
+<section id="id8">
+<h4>MinusNormalVector<a class="headerlink" href="#id8" title="Link to this heading">#</a></h4>
+<p>defines the outward normal vector to a boundary. It is an extension of the <strong>NormalVector</strong> object.</p>
+</section>
+<section id="tangentvector">
+<h4>TangentVector<a class="headerlink" href="#tangentvector" title="Link to this heading">#</a></h4>
+<p>defines the tangent vector to boundaries. It is an extension of the <strong>BoundaryVector</strong> object.</p>
+</section>
+</section>
+<section id="elementdomain">
+<h3>ElementDomain<a class="headerlink" href="#elementdomain" title="Link to this heading">#</a></h3>
+<p>defines an element of the domain.</p>
+</section>
+<section id="basicarea">
+<h3>BasicArea<a class="headerlink" href="#basicarea" title="Link to this heading">#</a></h3>
+<p>A basic class defining the concept of an area. It is an extension of the <strong>AtomicExpr</strong> object from <strong>sympy</strong>, this way we can use it in arithmetic expressions.</p>
+</section>
+<section id="domainarea">
+<h3>DomainArea<a class="headerlink" href="#domainarea" title="Link to this heading">#</a></h3>
+<p>defines the area of a <strong>Domain</strong>.</p>
+</section>
+<section id="elementarea">
+<h3>ElementArea<a class="headerlink" href="#elementarea" title="Link to this heading">#</a></h3>
+<p>defines the area of an element of a <strong>Domain</strong>.</p>
+</section>
+<section id="geometry-operations">
+<h3>Geometry Operations<a class="headerlink" href="#geometry-operations" title="Link to this heading">#</a></h3>
+<p><strong>TODO</strong></p>
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+<p><strong>TODO</strong></p>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Properties</a></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ncube">NCube</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#line">Line</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Properties</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#square">Square</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">Properties</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#cube">Cube</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id4">Properties</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#interiordomain">InteriorDomain</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id5">Examples</a><ul class="nav section-nav flex-column">
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+</ul>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ncubeinterior">NCubeInterior</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id7">Properties</a></li>
+</ul>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#additional-geometry-objects">Additional geometry objects</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#vectors">Vectors</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#boundaryvector">BoundaryVector</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#normalvector">NormalVector</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#minusnormalvector">MinusNormalVector</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#id8">MinusNormalVector</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#tangentvector">TangentVector</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#elementdomain">ElementDomain</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#basicarea">BasicArea</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#domainarea">DomainArea</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#elementarea">ElementArea</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#geometry-operations">Geometry Operations</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#basicgeometryoperator">BasicGeometryOperator</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#area">Area</a></li>
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+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
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+</li>
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+</ul>
+</li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.glt.html">psydac.api.ast.glt</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
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+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Stabilized advection-diffusion equation</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#build-a-manifactured-solution">Build a manifactured solution</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="stabilized-advection-diffusion-equation">
+<h1>Stabilized advection-diffusion equation<a class="headerlink" href="#stabilized-advection-diffusion-equation" title="Link to this heading">#</a></h1>
+<p>In the sequel we consider two different stabilization methods, leading to formulations of the form</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \text{find $u \in V$ such that} \quad 
+  a(u,v) + a_s(u,v) = l(v) + l_s(v) \quad \forall v \in V,
+\end{align}
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(a_s\)</span> and <span class="math notranslate nohighlight">\(l_s\)</span> are additional terms depending on the stabilization method.</p>
+<p>The streamline upwind Petrov-Galerkin (SUPG) method is given by</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+    a_s(u,v) = \int_{\Omega} \tau ~ L(u) ~ \left( \mathbf{b} \cdot \nabla v \right) ~d\Omega 
+    \quad \mbox{and} \quad
+    l_s(v) = \int_{\Omega} \tau ~ f ~ \left( \mathbf{b} \cdot \nabla v \right) ~d\Omega 
+  &amp;&amp; \text{[SUPG]}
+\end{align}
+\]</div>
+<p>while the Galerkin least squares (GLS) method is as follows,</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+    a_s(u,v) = \int_{\Omega} \tau ~ L(u) ~ L(v) ~d\Omega 
+    \quad \mbox{and} \quad
+    l_s(v) = \int_{\Omega} \tau ~ f ~ L(v) ~d\Omega 
+  &amp;&amp; \text{[GLS]}
+\end{align}
+\]</div>
+<p>where we introduced the differential operator <span class="math notranslate nohighlight">\(L(u) := - \kappa \nabla^2 u + \mathbf{b} \cdot \nabla u\)</span>.</p>
+<p>Notice that in these formulations <span class="math notranslate nohighlight">\(\tau\)</span> stands for a piecewise function that is constant on each element of the tessellation associated to the computational domain <span class="math notranslate nohighlight">\(\Omega\)</span>.</p>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<section id="build-a-manifactured-solution">
+<h3>Build a manifactured solution<a class="headerlink" href="#build-a-manifactured-solution" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">laplace</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span>     <span class="kn">import</span> <span class="n">Constant</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">Tuple</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">kappa</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;kappa&#39;</span><span class="p">,</span> <span class="n">is_real</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">b1</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">b2</span> <span class="o">=</span> <span class="mf">0.</span>
+<span class="n">b</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">b1</span><span class="p">,</span> <span class="n">b2</span><span class="p">)</span>
+
+<span class="n">L</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">w</span><span class="p">:</span> <span class="o">-</span><span class="n">kappa</span> <span class="o">*</span> <span class="n">laplace</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="n">grad</span><span class="p">(</span><span class="n">w</span><span class="p">))</span>
+
+<span class="n">ue</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="n">f</span> <span class="o">=</span> <span class="n">L</span><span class="p">(</span><span class="n">ue</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>You can have the full expression of <strong>f</strong> by calling the <strong>TerminalExpr</strong> as follows</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">TerminalExpr</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="n">TerminalExpr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">domain</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2*kappa*pi**2*sin(pi*x1)*sin(pi*x2) + 1.0*pi*sin(pi*x2)*cos(pi*x1)
+</pre></div>
+</div>
+</div>
+</div>
+<p>Let’s go back now to our formal model.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">V</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">expr</span> <span class="o">=</span> <span class="n">kappa</span> <span class="o">*</span> <span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))</span> <span class="o">*</span> <span class="n">v</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">expr</span><span class="p">))</span>
+
+<span class="c1"># linear form</span>
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">f</span><span class="o">*</span><span class="n">v</span><span class="p">))</span>
+
+<span class="n">tau</span>   <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;tau&#39;</span><span class="p">,</span> <span class="n">is_real</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="c1"># ...</span>
+<span class="n">s_supg</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">v</span><span class="p">,</span><span class="n">u</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">tau</span> <span class="o">*</span> <span class="n">L</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">*</span> <span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">))))</span>
+
+<span class="n">l_supg</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">tau</span> <span class="o">*</span> <span class="n">f</span> <span class="o">*</span> <span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">))))</span>
+<span class="c1"># ...</span>
+
+<span class="c1"># ...</span>
+<span class="n">s_gls</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">v</span><span class="p">,</span><span class="n">u</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">tau</span> <span class="o">*</span> <span class="n">L</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">*</span> <span class="n">L</span><span class="p">(</span><span class="n">v</span><span class="p">)))</span>
+
+<span class="n">l_gls</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">tau</span> <span class="o">*</span> <span class="n">f</span> <span class="o">*</span> <span class="n">L</span><span class="p">(</span><span class="n">v</span><span class="p">)))</span>
+<span class="c1"># ...</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation_supg</span> <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o">+</span> <span class="n">s_supg</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">+</span> <span class="n">l_supg</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+<span class="n">equation_gls</span>  <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o">+</span> <span class="n">s_gls</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">+</span> <span class="n">l_gls</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<p>We shall need the <strong>discretize</strong> function from <strong>PsyDAC</strong>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">Vh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation SUPG</span>
+<span class="n">equation_supg_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation_supg</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">])</span>
+
+<span class="c1"># Discretize equation GLS</span>
+<span class="n">equation_gls_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation_gls</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<p>Since the problem is not symmetric, we shall use <strong>gmres</strong> for the linear solver.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">equation_supg_h</span><span class="o">.</span><span class="n">set_solver</span><span class="p">(</span><span class="s1">&#39;gmres&#39;</span><span class="p">,</span> <span class="n">info</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-8</span><span class="p">)</span>
+<span class="n">equation_gls_h</span><span class="o">.</span><span class="n">set_solver</span><span class="p">(</span><span class="s1">&#39;gmres&#39;</span><span class="p">,</span> <span class="n">info</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-8</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">uh_supg</span> <span class="o">=</span> <span class="n">equation_supg_h</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">kappa</span><span class="o">=</span><span class="mf">1.e-1</span><span class="p">,</span> <span class="n">tau</span><span class="o">=</span><span class="mf">1.e-3</span><span class="p">)</span>
+<span class="n">uh_gls</span>  <span class="o">=</span> <span class="n">equation_gls_h</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">kappa</span><span class="o">=</span><span class="mf">1.e-1</span><span class="p">,</span> <span class="n">tau</span><span class="o">=</span><span class="mf">1.e-3</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">u</span> <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+
+<span class="c1"># create the formal Norm object</span>
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">l2norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">l2_error</span> <span class="o">=</span> <span class="n">l2norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh_supg</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;&gt;&gt;&gt; SUPG = &#39;</span><span class="p">,</span> <span class="n">l2_error</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">l2_error</span> <span class="o">=</span> <span class="n">l2norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh_gls</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;&gt;&gt;&gt; GLS = &#39;</span><span class="p">,</span> <span class="n">l2_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&gt;&gt;&gt; SUPG =  0.002139370600719921
+&gt;&gt;&gt; GLS =  0.00021873265193415864
+</pre></div>
+</div>
+</div>
+</div>
+</section>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
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+</li>
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+</ul>
+</li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.glt.html">psydac.api.ast.glt</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
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+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
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+document.write(`
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Advection-diffusion equation</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-variational-formulation">The Variational Formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#build-a-manifactured-solution">Build a manifactured solution</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="advection-diffusion-equation">
+<h1>Advection-diffusion equation<a class="headerlink" href="#advection-diffusion-equation" title="Link to this heading">#</a></h1>
+<p>We consider the advection-diffusion problem consisting of finding a scalar-valued function <span class="math notranslate nohighlight">\(u\)</span> such that</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+  \begin{cases}
+    - \kappa \nabla^2 u + \mathbf{b} \cdot \nabla u = f &amp;\text{in $\Omega$}, \\
+    u = 0                                               &amp;\text{on $\partial \Omega$}.
+  \end{cases}
+\end{align}
+\end{split}\]</div>
+<section id="the-variational-formulation">
+<h2>The Variational Formulation<a class="headerlink" href="#the-variational-formulation" title="Link to this heading">#</a></h2>
+<p>The variational formulation reads</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \text{find $u \in V$ such that} \quad a(u,v) = l(v) \quad \forall v \in V,
+\end{align}
+\]</div>
+<p>where</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(V \subset H_0^1(\Omega)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(a(u,v) := \int_{\Omega} \left( \kappa \nabla u \cdot \nabla v + \left( \mathbf{b} \cdot \nabla u \right) v \right) d\Omega\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(l(v) := \int_{\Omega} f v~d\Omega\)</span>.</p></li>
+</ul>
+</section>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<section id="build-a-manifactured-solution">
+<h3>Build a manifactured solution<a class="headerlink" href="#build-a-manifactured-solution" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">laplace</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span>     <span class="kn">import</span> <span class="n">Constant</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">Tuple</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">kappa</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;kappa&#39;</span><span class="p">,</span> <span class="n">is_real</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">b1</span> <span class="o">=</span> <span class="mf">1.</span>
+<span class="n">b2</span> <span class="o">=</span> <span class="mf">0.</span>
+<span class="n">b</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">b1</span><span class="p">,</span> <span class="n">b2</span><span class="p">)</span>
+
+<span class="n">L</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">w</span><span class="p">:</span> <span class="o">-</span><span class="n">kappa</span> <span class="o">*</span> <span class="n">laplace</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="n">grad</span><span class="p">(</span><span class="n">w</span><span class="p">))</span>
+
+<span class="n">ue</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="n">f</span> <span class="o">=</span> <span class="n">L</span><span class="p">(</span><span class="n">ue</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>You can have the full expression of <strong>f</strong> by calling the <strong>TerminalExpr</strong> as follows</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">TerminalExpr</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="n">TerminalExpr</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">domain</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2*kappa*pi**2*sin(pi*x1)*sin(pi*x2) + 1.0*pi*sin(pi*x2)*cos(pi*x1)
+</pre></div>
+</div>
+</div>
+</div>
+<p>Let’s go back now to our formal model.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">V</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">expr</span> <span class="o">=</span> <span class="n">kappa</span> <span class="o">*</span> <span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))</span> <span class="o">*</span> <span class="n">v</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">expr</span><span class="p">))</span>
+
+<span class="c1"># linear form</span>
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">f</span><span class="o">*</span><span class="n">v</span><span class="p">))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<p>We shall need the <strong>discretize</strong> function from <strong>PsyDAC</strong>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">Vh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<p>Since the problem is not symmetric, we shall use <strong>gmres</strong> for the linear solver.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">equation_h</span><span class="o">.</span><span class="n">set_solver</span><span class="p">(</span><span class="s1">&#39;gmres&#39;</span><span class="p">,</span> <span class="n">info</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-8</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">uh</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">kappa</span><span class="o">=</span><span class="mf">1.e-1</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">u</span> <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+
+<span class="c1"># create the formal Norm object</span>
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">l2norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">l2_error</span> <span class="o">=</span> <span class="n">l2norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">l2_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.00022156219679932395
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-semi-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm<a class="headerlink" href="#computing-the-h-1-semi-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">SemiNorm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">h1norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">h1_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.013033907531543579
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm<a class="headerlink" href="#computing-the-h-1-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">h1norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">h1_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.013035790553238334
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_field_2d.html">sympde.topology.tests.test_field_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.equation.html">sympde.expr.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_equation_2d.html">sympde.expr.tests.test_equation_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
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+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.form.html">sympde.exterior.form</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_compiler.html">sympde.exterior.tests.test_compiler</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
+</ul>
+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.utilities.utils.html">sympde.utilities.utils</a></li>
+</ul>
+</li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.expr.html">psydac.api.ast.expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.glt.html">psydac.api.ast.glt</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.utilities.html">psydac.api.ast.utilities</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.discretization.html">psydac.api.discretization</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.equation.html">psydac.api.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.essential_bc.html">psydac.api.essential_bc</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.expr.html">psydac.api.expr</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.feec.html">psydac.api.feec</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.fem.html">psydac.api.fem</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.glt.html">psydac.api.glt</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.grid.html">psydac.api.grid</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.postprocessing.html">psydac.api.postprocessing</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/psydac.api.printing.html">psydac.api.printing</a><input class="toctree-checkbox" id="toctree-checkbox-29" name="toctree-checkbox-29" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-29"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.printing.pycode.html">psydac.api.printing.pycode</a></li>
+</ul>
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+    <h1>The Biharmonic problem</h1>
+    <!-- Table of contents -->
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+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-variational-formulation">The Variational Formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-2-semi-norm">Computing the H^2 semi-norm</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="the-biharmonic-problem">
+<h1>The Biharmonic problem<a class="headerlink" href="#the-biharmonic-problem" title="Link to this heading">#</a></h1>
+<p>We consider the (inhomogeneous) biharmonic equation with homogeneous essential boundary conditions,</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \nabla^2 \nabla^2  u = f       \quad \text{in $\Omega$}         , \quad \quad 
+  u = 0                          \quad \text{on $\partial \Omega$}, \quad \quad
+  \nabla u \cdot \mathbf{n} = 0  \quad \text{on $\partial \Omega$}.
+\end{align}
+\]</div>
+<section id="the-variational-formulation">
+<h2>The Variational Formulation<a class="headerlink" href="#the-variational-formulation" title="Link to this heading">#</a></h2>
+<p>An <span class="math notranslate nohighlight">\(H^2\)</span>-conforming variational formulation reads</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \text{find $u \in V$ such that} \quad a(u,v) = l(v) \quad \forall v \in V,
+\end{align}
+\]</div>
+<p>where</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(V \subset H^2(\Omega)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(a(u,v) := \int_{\Omega} \nabla^2 u ~ \nabla^2 v ~ d\Omega\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(l(v) := \int_{\Omega} f v ~ d\Omega\)</span>.</p></li>
+</ul>
+</section>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<p>In this example, the analytical solution is given by</p>
+<div class="math notranslate nohighlight">
+\[
+u_e = \sin(\pi x)^2 \sin(\pi y)^2
+\]</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">laplace</span><span class="p">,</span> <span class="n">dot</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+
+<span class="n">V</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+
+<span class="c1"># the analytical solution and its rhs</span>
+<span class="n">ue</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
+<span class="n">f</span>  <span class="o">=</span> <span class="n">laplace</span><span class="p">(</span><span class="n">laplace</span><span class="p">(</span><span class="n">ue</span><span class="p">))</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span> <span class="p">,</span> <span class="n">laplace</span><span class="p">(</span><span class="n">u</span><span class="p">)</span><span class="o">*</span><span class="n">laplace</span><span class="p">(</span><span class="n">v</span><span class="p">)))</span>
+
+<span class="c1"># linear form</span>
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">f</span><span class="o">*</span><span class="n">v</span><span class="p">))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<p>We shall need the <strong>discretize</strong> function from <strong>PsyDAC</strong>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">Vh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">uh</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">u</span> <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+
+<span class="c1"># create the formal Norm object</span>
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">l2norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">l2_error</span> <span class="o">=</span> <span class="n">l2norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">l2_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.5367871671842632
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-semi-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm<a class="headerlink" href="#computing-the-h-1-semi-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">SemiNorm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">h1norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">h1_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2.4343747827631277
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm<a class="headerlink" href="#computing-the-h-1-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">h1norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">h1_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2.492853996086961
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-2-semi-norm">
+<h3>Computing the H^2 semi-norm<a class="headerlink" href="#computing-the-h-2-semi-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="n">h2norm</span> <span class="o">=</span> <span class="n">SemiNorm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h2&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">h2norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h2norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">h2_error</span> <span class="o">=</span> <span class="n">h2norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">h2_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>9.115861322114647
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+</section>
+
+    <script type="text/x-thebe-config">
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+    <h1>Elliptic-curl Problem</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-variational-formulation">The Variational Formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
+</ul>
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+        </div>
+    </div>
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+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="elliptic-curl-problem">
+<h1>Elliptic-curl Problem<a class="headerlink" href="#elliptic-curl-problem" title="Link to this heading">#</a></h1>
+<p>Let <span class="math notranslate nohighlight">\(\Omega \subset \mathbb{R}^d\)</span> be an open Liptschitz bounded set, and we look for the solution of the following problem</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+  \left\{ 
+  \begin{array}{rl}
+    \nabla \times \nabla \times \mathbf{u} + \mu \mathbf{u} &amp;= \mathbf{f}, \quad \Omega 
+    \\
+    \mathbf{u} \times \mathbf{n} &amp;= 0, \quad \partial\Omega
+  \end{array} \right.
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{f} \in \mathbf{L}^2(\Omega)\)</span>,  <span class="math notranslate nohighlight">\(\mu \in L^\infty(\Omega)\)</span> and there exists <span class="math notranslate nohighlight">\(\mu_0 &gt; 0\)</span> such that <span class="math notranslate nohighlight">\(\mu \geq \mu_0\)</span> almost everywhere.</p>
+<section id="the-variational-formulation">
+<h2>The Variational Formulation<a class="headerlink" href="#the-variational-formulation" title="Link to this heading">#</a></h2>
+<p>We take the Hilbert space <span class="math notranslate nohighlight">\(V := \mathbf{H}\_0(\mbox{curl}, \Omega)\)</span>, in which case the variational formulation writes</p>
+<p>Find <span class="math notranslate nohighlight">\(\mathbf{u} \in V\)</span> such that</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  a(\mathbf{u},\mathbf{v}) = l(\mathbf{v}) \quad \forall \mathbf{v} \in V 
+\end{align}
+\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\left\{ 
+\begin{array}{rll}
+a(\mathbf{u}, \mathbf{v}) &amp;:= \int_{\Omega} \nabla \times \mathbf{u} \cdot \nabla \times \mathbf{v} + \int_{\Omega} \mu \mathbf{u} \cdot \mathbf{v}, &amp; \forall \mathbf{u}, \mathbf{v} \in V  \\
+l(\mathbf{v}) &amp;:= \int_{\Omega} \mathbf{v} \cdot \mathbf{f}, &amp; \forall \mathbf{v} \in V  
+\end{array} \right.
+\end{align}
+\end{split}\]</div>
+<p>We recall that in <span class="math notranslate nohighlight">\(\mathbf{H}_0(\mbox{curl}, \Omega)\)</span>, the bilinear form <span class="math notranslate nohighlight">\(a\)</span> is equivalent to the inner product and is therefor continuous and coercive. Hence, our abstract theory applies and there exists a unique solution to the problem.</p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>This NB needs to be fixed</p>
+</div>
+</section>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">VectorFunctionSpace</span><span class="p">,</span> <span class="n">Cube</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">curl</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">grad</span><span class="p">,</span> <span class="n">div</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span>     <span class="kn">import</span> <span class="n">Constant</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">Tuple</span>
+
+<span class="n">mu</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;mu&#39;</span><span class="p">,</span> <span class="n">is_real</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Cube</span><span class="p">()</span>
+
+<span class="n">V</span> <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Hcurl&#39;</span><span class="p">)</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">curl</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">curl</span><span class="p">(</span><span class="n">u</span><span class="p">))</span> <span class="o">+</span> <span class="n">mu</span> <span class="o">*</span> <span class="n">dot</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">)))</span>
+
+<span class="c1"># linear form</span>
+<span class="n">f1</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">f2</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">f3</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">f</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="n">f2</span><span class="p">,</span> <span class="n">f3</span><span class="p">)</span>
+
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">v</span><span class="p">)))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<p>We shall need the <strong>discretize</strong> function from <strong>PsyDAC</strong>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">Vh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#uh = equation_h.solve()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<p>When the analytical solution is available, you might be interested in computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm or <span class="math notranslate nohighlight">\(H^1_0\)</span> semi-norm.
+SymPDE allows you to do so, by creating the <strong>Norm</strong> object.
+In this example, the analytical solution is given by</p>
+<div class="math notranslate nohighlight">
+\[
+u_e = \sin(\pi x) \sin(\pi y)
+\]</div>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#u = element_of(V, name=&#39;u&#39;)</span>
+
+<span class="c1"># create the formal Norm object</span>
+<span class="c1">#l2norm = Norm(u - ue, domain, kind=&#39;l2&#39;)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="c1">#l2norm_h = discretize(l2norm, domain_h, Vh)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="c1">#l2_error = l2norm_h.assemble(u=uh)</span>
+
+<span class="c1"># print the result</span>
+<span class="c1">#print(l2_error)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-semi-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm<a class="headerlink" href="#computing-the-h-1-semi-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="c1">#h1norm = SemiNorm(u - ue, domain, kind=&#39;h1&#39;)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="c1">#h1norm_h = discretize(h1norm, domain_h, Vh)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="c1">#h1_error = h1norm_h.assemble(u=uh)</span>
+
+<span class="c1"># print the result</span>
+<span class="c1">#print(h1_error)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm<a class="headerlink" href="#computing-the-h-1-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="c1">#h1norm = Norm(u - ue, domain, kind=&#39;h1&#39;)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="c1">#h1norm_h = discretize(h1norm, domain_h, Vh)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="c1">#h1_error = h1norm_h.assemble(u=uh)</span>
+
+<span class="c1"># print the result</span>
+<span class="c1">#print(h1_error)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+</section>
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+    <i class="fa-solid fa-list"></i> Contents
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-variational-formulation">The Variational Formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+    <h1>Elliptic-div Problem</h1>
+    <!-- Table of contents -->
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+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-variational-formulation">The Variational Formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="elliptic-div-problem">
+<h1>Elliptic-div Problem<a class="headerlink" href="#elliptic-div-problem" title="Link to this heading">#</a></h1>
+<p>Let <span class="math notranslate nohighlight">\(\Omega \subset \mathbb{R}^d\)</span> be an open Liptschitz bounded set, and we look for the solution of the following problem</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+  \left\{ 
+  \begin{array}{rl}
+    - \nabla \nabla \cdot \mathbf{u} + \mu \mathbf{u} &amp;= \mathbf{f}, \quad \Omega 
+    \\
+    \mathbf{u} \times \mathbf{n} &amp;= 0, \quad \partial\Omega
+  \end{array} \right.
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{f} \in \mathbf{L}^2(\Omega)\)</span>,  <span class="math notranslate nohighlight">\(\mu \in L^\infty(\Omega)\)</span> and there exists <span class="math notranslate nohighlight">\(\mu_0 &gt; 0\)</span> such that <span class="math notranslate nohighlight">\(\mu \geq \mu_0\)</span> almost everywhere.</p>
+<section id="the-variational-formulation">
+<h2>The Variational Formulation<a class="headerlink" href="#the-variational-formulation" title="Link to this heading">#</a></h2>
+<p>We take the Hilbert space <span class="math notranslate nohighlight">\(V := \mathbf{H}\_0(\mbox{div}, \Omega)\)</span>, in which case the variational formulation writes</p>
+<p>Find <span class="math notranslate nohighlight">\(\mathbf{u} \in V\)</span> such that</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  a(\mathbf{u},\mathbf{v}) = l(\mathbf{v}) \quad \forall \mathbf{v} \in V 
+\end{align}
+\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\left\{ 
+\begin{array}{rll}
+a(\mathbf{u}, \mathbf{v}) &amp;:= \int_{\Omega} \nabla \cdot \mathbf{u} ~ \nabla \cdot \mathbf{v} + \int_{\Omega} \mu \mathbf{u} \cdot \mathbf{v}, &amp; \forall \mathbf{u}, \mathbf{v} \in V  \\
+l(\mathbf{v}) &amp;:= \int_{\Omega} \mathbf{v} \cdot \mathbf{f}, &amp; \forall \mathbf{v} \in V  
+\end{array} \right.
+\end{align}
+\end{split}\]</div>
+<p>We recall that in <span class="math notranslate nohighlight">\(\mathbf{H}\_0(\mbox{div}, \Omega)\)</span>, the bilinear form <span class="math notranslate nohighlight">\(a\)</span> is equivalent to the inner product and is therefor continuous and coercive. Hence, our abstract theory applies and there exists a unique solution.</p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>This NB needs to be fixed</p>
+</div>
+</section>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">VectorFunctionSpace</span><span class="p">,</span> <span class="n">Cube</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">div</span><span class="p">,</span> <span class="n">dot</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span>     <span class="kn">import</span> <span class="n">Constant</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">Tuple</span>
+
+<span class="n">mu</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;mu&#39;</span><span class="p">,</span> <span class="n">is_real</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Cube</span><span class="p">()</span>
+
+<span class="n">V</span> <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Hdiv&#39;</span><span class="p">)</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">div</span><span class="p">(</span><span class="n">v</span><span class="p">)</span><span class="o">*</span><span class="n">div</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">+</span> <span class="n">mu</span> <span class="o">*</span> <span class="n">dot</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">)))</span>
+
+<span class="c1"># linear form</span>
+<span class="n">f1</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">f2</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">f3</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">f</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="n">f2</span><span class="p">,</span> <span class="n">f3</span><span class="p">)</span>
+
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">v</span><span class="p">)))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<p>We shall need the <strong>discretize</strong> function from <strong>PsyDAC</strong>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">Vh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#uh = equation_h.solve()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#ue = sin(pi*x)*sin(pi*y)</span>
+
+<span class="c1">#u = element_of(V, name=&#39;u&#39;)</span>
+
+<span class="c1"># create the formal Norm object</span>
+<span class="c1">#l2norm = Norm(u - ue, domain, kind=&#39;l2&#39;)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="c1">#l2norm_h = discretize(l2norm, domain_h, Vh)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="c1">#l2_error = l2norm_h.assemble(u=uh)</span>
+
+<span class="c1"># print the result</span>
+<span class="c1">#print(l2_error)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-semi-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm<a class="headerlink" href="#computing-the-h-1-semi-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="c1">#h1norm = SemiNorm(u - ue, domain, kind=&#39;h1&#39;)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="c1">#h1norm_h = discretize(h1norm, domain_h, Vh)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="c1">#h1_error = h1norm_h.assemble(u=uh)</span>
+
+<span class="c1"># print the result</span>
+<span class="c1">#print(h1_error)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm<a class="headerlink" href="#computing-the-h-1-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="c1">#h1norm = Norm(u - ue, domain, kind=&#39;h1&#39;)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="c1">#h1norm_h = discretize(h1norm, domain_h, Vh)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="c1">#h1_error = h1norm_h.assemble(u=uh)</span>
+
+<span class="c1"># print the result</span>
+<span class="c1">#print(h1_error)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+</section>
+
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-variational-formulation">The Variational Formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
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+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter0/bezier-curves.html">Bézier curves</a></li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter0/bsplines.html">B-Splines</a><input class="toctree-checkbox" id="toctree-checkbox-3" name="toctree-checkbox-3" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-3"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter0/bsplines-curves.html">B-Splines curves</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter0/bsplines-surfaces.html">B-Splines surfaces</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter0/iga.html">Isogeometric Analysis</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/data-structure.html">Data Structure</a></li>
+
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/install.html">Installation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter1/sympde.html">SymPDE concepts and their mathematical meaning</a><input class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-5"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/topology.html">Topology concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/space.html">Function Space concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/sympde.html">SymPDE API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-13" name="toctree-checkbox-13" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-13"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.algebra.html">sympde.core.algebra</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.basic.html">sympde.core.basic</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.core.tests.html">sympde.core.tests</a><input class="toctree-checkbox" id="toctree-checkbox-15" name="toctree-checkbox-15" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-15"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.core.utils.html">sympde.core.utils</a></li>
+</ul>
+</li>
+<li class="toctree-l2"><a class="reference internal" href="../documentation/_autosummary/sympde.calculus.html">sympde.calculus</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.analytical_mapping.html">sympde.topology.analytical_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.basic.html">sympde.topology.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.callable_mapping.html">sympde.topology.callable_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.datatype.html">sympde.topology.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.derivatives.html">sympde.topology.derivatives</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.domain.html">sympde.topology.domain</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.html">sympde.topology.tests</a><input class="toctree-checkbox" id="toctree-checkbox-17" name="toctree-checkbox-17" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-17"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_callable_mapping.html">sympde.topology.tests.test_callable_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_derivatives.html">sympde.topology.tests.test_derivatives</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_field_2d.html">sympde.topology.tests.test_field_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.html">sympde.expr</a><input class="toctree-checkbox" id="toctree-checkbox-18" name="toctree-checkbox-18" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-18"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.basic.html">sympde.expr.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.equation.html">sympde.expr.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_equation_2d.html">sympde.expr.tests.test_equation_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.html">sympde.printing</a><input class="toctree-checkbox" id="toctree-checkbox-20" name="toctree-checkbox-20" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-20"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.latex.html">sympde.printing.latex</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.tests.html">sympde.printing.tests</a><input class="toctree-checkbox" id="toctree-checkbox-21" name="toctree-checkbox-21" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-21"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.tests.test_latex.html">sympde.printing.tests.test_latex</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.html">sympde.exterior</a><input class="toctree-checkbox" id="toctree-checkbox-22" name="toctree-checkbox-22" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-22"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.calculus.html">sympde.exterior.calculus</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.form.html">sympde.exterior.form</a></li>
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+    <h1>Elliptic equation in the general form</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#variational-formulation">Variational Formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#manifactured-solution">Manifactured solution</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-equation">Formal Equation</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="elliptic-equation-in-the-general-form">
+<h1>Elliptic equation in the general form<a class="headerlink" href="#elliptic-equation-in-the-general-form" title="Link to this heading">#</a></h1>
+<p>We consider here, the following general form of an elliptic partial differential equation,</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+- \nabla \cdot \left( A \nabla u \right) + \mathbf{b} \cdot \nabla u + c u &amp;= f, \quad \Omega \\
+u &amp;= 0, \quad \partial \Omega
+\end{align}
+\end{split}\]</div>
+<section id="variational-formulation">
+<h2>Variational Formulation<a class="headerlink" href="#variational-formulation" title="Link to this heading">#</a></h2>
+<p>An <span class="math notranslate nohighlight">\(H^1\)</span>-conforming variational formulation of reads</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \text{find $u \in V$ such that} \quad a(u,v) = l(v) \quad \forall v \in V,
+\end{align}
+\]</div>
+<p>where</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(V \subset H^1(\Omega)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(a(u,v) := \int_{\Omega} \left( \left( A \nabla u \right) \cdot \nabla v + \left( \mathbf{b} \cdot \nabla u \right) v + cuv \right) ~ d\Omega\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(l(v) := \int_{\Omega} f v ~ d\Omega\)</span>.</p></li>
+</ul>
+</section>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">div</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span> <span class="kn">import</span> <span class="n">Vector</span><span class="p">,</span> <span class="n">Matrix</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span>
+
+<span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+
+<span class="n">V</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+
+<span class="n">c</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">y</span>
+<span class="n">b</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">([</span><span class="mf">1e-2</span><span class="p">,</span> <span class="mf">1e-1</span><span class="p">],</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">)</span>
+<span class="n">A</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]],</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;A&#39;</span><span class="p">)</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">expr</span> <span class="o">=</span> <span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">A</span> <span class="o">*</span> <span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))</span><span class="o">*</span><span class="n">v</span> <span class="o">+</span> <span class="n">c</span><span class="o">*</span><span class="n">u</span><span class="o">*</span><span class="n">v</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">expr</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
+<span class="ne">ImportError</span><span class="g g-Whitespace">                               </span>Traceback (most recent call last)
+<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">line</span> <span class="mi">5</span>
+<span class="g g-Whitespace">      </span><span class="mi">3</span> <span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="g g-Whitespace">      </span><span class="mi">4</span> <span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">div</span>
+<span class="ne">----&gt; </span><span class="mi">5</span> <span class="kn">from</span> <span class="nn">sympde.core</span> <span class="kn">import</span> <span class="n">Vector</span><span class="p">,</span> <span class="n">Matrix</span>
+<span class="g g-Whitespace">      </span><span class="mi">7</span> <span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span>
+<span class="g g-Whitespace">      </span><span class="mi">9</span> <span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+
+<span class="ne">ImportError</span>: cannot import name &#39;Vector&#39; from &#39;sympde.core&#39; (/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympde/core/__init__.py)
+</pre></div>
+</div>
+</div>
+</div>
+<section id="manifactured-solution">
+<h3>Manifactured solution<a class="headerlink" href="#manifactured-solution" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># the analytical solution and its rhs</span>
+<span class="n">ue</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span>
+
+<span class="n">L</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">u</span><span class="p">:</span> <span class="o">-</span> <span class="n">div</span><span class="p">(</span><span class="n">A</span><span class="o">*</span><span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">b</span><span class="p">,</span><span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))</span> <span class="o">+</span> <span class="n">c</span><span class="o">*</span><span class="n">u</span>
+<span class="n">f</span>  <span class="o">=</span> <span class="n">L</span><span class="p">(</span><span class="n">ue</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="formal-equation">
+<h3>Formal Equation<a class="headerlink" href="#formal-equation" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">f</span><span class="o">*</span><span class="n">v</span><span class="p">))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">Vh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/home/ranania/PYCCEL/IGA-Python/.iga-python/lib/python3.10/site-packages/sympy/matrices/repmatrix.py:90: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  ).warn()
+</pre></div>
+</div>
+</div>
+</div>
+<section id="solving-the-pde">
+<h3>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">equation_h</span><span class="o">.</span><span class="n">set_solver</span><span class="p">(</span><span class="s1">&#39;gmres&#39;</span><span class="p">,</span> <span class="n">info</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-8</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">uh</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">u</span> <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+
+<span class="c1"># create the formal Norm object</span>
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">l2norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">l2_error</span> <span class="o">=</span> <span class="n">l2norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">l2_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.00021948770944141364
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+</section>
+
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#manifactured-solution">Manifactured solution</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_compiler.html">sympde.exterior.tests.test_compiler</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
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+</li>
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+</ul>
+</li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.expr.html">psydac.api.ast.expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.glt.html">psydac.api.ast.glt</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
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+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
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+    <h1>Linear Elasticity Problem</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-pde-problem">The PDE problem</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-variational-formulation">The Variational Formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+</ul>
+            </nav>
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+    </div>
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+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="linear-elasticity-problem">
+<h1>Linear Elasticity Problem<a class="headerlink" href="#linear-elasticity-problem" title="Link to this heading">#</a></h1>
+<p>Analysis of deformable structures is essential in engineering, with the equations of linear elasticity being fundamental in this field. In this section, we present the variational formulation of linear elasticity equations using the principle of virtual work.</p>
+<section id="the-pde-problem">
+<h2>The PDE problem<a class="headerlink" href="#the-pde-problem" title="Link to this heading">#</a></h2>
+<p>The governing equations for small elastic deformations of a body <span class="math notranslate nohighlight">\( \Omega \)</span> can be expressed as:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+    -\nabla \cdot \sigma(u) &amp;= f &amp; \text{in } \Omega \\
+    \sigma(u) &amp;= \kappa \text{tr}(\epsilon(u))I + 2 \mu \epsilon(u)
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\( \sigma \)</span> is the stress tensor, <span class="math notranslate nohighlight">\( f \)</span> represents the body force per unit volume, <span class="math notranslate nohighlight">\( \kappa \)</span> and <span class="math notranslate nohighlight">\( \mu \)</span> are Lamé’s elasticity parameters for the material, <span class="math notranslate nohighlight">\( I \)</span> denotes the identity tensor, and <span class="math notranslate nohighlight">\( \epsilon \)</span> is the symmetric strain tensor. The displacement vector field is denoted by <span class="math notranslate nohighlight">\( u \)</span>.</p>
+<p>By substituting <span class="math notranslate nohighlight">\( \epsilon(u) \)</span> into <span class="math notranslate nohighlight">\( \sigma \)</span>, we obtain:</p>
+<div class="math notranslate nohighlight">
+\[
+\sigma(u) = \kappa (\nabla \cdot u)I + \mu(\nabla u + (\nabla u)^T)
+\]</div>
+</section>
+<section id="the-variational-formulation">
+<h2>The Variational Formulation<a class="headerlink" href="#the-variational-formulation" title="Link to this heading">#</a></h2>
+<p>The variational formulation of the linear elasticity equations involves forming the inner product of the PDE with a vector test function <span class="math notranslate nohighlight">\( v \in \hat{V} \)</span> and integrating over the domain <span class="math notranslate nohighlight">\( \Omega \)</span>. This yields:</p>
+<div class="math notranslate nohighlight">
+\[
+\int_{\Omega} \sigma : \nabla v \, \mathrm{d} x = \int_{\Omega} f \cdot v \, \mathrm{d} x
+\]</div>
+<p>Integrating the term <span class="math notranslate nohighlight">\( \nabla \cdot \sigma \cdot v \)</span> by parts, considering boundary conditions, we obtain:</p>
+<div class="math notranslate nohighlight">
+\[
+\int_{\Omega} \sigma : \nabla v \, \mathrm{d} x = \int_{\Omega} f \cdot v \, \mathrm{d} x + \int_{\partial \Omega_T} T \cdot v \, \mathrm{d} s
+\]</div>
+<p>where <span class="math notranslate nohighlight">\( T \)</span> represents the traction vector on the part <span class="math notranslate nohighlight">\( \partial \Omega_T \)</span> of the boundary where it’s prescribed.</p>
+<p>This leads to the variational formulation: Find <span class="math notranslate nohighlight">\( u \in V \)</span> such that</p>
+<div class="math notranslate nohighlight">
+\[
+a(u, v) = L(v) \quad \forall v \in \hat{V}
+\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+    a(u, v) &amp;= \int_{\Omega} \sigma(u) : \nabla v \, \mathrm{d} x \\
+    L(v) &amp;= \int_{\Omega} f \cdot v \, \mathrm{d} x + \int_{\partial \Omega_T} T \cdot v \, \mathrm{d} s
+\end{align}
+\end{split}\]</div>
+<p>This formulation can be alternatively expressed as:</p>
+<div class="math notranslate nohighlight">
+\[
+a(u, v) = \int_{\Omega} \sigma(u) : \epsilon(v) \, \mathrm{d} x
+\]</div>
+<p>where <span class="math notranslate nohighlight">\( \epsilon(v) = \frac{1}{2} (\nabla v + (\nabla v)^T) \)</span> is the symmetric strain tensor.</p>
+<p>This variational formulation is essential for solving linear elasticity problems numerically using methods like the finite element method (FEM).</p>
+</section>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">VectorFunctionSpace</span><span class="p">,</span> <span class="n">Cube</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">inner</span><span class="p">,</span> <span class="n">outer</span><span class="p">,</span> <span class="n">cross</span><span class="p">,</span> <span class="n">div</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span> <span class="kn">import</span> <span class="n">Constant</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Vector</span><span class="p">,</span> <span class="n">Transpose</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Cube</span><span class="p">()</span>
+
+<span class="n">I</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;I&#39;</span><span class="p">)</span>
+
+<span class="n">kappa</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;kappa&#39;</span><span class="p">,</span> <span class="n">is_real</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">mu</span>    <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;mu&#39;</span><span class="p">,</span>    <span class="n">is_real</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+<span class="n">rho</span>   <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;rho&#39;</span><span class="p">,</span>   <span class="n">is_real</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="n">epsilon</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">w</span><span class="p">:</span> <span class="mf">0.5</span><span class="o">*</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="o">+</span> <span class="n">Transpose</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">w</span><span class="p">)))</span>
+<span class="n">sigma</span>   <span class="o">=</span> <span class="k">lambda</span> <span class="n">w</span><span class="p">:</span> <span class="n">kappa</span> <span class="o">*</span> <span class="n">div</span><span class="p">(</span><span class="n">w</span><span class="p">)</span> <span class="o">*</span> <span class="n">I</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">mu</span> <span class="o">*</span> <span class="n">epsilon</span><span class="p">(</span><span class="n">w</span><span class="p">)</span>
+
+<span class="n">V</span> <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
+<span class="ne">ImportError</span><span class="g g-Whitespace">                               </span>Traceback (most recent call last)
+<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">line</span> <span class="mi">6</span>
+<span class="g g-Whitespace">      </span><span class="mi">4</span> <span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">inner</span><span class="p">,</span> <span class="n">outer</span><span class="p">,</span> <span class="n">cross</span><span class="p">,</span> <span class="n">div</span>
+<span class="g g-Whitespace">      </span><span class="mi">5</span> <span class="kn">from</span> <span class="nn">sympde.core</span> <span class="kn">import</span> <span class="n">Constant</span>
+<span class="ne">----&gt; </span><span class="mi">6</span> <span class="kn">from</span> <span class="nn">sympde.core</span> <span class="kn">import</span> <span class="n">Matrix</span><span class="p">,</span> <span class="n">Vector</span><span class="p">,</span> <span class="n">Transpose</span>
+<span class="g g-Whitespace">      </span><span class="mi">8</span> <span class="n">domain</span> <span class="o">=</span> <span class="n">Cube</span><span class="p">()</span>
+<span class="g g-Whitespace">     </span><span class="mi">10</span> <span class="n">I</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;I&#39;</span><span class="p">)</span>
+
+<span class="ne">ImportError</span>: cannot import name &#39;Vector&#39; from &#39;sympde.core&#39; (/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympde/core/__init__.py)
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># bilinear form</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span> <span class="p">,</span> <span class="n">inner</span><span class="p">(</span><span class="n">sigma</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">epsilon</span><span class="p">(</span><span class="n">v</span><span class="p">))))</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">L</span> <span class="o">=</span> <span class="mi">1</span>
+<span class="n">W</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1">#0.2</span>
+<span class="n">delta</span> <span class="o">=</span> <span class="n">W</span> <span class="o">/</span> <span class="n">L</span>
+
+<span class="c1">#mu = 1</span>
+<span class="c1">#rho = 1</span>
+<span class="c1">#kappa = 1.25</span>
+<span class="n">g</span> <span class="o">=</span> <span class="mf">0.4</span> <span class="o">*</span> <span class="n">delta</span><span class="o">**</span><span class="mi">2</span>
+
+<span class="c1"># linear form</span>
+<span class="n">f</span> <span class="o">=</span> <span class="n">Vector</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">-</span><span class="n">rho</span><span class="o">*</span><span class="n">g</span><span class="p">],</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;f&#39;</span><span class="p">)</span>
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">v</span><span class="p">)))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<p>We shall need the <strong>discretize</strong> function from <strong>PsyDAC</strong>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">Vh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/home/ranania/PYCCEL/IGA-Python/.iga-python/lib/python3.10/site-packages/sympy/matrices/repmatrix.py:90: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  ).warn()
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">uh</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">mu</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">rho</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">kappa</span><span class="o">=</span><span class="mf">1.25</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+
+    <script type="text/x-thebe-config">
+    {
+        requestKernel: true,
+        binderOptions: {
+            repo: "binder-examples/jupyter-stacks-datascience",
+            ref: "master",
+        },
+        codeMirrorConfig: {
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+    <h1>First mixed formulation of the Poisson problem</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+</ul>
+</li>
+</ul>
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+        </div>
+    </div>
+</div>
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+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="first-mixed-formulation-of-the-poisson-problem">
+<h1>First mixed formulation of the Poisson problem<a class="headerlink" href="#first-mixed-formulation-of-the-poisson-problem" title="Link to this heading">#</a></h1>
+<p>Instead of having one unknown, we now have two, along with the above two equations.
+In order to get a mixed variational formulation, we first take the dot product of the first one by <span class="math notranslate nohighlight">\( \mathbf{v}\)</span> and integrate by parts</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\int_{\Omega} \mathbf{u}\cdot \mathbf{v}~\mathrm{d} \mathbf{x} -\int_{\Omega} p ~ \nabla\cdot \mathbf{v}~\mathrm{d} \mathbf{x} + \int_{\partial\Omega} p ~  \mathbf{v}\cdot \mathbf{n}~\mathrm{d} \sigma = \int_{\Omega} \mathbf{u}\cdot \mathbf{v}~\mathrm{d} \mathbf{x} -\int_{\Omega} p ~ \nabla\cdot \mathbf{v}~\mathrm{d} \mathbf{x}=0,
+\end{align}
+\]</div>
+<p>using <span class="math notranslate nohighlight">\(p=0\)</span> as a natural boundary condition. Then multiplying the second equation by <span class="math notranslate nohighlight">\(q\)</span> and integrating yields</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\int_{\Omega} \nabla\cdot\mathbf{u} ~ q ~\mathrm{d} \mathbf{x} = \int_{\Omega} f q ~\mathrm{d} \mathbf{x}.
+\end{align}
+\]</div>
+<p>No integration by parts is necessary here. And we thus get the following mixed variational formulation:</p>
+<p>Find <span class="math notranslate nohighlight">\((\mathbf{u},p) \in H(\mathrm{div},\Omega)\times L^2(\Omega)\)</span> such that</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\left\{ 
+\begin{array}{llll}
+  \int_{\Omega} \mathbf{u}\cdot \mathbf{v}~\mathrm{d} \mathbf{x} &amp;- \int_{\Omega} p ~ \nabla\cdot \mathbf{v}~\mathrm{d} \mathbf{x} &amp;=0, &amp; \forall \mathbf{v}\in H(\mathrm{div},\Omega) \\
+  \int_{\Omega} \nabla\cdot\mathbf{u} ~ q ~\mathrm{d} \mathbf{x} &amp;  &amp;= \int_{\Omega} f q ~\mathrm{d} \mathbf{x}, &amp; \forall q\in L^2(\Omega)
+\end{array} \right.
+\end{align}
+\end{split}\]</div>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="p">(</span><span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">VectorFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span>
+                             <span class="n">element_of</span><span class="p">)</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">div</span><span class="p">,</span> <span class="n">inner</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span>     <span class="kn">import</span> <span class="n">Constant</span>
+
+<span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">Tuple</span>
+
+<span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+
+<span class="n">V1</span> <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V1&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;Hdiv&#39;</span><span class="p">)</span>
+<span class="n">V2</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V2&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;L2&#39;</span><span class="p">)</span>
+<span class="n">X</span>  <span class="o">=</span> <span class="n">V1</span><span class="o">*</span><span class="n">V2</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="c1"># rhs</span>
+<span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="c1"># exact solution</span>
+<span class="n">pe</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="c1">#ue = -grad(pe) # not working when computing the norm</span>
+<span class="n">ue</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">-</span><span class="n">pi</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="o">-</span><span class="n">pi</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">))</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V1</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+<span class="n">p</span><span class="p">,</span><span class="n">q</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V2</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;p&#39;</span><span class="p">,</span> <span class="s1">&#39;q&#39;</span><span class="p">]]</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">a</span>  <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">(((</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">),(</span><span class="n">v</span><span class="p">,</span><span class="n">q</span><span class="p">)),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">)</span> <span class="o">-</span> <span class="n">p</span><span class="o">*</span><span class="n">div</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">+</span> <span class="n">div</span><span class="p">(</span><span class="n">u</span><span class="p">)</span><span class="o">*</span><span class="n">q</span><span class="p">)</span> <span class="p">)</span>
+
+<span class="c1"># linear form</span>
+<span class="n">l</span>  <span class="o">=</span> <span class="n">LinearForm</span><span class="p">((</span><span class="n">v</span><span class="p">,</span><span class="n">q</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">f</span><span class="o">*</span><span class="n">q</span><span class="p">))</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span> <span class="o">=</span> <span class="n">find</span><span class="p">([</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">],</span> <span class="n">forall</span><span class="o">=</span><span class="p">[</span><span class="n">v</span><span class="p">,</span><span class="n">q</span><span class="p">],</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">),(</span><span class="n">v</span><span class="p">,</span><span class="n">q</span><span class="p">)),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">q</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">V1h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V1</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+<span class="n">V2h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V2</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+<span class="n">Xh</span>  <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Xh</span><span class="p">,</span> <span class="n">Xh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">equation_h</span><span class="o">.</span><span class="n">set_solver</span><span class="p">(</span><span class="s1">&#39;gmres&#39;</span><span class="p">,</span> <span class="n">info</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-8</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#uh, ph = equation_h.solve()</span>
+
+<span class="kn">from</span> <span class="nn">psydac.fem.basic</span>          <span class="kn">import</span> <span class="n">FemField</span>
+
+<span class="n">phi_h</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+
+<span class="n">uh</span> <span class="o">=</span> <span class="n">FemField</span><span class="p">(</span><span class="n">V1h</span><span class="p">)</span>
+<span class="n">uh</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">][:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">][:]</span>
+<span class="n">uh</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">1</span><span class="p">][:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">1</span><span class="p">][:]</span>
+
+<span class="n">ph</span> <span class="o">=</span> <span class="n">FemField</span><span class="p">(</span><span class="n">V2h</span><span class="p">)</span>
+<span class="n">ph</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">2</span><span class="p">][:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># L2 error norm of the velocity field</span>
+<span class="n">error_u</span>   <span class="o">=</span> <span class="p">[</span><span class="n">ue</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ue</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
+<span class="n">l2norm_u</span>  <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">error_u</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+<span class="n">l2norm_uh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm_u</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">V1h</span><span class="p">)</span>
+
+<span class="c1"># L2 error norm of the pressure, after removing the average value from the field</span>
+<span class="n">l2norm_p</span>  <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">pe</span> <span class="o">-</span> <span class="n">p</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+<span class="n">l2norm_ph</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm_p</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">V2h</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">l2norm</span> <span class="o">=</span> <span class="n">l2norm_uh</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;&gt;&gt;&gt; norm-l2 uh = &#39;</span><span class="p">,</span> <span class="n">l2norm</span><span class="p">)</span>
+
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">l2norm_ph</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">p</span><span class="o">=</span><span class="n">ph</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;&gt;&gt;&gt; norm-l2 ph = &#39;</span><span class="p">,</span> <span class="n">l2norm</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&gt;&gt;&gt; norm-l2 uh =  0.012985071480053038
+&gt;&gt;&gt; norm-l2 ph =  0.004127522959699811
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+</section>
+
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+    <i class="fa-solid fa-list"></i> Contents
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+</ul>
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+</ul>
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+By Ahmed Ratnani
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+    <h1>Second mixed formulation of the Poisson problem</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="second-mixed-formulation-of-the-poisson-problem">
+<h1>Second mixed formulation of the Poisson problem<a class="headerlink" href="#second-mixed-formulation-of-the-poisson-problem" title="Link to this heading">#</a></h1>
+<p>Here, we get an alternative formulation by not integrating by parts, the mixed term in the first formulation but in the second. The first formulation simply becomes</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\int_{\Omega} \mathbf{u}\cdot \mathbf{v}~\mathrm{d} \mathbf{x} +\int_{\Omega} \nabla p \cdot \mathbf{v}~\mathrm{d} \mathbf{x}=0,
+\end{align}
+\]</div>
+<p>and the second, removing immediately the boundary term due to the essential boundary condition <span class="math notranslate nohighlight">\(q=0\)</span></p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\int_{\Omega}\nabla \cdot\mathbf{u}  ~ q ~\mathrm{d} \mathbf{x} = -\int_{\Omega}  \mathbf{u} \cdot \nabla q  ~\mathrm{d} \mathbf{x} = \int_{\Omega} f q ~\mathrm{d} \mathbf{x},
+\end{align}
+\]</div>
+<p>which leads to the variational formulation</p>
+<p>Find <span class="math notranslate nohighlight">\((\mathbf{u},p) \in L^2(\Omega)^3 \times H^1_0(\Omega)\)</span> such that</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\left\{ 
+\begin{array}{llll}
+  \int_{\Omega} \mathbf{u}\cdot \mathbf{v}~\mathrm{d} \mathbf{x} &amp;+ \int_{\Omega} \nabla p \cdot \mathbf{v}~\mathrm{d} \mathbf{x} &amp;=0, &amp; \forall \mathbf{v}\in L^2(\Omega)^3 \\
+  \int_{\Omega}  \mathbf{u} \cdot \nabla q  ~\mathrm{d} \mathbf{x} &amp; &amp; = -\int_{\Omega} f q ~\mathrm{d} \mathbf{x}, &amp; \forall q\in H^1_0(\Omega)
+\end{array} \right.
+\end{align}
+\end{split}\]</div>
+<p>Note that this formulation actually contains the classical variational formulation for the Poisson equation. Indeed for <span class="math notranslate nohighlight">\(q\in H^1_0(\Omega)\)</span>, <span class="math notranslate nohighlight">\(\nabla q \in L^2(\Omega)^3\)</span> can be used as a test function in the first equation. And plugging this into the second we get</p>
+<div class="math notranslate nohighlight">
+\[
+\int_{\Omega}  \nabla p \cdot \nabla q  ~\mathrm{d} \mathbf{x}  = \int_{\Omega} f q ~\mathrm{d} \mathbf{x}, \quad \forall q\in H^1_0(\Omega).
+\]</div>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="p">(</span><span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">VectorFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span>
+                             <span class="n">element_of</span><span class="p">)</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">div</span><span class="p">,</span> <span class="n">inner</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span>     <span class="kn">import</span> <span class="n">Constant</span>
+
+<span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">Tuple</span>
+
+<span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+
+<span class="n">V1</span> <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V1&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;L2&#39;</span><span class="p">)</span>
+<span class="n">V2</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V2&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;H1&#39;</span><span class="p">)</span>
+<span class="n">X</span>  <span class="o">=</span> <span class="n">V1</span><span class="o">*</span><span class="n">V2</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="c1"># rhs</span>
+<span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="c1"># exact solution</span>
+<span class="n">pe</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="c1">#ue = -grad(pe) # not working when computing the norm</span>
+<span class="n">ue</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="o">-</span><span class="n">pi</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">),</span> <span class="o">-</span><span class="n">pi</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">))</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V1</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+<span class="n">p</span><span class="p">,</span><span class="n">q</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V2</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;p&#39;</span><span class="p">,</span> <span class="s1">&#39;q&#39;</span><span class="p">]]</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">a</span>  <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">(((</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">),(</span><span class="n">v</span><span class="p">,</span><span class="n">q</span><span class="p">)),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">)</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">p</span><span class="p">),</span><span class="n">v</span><span class="p">)</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">grad</span><span class="p">(</span><span class="n">q</span><span class="p">)))</span> <span class="p">)</span>
+
+<span class="c1"># linear form</span>
+<span class="n">l</span>  <span class="o">=</span> <span class="n">LinearForm</span><span class="p">((</span><span class="n">v</span><span class="p">,</span><span class="n">q</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="o">-</span><span class="n">f</span><span class="o">*</span><span class="n">q</span><span class="p">))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span> <span class="o">=</span> <span class="n">find</span><span class="p">([</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">],</span> <span class="n">forall</span><span class="o">=</span><span class="p">[</span><span class="n">v</span><span class="p">,</span><span class="n">q</span><span class="p">],</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">p</span><span class="p">),(</span><span class="n">v</span><span class="p">,</span><span class="n">q</span><span class="p">)),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">q</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">V1h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V1</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+<span class="n">V2h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V2</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+<span class="n">Xh</span>  <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Xh</span><span class="p">,</span> <span class="n">Xh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">equation_h</span><span class="o">.</span><span class="n">set_solver</span><span class="p">(</span><span class="s1">&#39;gmres&#39;</span><span class="p">,</span> <span class="n">info</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-8</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#uh, ph = equation_h.solve()</span>
+
+<span class="kn">from</span> <span class="nn">psydac.fem.basic</span>          <span class="kn">import</span> <span class="n">FemField</span>
+
+<span class="n">phi_h</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+
+<span class="n">uh</span> <span class="o">=</span> <span class="n">FemField</span><span class="p">(</span><span class="n">V1h</span><span class="p">)</span>
+<span class="n">uh</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">][:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">][:]</span>
+<span class="n">uh</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">1</span><span class="p">][:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">1</span><span class="p">][:]</span>
+
+<span class="n">ph</span> <span class="o">=</span> <span class="n">FemField</span><span class="p">(</span><span class="n">V2h</span><span class="p">)</span>
+<span class="n">ph</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">2</span><span class="p">][:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># L2 error norm of the velocity field</span>
+<span class="n">error_u</span>   <span class="o">=</span> <span class="p">[</span><span class="n">ue</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ue</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
+<span class="n">l2norm_u</span>  <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">error_u</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+<span class="n">l2norm_uh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm_u</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">V1h</span><span class="p">)</span>
+
+<span class="c1"># L2 error norm of the pressure, after removing the average value from the field</span>
+<span class="n">l2norm_p</span>  <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">pe</span> <span class="o">-</span> <span class="n">p</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+<span class="n">l2norm_ph</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm_p</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">V2h</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">l2norm</span> <span class="o">=</span> <span class="n">l2norm_uh</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;&gt;&gt;&gt; norm-l2 uh = &#39;</span><span class="p">,</span> <span class="n">l2norm</span><span class="p">)</span>
+
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">l2norm_ph</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">p</span><span class="o">=</span><span class="n">ph</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;&gt;&gt;&gt; norm-l2 ph = &#39;</span><span class="p">,</span> <span class="n">l2norm</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&gt;&gt;&gt; norm-l2 uh =  64.20821457571618
+&gt;&gt;&gt; norm-l2 ph =  3254706679445337.5
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+</section>
+
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+    <i class="fa-solid fa-list"></i> Contents
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="poisson.html">The Poisson equation</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="linear-elasticity.html">Linear Elasticity Problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+    <h1>Mixed FEM for the Poisson problem</h1>
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+<h1>Mixed FEM for the Poisson problem<a class="headerlink" href="#mixed-fem-for-the-poisson-problem" title="Link to this heading">#</a></h1>
+<p>Let <span class="math notranslate nohighlight">\(\Omega \subset \mathbb{R}^3\)</span> and consider the Poisson problem</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+  \left\{ 
+  \begin{array}{clr}
+    -\Delta p &amp; =f &amp; ,~\Omega    \\
+    p         &amp; =0 &amp; ,~\partial \Omega
+  \end{array} \right.
+\end{align}
+\end{split}\]</div>
+<p>Using that <span class="math notranslate nohighlight">\(\Delta p = \nabla\cdot\nabla p\)</span>, we set <span class="math notranslate nohighlight">\( \mathbf{u}=\nabla p\)</span>, then the Poisson equation can be written equivalently</p>
+<div class="math notranslate nohighlight">
+\[ \mathbf{u}=-\nabla p, ~~~ \nabla\cdot \mathbf{u}= f.\]</div>
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+    <h1>The Poisson equation with weak imposition of Dirichlet conditions</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-variational-formulation">The variational formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+</ul>
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+</ul>
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+              
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+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="the-poisson-equation-with-weak-imposition-of-dirichlet-conditions">
+<h1>The Poisson equation with weak imposition of Dirichlet conditions<a class="headerlink" href="#the-poisson-equation-with-weak-imposition-of-dirichlet-conditions" title="Link to this heading">#</a></h1>
+<p>In this example, we consider the following Poisson problem</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  - \nabla^2 u = f \quad \text{in~$\Omega$}, \quad \quad 
+  u = g            \quad \text{on~$\partial \Omega$}.
+\end{align}
+\]</div>
+<section id="the-variational-formulation">
+<h2>The variational formulation<a class="headerlink" href="#the-variational-formulation" title="Link to this heading">#</a></h2>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \text{find $u \in V$ such that} \quad a(u,v) = l(v) \quad \forall v \in V,
+\end{align}
+\]</div>
+<p>where</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(V \subset H^1(\Omega)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(a(u,v) := \int_{\Omega} \nabla u \cdot \nabla v ~ d\Omega + \int_{\partial \Omega} \left( \kappa u v - u \partial_n v - v \partial_n u \right)~d\partial\Omega \)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(l(v) := \int_{\Omega} f v ~ d\Omega + \int_{\partial\Omega} \left( \kappa g v - g \partial_n v \right) ~ d\Gamma\)</span>.</p></li>
+</ul>
+</section>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">Dn</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span> <span class="kn">import</span> <span class="n">Constant</span>
+
+<span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span>
+
+<span class="n">kappa</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;kappa&#39;</span><span class="p">,</span> <span class="n">is_real</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+
+<span class="n">V</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))</span> <span class="o">+</span> <span class="n">kappa</span><span class="o">*</span><span class="n">u</span><span class="o">*</span><span class="n">v</span> <span class="o">-</span> <span class="n">u</span><span class="o">*</span><span class="n">Dn</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">-</span> <span class="n">v</span><span class="o">*</span><span class="n">Dn</span><span class="p">(</span><span class="n">u</span><span class="p">)))</span>
+
+<span class="c1"># linear form</span>
+<span class="n">g</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">f</span><span class="o">*</span><span class="n">v</span> <span class="o">+</span> <span class="n">kappa</span><span class="o">*</span><span class="n">g</span><span class="o">*</span><span class="n">v</span> <span class="o">-</span> <span class="n">g</span><span class="o">*</span><span class="n">Dn</span><span class="p">(</span><span class="n">v</span><span class="p">)))</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">Vh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="c1">#equation_h = discretize(equation, domain_h, [Vh, Vh])</span>
+</pre></div>
+</div>
+</div>
+</div>
+<section id="solving-the-pde">
+<h3>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
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+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#uh = equation_h.solve()</span>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a><ul class="nav section-nav flex-column">
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.fem.html">psydac.api.fem</a></li>
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+    <!-- Table of contents -->
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+                <h2> Contents </h2>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#variational-formulation">Variational Formulation</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
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+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="the-poisson-equation">
+<h1>The Poisson equation<a class="headerlink" href="#the-poisson-equation" title="Link to this heading">#</a></h1>
+<p>As a first example, we consider the Poisson equation</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  - \nabla^2 u = f \quad \text{in $\Omega$}, \quad \quad 
+  u = 0            \quad \text{on $\Gamma_D$}, \quad \quad 
+  \partial_n u = g \quad \text{on $\Gamma_N$}.
+\end{align}
+\]</div>
+<section id="variational-formulation">
+<h2>Variational Formulation<a class="headerlink" href="#variational-formulation" title="Link to this heading">#</a></h2>
+<p>An <span class="math notranslate nohighlight">\(H^1\)</span>-conforming variational formulation of reads</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \text{find $u \in V$ such that} \quad a(u,v) = l(v) \quad \forall v \in V,
+\end{align}
+\]</div>
+<p>where</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(V \subset H^1(\Omega)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(a(u,v) := \int_{\Omega} \nabla u \cdot \nabla v ~ d\Omega\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(l(v) := \int_{\Omega} f v ~ d\Omega + \int_{\Gamma_N} g v ~ d\Gamma\)</span>.</p></li>
+</ul>
+</section>
+<section id="examples">
+<h2>Examples<a class="headerlink" href="#examples" title="Link to this heading">#</a></h2>
+<ul class="simple">
+<li><p><a class="reference external" href="https://github.com/pyccel/sympde/blob/devel-documentation/docs/examples/2d_poisson_dir0.py">Poisson on a square domain with Homogeneous Boundary Conditions</a></p></li>
+<li><p><a class="reference external" href="https://github.com/pyccel/sympde/blob/devel-documentation/docs/examples/2d_poisson_dir0_neu.py">Poisson on a square domain with Homogeneous and Neumann Boundary Conditions</a></p></li>
+<li><p><a class="reference external" href="https://github.com/pyccel/sympde/blob/devel-documentation/docs/examples/2d_poisson_dir_neu.py">Poisson on a square domain with Homogeneous, Inhomogeneous and Neumann Boundary Conditions</a></p></li>
+</ul>
+</section>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+
+<span class="n">V</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span> <span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))))</span>
+
+<span class="c1"># linear form</span>
+<span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">f</span><span class="o">*</span><span class="n">v</span><span class="p">))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<p>This very simple Python code reflects well the abstract mathematical framework needed for variational formulations.
+The structure of the code is as follows,</p>
+<ol class="arabic simple">
+<li><p>Create a domain.</p></li>
+<li><p>Create a space of <em>scalar</em> functions over the domain.</p></li>
+<li><p>Create elements from this function space. These elements will denote the test and trial functions.</p></li>
+<li><p>Create the Bilinear and Linear forms, <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(l\)</span> respectively.</p></li>
+<li><p>Create Essential Boundary Conditions.</p></li>
+<li><p>Create the variational problem.</p></li>
+</ol>
+<p>Most of the time, you will need to follow the same steps, with some minor variants depending on the problem you’re considering.</p>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<p>We shall need the <strong>discretize</strong> function from <strong>PsyDAC</strong>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">Vh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">uh</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<p>When the analytical solution is available, you might be interested in computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm or <span class="math notranslate nohighlight">\(H^1_0\)</span> semi-norm.
+SymPDE allows you to do so, by creating the <strong>Norm</strong> object.
+In this example, the analytical solution is given by</p>
+<div class="math notranslate nohighlight">
+\[
+u_e = \sin(\pi x) \sin(\pi y)
+\]</div>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ue</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+
+<span class="n">u</span> <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+
+<span class="c1"># create the formal Norm object</span>
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">l2norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">l2_error</span> <span class="o">=</span> <span class="n">l2norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">l2_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.00021808678603242297
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-semi-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm<a class="headerlink" href="#computing-the-h-1-semi-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">SemiNorm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">h1norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">h1_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.013023570720372876
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm<a class="headerlink" href="#computing-the-h-1-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">u</span> <span class="o">-</span> <span class="n">ue</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">h1norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">h1_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.013025396583398467
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+</section>
+
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+    <i class="fa-solid fa-list"></i> Contents
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#variational-formulation">Variational Formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#examples">Examples</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+    <h1>First mixed formulation of the Stokes problem</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
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+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="first-mixed-formulation-of-the-stokes-problem">
+<h1>First mixed formulation of the Stokes problem<a class="headerlink" href="#first-mixed-formulation-of-the-stokes-problem" title="Link to this heading">#</a></h1>
+<p>For the variational formulation, we take the dot product of the first equation with <span class="math notranslate nohighlight">\(v\)</span> and integrate over the whole domain</p>
+<div class="math notranslate nohighlight">
+\[
+\int_{\Omega} (-\Delta \mathbf{u} + \nabla p)\cdot \mathbf{v} ~\mathrm{d} \mathbf{x} = \int_{\Omega}\nabla \mathbf{u}:\nabla \mathbf{v} ~\mathrm{d} \mathbf{x} + \int_{\Omega} \nabla p \cdot \mathbf{v} ~\mathrm{d} \mathbf{x} = \int_{\Omega} \mathbf{f}\cdot \mathbf{v} ~\mathrm{d} \mathbf{x}
+\]</div>
+<p>The integration by parts is performed component by component. We impose the essential boundary condition <span class="math notranslate nohighlight">\(\mathbf{v}=0\)</span> on <span class="math notranslate nohighlight">\(\partial\Omega\)</span>, and we denote by</p>
+<div class="math notranslate nohighlight">
+\[
+\int_{\Omega}\nabla \mathbf{u}:\nabla \mathbf{v} ~\mathrm{d} \mathbf{x} =\sum_{i=1}^3 \int_{\Omega}\nabla u_i\cdot\nabla v_i ~\mathrm{d} \mathbf{x} =\sum_{i,j=1}^3 \int_{\Omega}\partial_j u_i\partial_j v_i ~\mathrm{d} \mathbf{x} .
+\]</div>
+<p>We now need to deal with the constraint <span class="math notranslate nohighlight">\(\nabla\cdot \mathbf{u}=0\)</span>. The theoretical framework for saddle point problems requires that the corresponding bilinear form is the same as the second one appearing in the first part of the variational formulation. To this aim we multiply  <span class="math notranslate nohighlight">\(\nabla\cdot \mathbf{u}=0\)</span> by a scalar test function (which will be associated to <span class="math notranslate nohighlight">\(p\)</span>) and integrate on the full domain, with an integration by parts in order to get the same bilinear form as in the first equation</p>
+<div class="math notranslate nohighlight">
+\[
+\int_{\Omega} \nabla\cdot \mathbf{u} ~ q ~\mathrm{d} \mathbf{x}= - \int_{\Omega} \mathbf{u}\cdot \nabla q~\mathrm{d} \mathbf{x}=0,
+\]</div>
+<p>using that <span class="math notranslate nohighlight">\(q=0\)</span> on the boundary as an essential boundary condition.
+We finally obtain the mixed variational formulation:</p>
+<p>Find <span class="math notranslate nohighlight">\(( \mathbf{u},p)\in H^1_0(\Omega)^3\times H^1_0(\Omega)\)</span> such that</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\left\{
+  \begin{array}{llll}
+    \int_{\Omega}\nabla \mathbf{u}:\nabla \mathbf{v} ~\mathrm{d} \mathbf{x} &amp;+ \int_{\Omega} \nabla p \cdot \mathbf{v} ~\mathrm{d} \mathbf{x}
+    &amp;= \int_{\Omega} \mathbf{f}\cdot \mathbf{v} ~\mathrm{d} \mathbf{x}, &amp; \forall \mathbf{v}\in H_0^1(\Omega)^3 \\
+    \int_{\Omega} \mathbf{u}\cdot \nabla q~\mathrm{d} \mathbf{x} &amp; &amp;=0, &amp; \forall p\in H^1_0(\Omega)
+  \end{array}
+  \right.
+\end{align}
+\end{split}\]</div>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="p">(</span><span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">VectorFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span>
+                             <span class="n">elements_of</span><span class="p">)</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">div</span><span class="p">,</span> <span class="n">inner</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span>     <span class="kn">import</span> <span class="n">Constant</span>
+
+<span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">Tuple</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+<span class="n">x</span><span class="p">,</span> <span class="n">y</span>   <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">V1</span> <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V1&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;H1&#39;</span><span class="p">)</span>
+<span class="n">V2</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V2&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;H1&#39;</span><span class="p">)</span>
+<span class="n">X</span>  <span class="o">=</span> <span class="n">V1</span><span class="o">*</span><span class="n">V2</span>
+
+<span class="c1"># rhs</span>
+<span class="n">fx</span> <span class="o">=</span> <span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">24</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">12</span><span class="p">)</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
+<span class="n">fy</span> <span class="o">=</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">24</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">12</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="n">f</span>  <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">fx</span><span class="p">,</span> <span class="n">fy</span><span class="p">)</span>
+
+<span class="c1"># exact solution</span>
+<span class="n">ue1</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="n">ue2</span> <span class="o">=-</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
+<span class="n">ue</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">ue1</span><span class="p">,</span> <span class="n">ue2</span><span class="p">)</span>
+<span class="n">pe</span> <span class="o">=</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+
+<span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">elements_of</span><span class="p">(</span><span class="n">V1</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="s1">&#39;u, v&#39;</span><span class="p">)</span>
+<span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">elements_of</span><span class="p">(</span><span class="n">V2</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="s1">&#39;p, q&#39;</span><span class="p">)</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">a</span>  <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">(((</span><span class="n">u</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">q</span><span class="p">)),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">inner</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">))</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="n">v</span><span class="p">)</span> <span class="o">+</span> <span class="n">dot</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">grad</span><span class="p">(</span><span class="n">q</span><span class="p">))</span> <span class="p">)</span> <span class="p">)</span>
+
+<span class="c1"># linear form</span>
+<span class="n">l</span>  <span class="o">=</span> <span class="n">LinearForm</span><span class="p">((</span><span class="n">v</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">v</span><span class="p">)))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)</span>
+
+<span class="n">equation</span> <span class="o">=</span> <span class="n">find</span><span class="p">((</span><span class="n">u</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="n">forall</span><span class="o">=</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">((</span><span class="n">u</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">q</span><span class="p">)),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">V1h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V1</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+<span class="n">V2h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V2</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+<span class="n">Xh</span>  <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Xh</span><span class="p">,</span> <span class="n">Xh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">equation_h</span><span class="o">.</span><span class="n">set_solver</span><span class="p">(</span><span class="s1">&#39;gmres&#39;</span><span class="p">,</span> <span class="n">info</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-8</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#uh, ph = equation_h.solve()</span>
+
+<span class="kn">from</span> <span class="nn">psydac.fem.basic</span>          <span class="kn">import</span> <span class="n">FemField</span>
+
+<span class="n">phi_h</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+
+<span class="n">uh</span> <span class="o">=</span> <span class="n">FemField</span><span class="p">(</span><span class="n">V1h</span><span class="p">)</span>
+<span class="n">uh</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">][:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">][:]</span>
+<span class="n">uh</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">1</span><span class="p">][:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">1</span><span class="p">][:]</span>
+
+<span class="n">ph</span> <span class="o">=</span> <span class="n">FemField</span><span class="p">(</span><span class="n">V2h</span><span class="p">)</span>
+<span class="n">ph</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">2</span><span class="p">][:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># L2 error norm of the velocity field</span>
+<span class="n">error_u</span>   <span class="o">=</span> <span class="p">[</span><span class="n">ue</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ue</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
+<span class="n">l2norm_u</span>  <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">error_u</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+<span class="n">l2norm_uh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm_u</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">V1h</span><span class="p">)</span>
+
+<span class="c1"># L2 error norm of the pressure, after removing the average value from the field</span>
+<span class="n">l2norm_p</span>  <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">pe</span> <span class="o">-</span> <span class="n">p</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+<span class="n">l2norm_ph</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm_p</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">V2h</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">l2norm</span> <span class="o">=</span> <span class="n">l2norm_uh</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;&gt;&gt;&gt; norm-l2 uh = &#39;</span><span class="p">,</span> <span class="n">l2norm</span><span class="p">)</span>
+
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">l2norm_ph</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">p</span><span class="o">=</span><span class="n">ph</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;&gt;&gt;&gt; norm-l2 ph = &#39;</span><span class="p">,</span> <span class="n">l2norm</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&gt;&gt;&gt; norm-l2 uh =  5.852908166372369e-05
+&gt;&gt;&gt; norm-l2 ph =  0.01411215698700372
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
+</section>
+
+    <script type="text/x-thebe-config">
+    {
+        requestKernel: true,
+        binderOptions: {
+            repo: "binder-examples/jupyter-stacks-datascience",
+            ref: "master",
+        },
+        codeMirrorConfig: {
+            theme: "abcdef",
+            mode: "python"
+        },
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+    <h1>Second mixed formulation of the Stokes problem</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="second-mixed-formulation-of-the-stokes-problem">
+<h1>Second mixed formulation of the Stokes problem<a class="headerlink" href="#second-mixed-formulation-of-the-stokes-problem" title="Link to this heading">#</a></h1>
+<p>Another possibility to obtained a well posed variational formulation, is to integrate by parts the
+<span class="math notranslate nohighlight">\(\int_{\Omega} \nabla p \cdot \mathbf{v} ~\mathrm{d} \mathbf{x}\)</span> term in the first formulation:</p>
+<div class="math notranslate nohighlight">
+\[ 
+\int_{\Omega} \nabla p \cdot \mathbf{v} ~\mathrm{d} \mathbf{x} = - \int_{\Omega} p \nabla \cdot \mathbf{v} ~\mathrm{d} \mathbf{x} + \int_{\partial\Omega} p \mathbf{v} \cdot \mathbf{n}~\mathrm{d} \sigma= -\int_{\Omega} p ~ \nabla \cdot \mathbf{v} ~\mathrm{d} \mathbf{x} ,
+\]</div>
+<p>using here <span class="math notranslate nohighlight">\(p=0\)</span> as a natural boundary condition. Note that in the other variational formulation the same boundary condition was essential. In this case, for the second variational formulation, we just multiply <span class="math notranslate nohighlight">\(\nabla\cdot \mathbf{u}=0\)</span> by <span class="math notranslate nohighlight">\(q\)</span> and integrate. No integration by parts is needed in this case.</p>
+<div class="math notranslate nohighlight">
+\[
+\int_{\Omega} \nabla \cdot \mathbf{u} ~ q ~\mathrm{d} \mathbf{x} =0.
+\]</div>
+<p>This then leads to the following variational formulation:</p>
+<p>Find <span class="math notranslate nohighlight">\(( \mathbf{u},p)\in H^1(\Omega)^3\times L^2(\Omega)\)</span> such that</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\left\{
+  \begin{array}{llll}
+    \int_{\Omega}\nabla \mathbf{u}:\nabla \mathbf{v} ~\mathrm{d} \mathbf{x} &amp;- \int_{\Omega}  p ~ \nabla\cdot \mathbf{v} ~\mathrm{d} \mathbf{x}
+    &amp;= \int_{\Omega} \mathbf{f}\cdot \mathbf{v} ~\mathrm{d} \mathbf{x}, &amp;\forall \mathbf{v}\in H^1(\Omega)^3
+    \\
+    \int_{\Omega}  \nabla\cdot\mathbf{u} ~ q~\mathrm{d} \mathbf{x} &amp; &amp;=0,  &amp;\forall q\in L^2(\Omega)
+  \end{array}
+  \right.
+\end{align}
+\end{split}\]</div>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="p">(</span><span class="n">ScalarFunctionSpace</span><span class="p">,</span> <span class="n">VectorFunctionSpace</span><span class="p">,</span> <span class="n">Square</span><span class="p">,</span>
+                             <span class="n">elements_of</span><span class="p">)</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span><span class="p">,</span> <span class="n">div</span><span class="p">,</span> <span class="n">inner</span>
+<span class="kn">from</span> <span class="nn">sympde.core</span>     <span class="kn">import</span> <span class="n">Constant</span>
+
+<span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">Tuple</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+<span class="n">x</span><span class="p">,</span> <span class="n">y</span>   <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">V1</span> <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V1&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;H1&#39;</span><span class="p">)</span>
+<span class="n">V2</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V2&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;L2&#39;</span><span class="p">)</span>
+<span class="n">X</span> <span class="o">=</span> <span class="n">V1</span><span class="o">*</span><span class="n">V2</span>
+
+<span class="c1"># rhs</span>
+<span class="n">fx</span> <span class="o">=</span> <span class="o">-</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">24</span><span class="o">*</span><span class="n">y</span> <span class="o">-</span> <span class="mi">12</span><span class="p">)</span> <span class="o">-</span> <span class="mi">4</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
+<span class="n">fy</span> <span class="o">=</span> <span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">4</span><span class="o">*</span><span class="n">y</span><span class="o">*</span><span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">24</span><span class="o">*</span><span class="n">x</span> <span class="o">-</span> <span class="mi">12</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">y</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="n">f</span>  <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">fx</span><span class="p">,</span> <span class="n">fy</span><span class="p">)</span>
+
+<span class="c1"># exact solution</span>
+<span class="n">ue1</span> <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="n">ue2</span> <span class="o">=-</span><span class="n">y</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="o">-</span><span class="n">y</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">6</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">*</span><span class="n">x</span><span class="p">)</span>
+<span class="n">ue</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">ue1</span><span class="p">,</span> <span class="n">ue2</span><span class="p">)</span>
+<span class="n">pe</span> <span class="o">=</span> <span class="o">-</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">sin</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+
+<span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">elements_of</span><span class="p">(</span><span class="n">V1</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="s1">&#39;u, v&#39;</span><span class="p">)</span>
+<span class="n">p</span><span class="p">,</span> <span class="n">q</span> <span class="o">=</span> <span class="n">elements_of</span><span class="p">(</span><span class="n">V2</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="s1">&#39;p, q&#39;</span><span class="p">)</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">a</span>  <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">(((</span><span class="n">u</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">q</span><span class="p">)),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">inner</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">))</span> <span class="o">-</span> <span class="n">div</span><span class="p">(</span><span class="n">u</span><span class="p">)</span><span class="o">*</span><span class="n">q</span> <span class="o">-</span> <span class="n">p</span><span class="o">*</span><span class="n">div</span><span class="p">(</span><span class="n">v</span><span class="p">))</span> <span class="p">)</span>
+
+<span class="c1"># linear form</span>
+<span class="n">l</span>  <span class="o">=</span> <span class="n">LinearForm</span><span class="p">((</span><span class="n">v</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span> <span class="n">v</span><span class="p">)))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)</span>
+
+<span class="n">equation</span> <span class="o">=</span> <span class="n">find</span><span class="p">((</span><span class="n">u</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="n">forall</span><span class="o">=</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">((</span><span class="n">u</span><span class="p">,</span> <span class="n">p</span><span class="p">),</span> <span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">q</span><span class="p">)),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">q</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">V1h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V1</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+<span class="n">V2h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V2</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+<span class="n">Xh</span>  <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Xh</span><span class="p">,</span> <span class="n">Xh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">equation_h</span><span class="o">.</span><span class="n">set_solver</span><span class="p">(</span><span class="s1">&#39;gmres&#39;</span><span class="p">,</span> <span class="n">info</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-8</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#uh, ph = equation_h.solve()</span>
+
+<span class="kn">from</span> <span class="nn">psydac.fem.basic</span>          <span class="kn">import</span> <span class="n">FemField</span>
+
+<span class="n">phi_h</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+
+<span class="n">uh</span> <span class="o">=</span> <span class="n">FemField</span><span class="p">(</span><span class="n">V1h</span><span class="p">)</span>
+<span class="n">uh</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">][:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">0</span><span class="p">][:]</span>
+<span class="n">uh</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">1</span><span class="p">][:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">1</span><span class="p">][:]</span>
+
+<span class="n">ph</span> <span class="o">=</span> <span class="n">FemField</span><span class="p">(</span><span class="n">V2h</span><span class="p">)</span>
+<span class="n">ph</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[:]</span> <span class="o">=</span> <span class="n">phi_h</span><span class="o">.</span><span class="n">coeffs</span><span class="p">[</span><span class="mi">2</span><span class="p">][:]</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># L2 error norm of the velocity field</span>
+<span class="n">error_u</span>   <span class="o">=</span> <span class="p">[</span><span class="n">ue</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ue</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]]</span>
+<span class="n">l2norm_u</span>  <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">error_u</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+<span class="n">l2norm_uh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm_u</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">V1h</span><span class="p">)</span>
+
+<span class="c1"># L2 error norm of the pressure, after removing the average value from the field</span>
+<span class="n">l2norm_p</span>  <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">pe</span> <span class="o">-</span> <span class="n">p</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+<span class="n">l2norm_ph</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm_p</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">V2h</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">l2norm</span> <span class="o">=</span> <span class="n">l2norm_uh</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;&gt;&gt;&gt; norm-l2 uh = &#39;</span><span class="p">,</span> <span class="n">l2norm</span><span class="p">)</span>
+
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">l2norm_ph</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">p</span><span class="o">=</span><span class="n">ph</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;&gt;&gt;&gt; norm-l2 ph = &#39;</span><span class="p">,</span> <span class="n">l2norm</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&gt;&gt;&gt; norm-l2 uh =  1.860724404008467e-05
+&gt;&gt;&gt; norm-l2 ph =  0.024428172461472355
+</pre></div>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.html">sympde.printing</a><input class="toctree-checkbox" id="toctree-checkbox-20" name="toctree-checkbox-20" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-20"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.form.html">sympde.exterior.form</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
+</ul>
+</li>
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+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.utilities.utils.html">sympde.utilities.utils</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.utilities.html">psydac.api.ast.utilities</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.discretization.html">psydac.api.discretization</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.equation.html">psydac.api.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.essential_bc.html">psydac.api.essential_bc</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.expr.html">psydac.api.expr</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.feec.html">psydac.api.feec</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.fem.html">psydac.api.fem</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.glt.html">psydac.api.glt</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.grid.html">psydac.api.grid</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.postprocessing.html">psydac.api.postprocessing</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/psydac.api.printing.html">psydac.api.printing</a><input class="toctree-checkbox" id="toctree-checkbox-29" name="toctree-checkbox-29" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-29"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.printing.pycode.html">psydac.api.printing.pycode</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.settings.html">psydac.api.settings</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/psydac.api.tests.html">psydac.api.tests</a><input class="toctree-checkbox" id="toctree-checkbox-30" name="toctree-checkbox-30" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-30"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.tests.build_domain.html">psydac.api.tests.build_domain</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.tests.test_2d_biharmonic.html">psydac.api.tests.test_2d_biharmonic</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.tests.test_2d_complex.html">psydac.api.tests.test_2d_complex</a></li>
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+    <h1>Mixed FEM for the Stokes problem</h1>
+    <!-- Table of contents -->
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+  <section class="tex2jax_ignore mathjax_ignore" id="mixed-fem-for-the-stokes-problem">
+<h1>Mixed FEM for the Stokes problem<a class="headerlink" href="#mixed-fem-for-the-stokes-problem" title="Link to this heading">#</a></h1>
+<p>We consider now the Stokes problem for the steady-state modelling of an incompressible fluid</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+  \left\{
+    \begin{array}{rl}
+      - \nabla^2 \mathbf{u} + \nabla p = \mathbf{f} &amp; \mbox{in} ~ \Omega,  \\
+      \nabla \cdot \mathbf{u}          = 0   &amp; \mbox{in} ~ \Omega,  \\
+      \mathbf{u}               = 0   &amp; \mbox{on} ~ \partial \Omega,
+    \end{array}
+    \right.
+\end{align}
+\end{split}\]</div>
+<div class="toctree-wrapper compound">
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+    <h1>Vector Poisson equation</h1>
+    <!-- Table of contents -->
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+            
+            <div>
+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-variational-formulation">The Variational Formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#formal-model">Formal Model</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#solving-the-pde">Solving the PDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-error-norm">Computing the error norm</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-l-2-norm">Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
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+    </div>
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+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="vector-poisson-equation">
+<h1>Vector Poisson equation<a class="headerlink" href="#vector-poisson-equation" title="Link to this heading">#</a></h1>
+<p>In this example we consider the vector Poisson equation with homogeneous Dirichlet boundary conditions:</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  - \nabla^2 \mathbf{u} = \mathbf{f} \quad \mbox{in} ~ \Omega, \quad \quad 
+  \mathbf{u} = 0            \quad \mbox{on} ~ \partial \Omega.
+\end{align}
+\]</div>
+<section id="the-variational-formulation">
+<h2>The Variational Formulation<a class="headerlink" href="#the-variational-formulation" title="Link to this heading">#</a></h2>
+<p>The corresponding variational formulation, using <span class="math notranslate nohighlight">\(\mathbf{H}^1\)</span> formulation, <em>i.e.</em> all components are in <span class="math notranslate nohighlight">\(H^1\)</span>, reads</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \text{find $\mathbf{u} \in V$ such that} \quad 
+  a(\mathbf{u},\mathbf{v}) = l(\mathbf{v}) \quad \forall \mathbf{v} \in V,
+\end{align}
+\]</div>
+<p>where</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(V \subset \mathbf{H}_0^1(\Omega)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(a(\mathbf{u},\mathbf{v}) := \int_{\Omega} \nabla \mathbf{u} : \nabla \mathbf{v} ~ d\Omega\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(l(\mathbf{v}) := \int_{\Omega} \mathbf{f} \cdot \mathbf{v} ~ d\Omega\)</span>.</p></li>
+</ul>
+</section>
+<section id="formal-model">
+<h2>Formal Model<a class="headerlink" href="#formal-model" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.expr</span> <span class="kn">import</span> <span class="n">BilinearForm</span><span class="p">,</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">integral</span>
+<span class="kn">from</span> <span class="nn">sympde.expr</span>     <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span><span class="p">,</span> <span class="n">Norm</span><span class="p">,</span> <span class="n">SemiNorm</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">VectorFunctionSpace</span><span class="p">,</span> <span class="n">Cube</span><span class="p">,</span> <span class="n">element_of</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">inner</span><span class="p">,</span> <span class="n">dot</span>
+
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">Tuple</span><span class="p">,</span> <span class="n">Matrix</span>
+
+<span class="n">domain</span> <span class="o">=</span> <span class="n">Cube</span><span class="p">()</span>
+
+<span class="n">V</span> <span class="o">=</span> <span class="n">VectorFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+
+<span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">z</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">u</span><span class="p">,</span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="p">[</span><span class="s1">&#39;u&#39;</span><span class="p">,</span> <span class="s1">&#39;v&#39;</span><span class="p">]]</span>
+
+<span class="c1"># bilinear form</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span> <span class="p">,</span> <span class="n">inner</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">))))</span>
+
+<span class="c1"># linear form</span>
+<span class="n">f1</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">f2</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">f3</span> <span class="o">=</span> <span class="mi">3</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">f</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">f1</span><span class="p">,</span> <span class="n">f2</span><span class="p">,</span> <span class="n">f3</span><span class="p">)</span>
+
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">dot</span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="n">v</span><span class="p">)))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="discretization">
+<h2>Discretization<a class="headerlink" href="#discretization" title="Link to this heading">#</a></h2>
+<p>We shall need the <strong>discretize</strong> function from <strong>PsyDAC</strong>.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">,</span><span class="mi">8</span><span class="p">]</span>
+</pre></div>
+</div>
+</div>
+</div>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Create computational domain from topological domain</span>
+<span class="n">domain_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">comm</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="c1"># Create discrete spline space</span>
+<span class="n">Vh</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+
+<span class="c1"># Discretize equation</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">])</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="solving-the-pde">
+<h2>Solving the PDE<a class="headerlink" href="#solving-the-pde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">uh</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
+<span class="ne">KeyboardInterrupt</span><span class="g g-Whitespace">                         </span>Traceback (most recent call last)
+<span class="n">Cell</span> <span class="n">In</span><span class="p">[</span><span class="mi">5</span><span class="p">],</span> <span class="n">line</span> <span class="mi">1</span>
+<span class="ne">----&gt; </span><span class="mi">1</span> <span class="n">uh</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+
+<span class="nn">File /opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/psydac/api/equation.py:198,</span> in <span class="ni">DiscreteEquation.solve</span><span class="nt">(self, **kwargs)</span>
+<span class="g g-Whitespace">    </span><span class="mi">196</span> <span class="k">def</span> <span class="nf">solve</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
+<span class="ne">--&gt; </span><span class="mi">198</span>     <span class="bp">self</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+<span class="g g-Whitespace">    </span><span class="mi">200</span>     <span class="c1"># Free arguments of current equation</span>
+<span class="g g-Whitespace">    </span><span class="mi">201</span>     <span class="n">free_args</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">free_args</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">rhs</span><span class="o">.</span><span class="n">free_args</span><span class="p">)</span>
+
+<span class="nn">File /opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/psydac/api/equation.py:178,</span> in <span class="ni">DiscreteEquation.assemble</span><span class="nt">(self, **kwargs)</span>
+<span class="g g-Whitespace">    </span><span class="mi">176</span> <span class="c1"># Matrix (left-hand side)</span>
+<span class="g g-Whitespace">    </span><span class="mi">177</span> <span class="k">if</span> <span class="n">assemble_lhs</span><span class="p">:</span>
+<span class="ne">--&gt; </span><span class="mi">178</span>     <span class="n">A</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">lhs</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">reset</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>
+<span class="g g-Whitespace">    </span><span class="mi">179</span>     <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">bc</span><span class="p">:</span>
+<span class="g g-Whitespace">    </span><span class="mi">180</span>         <span class="n">apply_essential_bc</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">bc</span><span class="p">)</span>
+
+<span class="nn">File /opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/psydac/api/fem.py:496,</span> in <span class="ni">DiscreteBilinearForm.assemble</span><span class="nt">(self, reset, **kwargs)</span>
+<span class="g g-Whitespace">    </span><span class="mi">493</span> <span class="k">if</span> <span class="n">reset</span><span class="p">:</span>
+<span class="g g-Whitespace">    </span><span class="mi">494</span>     <span class="n">reset_arrays</span><span class="p">(</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">global_matrices</span><span class="p">)</span>
+<span class="ne">--&gt; </span><span class="mi">496</span> <span class="bp">self</span><span class="o">.</span><span class="n">_func</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">_threads_args</span><span class="p">)</span>
+<span class="g g-Whitespace">    </span><span class="mi">497</span> <span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_matrix</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">_update_ghost_regions</span><span class="p">:</span>
+<span class="g g-Whitespace">    </span><span class="mi">498</span>     <span class="bp">self</span><span class="o">.</span><span class="n">_matrix</span><span class="o">.</span><span class="n">exchange_assembly_data</span><span class="p">()</span>
+
+<span class="nn">File ~/work/IGA-Python/IGA-Python/chapter2/__psydac__/dependencies_bkggd7ac.py:58,</span> in <span class="ni">assemble_matrix_bkggd7ac</span><span class="nt">(global_test_basis_v_1, global_test_basis_v_2, global_test_basis_v_3, global_trial_basis_u_1, global_trial_basis_u_2, global_trial_basis_u_3, global_span_v_1, global_span_v_2, global_span_v_3, global_x1, global_x2, global_x3, test_v_p1, test_v_p2, test_v_p3, trial_u_p1, trial_u_p2, trial_u_p3, n_element_1, n_element_2, n_element_3, k1, k2, k3, pad1, pad2, pad3, g_mat_u_0_v_0_bkggd7ac, g_mat_u_1_v_1_bkggd7ac, g_mat_u_2_v_2_bkggd7ac)</span>
+<span class="g g-Whitespace">     </span><span class="mi">56</span> <span class="n">u_2_2_x2</span> <span class="o">=</span> <span class="n">global_trial_basis_u_2</span><span class="p">[</span><span class="n">i_element_2</span><span class="p">,</span><span class="n">j_basis_2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">i_quad_2</span><span class="p">]</span>
+<span class="g g-Whitespace">     </span><span class="mi">57</span> <span class="k">for</span> <span class="n">i_quad_3</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">):</span>
+<span class="ne">---&gt; </span><span class="mi">58</span>     <span class="n">x3</span> <span class="o">=</span> <span class="n">local_x3</span><span class="p">[</span><span class="n">i_quad_3</span><span class="p">]</span>
+<span class="g g-Whitespace">     </span><span class="mi">59</span>     <span class="n">v_0_3</span> <span class="o">=</span> <span class="n">global_test_basis_v_3</span><span class="p">[</span><span class="n">i_element_3</span><span class="p">,</span><span class="n">i_basis_3</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="n">i_quad_3</span><span class="p">]</span>
+<span class="g g-Whitespace">     </span><span class="mi">60</span>     <span class="n">v_0_3_x3</span> <span class="o">=</span> <span class="n">global_test_basis_v_3</span><span class="p">[</span><span class="n">i_element_3</span><span class="p">,</span><span class="n">i_basis_3</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">i_quad_3</span><span class="p">]</span>
+
+<span class="ne">KeyboardInterrupt</span>: 
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-error-norm">
+<h2>Computing the error norm<a class="headerlink" href="#computing-the-error-norm" title="Link to this heading">#</a></h2>
+<p>When the analytical solution is available, you might be interested in computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm or <span class="math notranslate nohighlight">\(H^1_0\)</span> semi-norm.
+SymPDE allows you to do so, by creating the <strong>Norm</strong> object.
+In this example, the analytical solution is given by</p>
+<div class="math notranslate nohighlight">
+\[
+u_e = \sin(\pi x) \sin(\pi y) \sin(\pi z)
+\]</div>
+<section id="computing-the-l-2-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(L^2\)</span> norm<a class="headerlink" href="#computing-the-l-2-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ue1</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">ue2</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">ue3</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span>
+<span class="n">ue</span> <span class="o">=</span> <span class="n">Tuple</span><span class="p">(</span><span class="n">ue1</span><span class="p">,</span> <span class="n">ue2</span><span class="p">,</span> <span class="n">ue3</span><span class="p">)</span>
+
+<span class="n">u</span> <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+
+<span class="n">error</span> <span class="o">=</span> <span class="n">Matrix</span><span class="p">([</span><span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">-</span><span class="n">ue</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">-</span><span class="n">ue</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">u</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="o">-</span><span class="n">ue</span><span class="p">[</span><span class="mi">2</span><span class="p">]])</span>
+
+<span class="c1"># create the formal Norm object</span>
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">error</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">l2norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">l2_error</span> <span class="o">=</span> <span class="n">l2norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">l2_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.0003267962376349343
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-semi-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm<a class="headerlink" href="#computing-the-h-1-semi-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">SemiNorm</span><span class="p">(</span><span class="n">error</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">h1norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">h1_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.01956220393797005
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="computing-the-h-1-norm">
+<h3>Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm<a class="headerlink" href="#computing-the-h-1-norm" title="Link to this heading">#</a></h3>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># create the formal Norm object</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">error</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+
+<span class="c1"># discretize the norm</span>
+<span class="n">h1norm_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">domain_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">)</span>
+
+<span class="c1"># assemble the norm</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="c1"># print the result</span>
+<span class="nb">print</span><span class="p">(</span><span class="n">h1_error</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.01956493339348906
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+</section>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discretization">Discretization</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-semi-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> semi-norm</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-the-h-1-norm">Computing the <span class="math notranslate nohighlight">\(H^1\)</span> norm</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
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+</li>
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+</li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#time-scheme">Time scheme</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weak-formulation">Weak formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sympde-code">SymPDE code</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-linear-form-g">Defining the Linear form <span class="math notranslate nohighlight">\(G\)</span></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-linear-form-l">Defining the Linear form <span class="math notranslate nohighlight">\(l\)</span></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#picard-method">Picard Method</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#picard-iteration">Picard iteration</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#newton-method">Newton Method</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#newton-iteration">Newton iteration</a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="d-burgers-equation">
+<h1>1D Burgers equation<a class="headerlink" href="#d-burgers-equation" title="Link to this heading">#</a></h1>
+<p>We consider the 1d Burgers equation</p>
+<div class="math notranslate nohighlight">
+\[
+\partial_t u + u \partial_x u = \nu \frac{\partial ^2u}{\partial x^2}
+\]</div>
+<p><span class="math notranslate nohighlight">\(u_0(x) := u(x,t)\)</span> denotes the initial condition.
+We choose homogeneous neumann boundary conditions in this example, i.e.
+$<span class="math notranslate nohighlight">\(\partial_n u = 0, \partial \Omega\)</span><span class="math notranslate nohighlight">\( with \)</span>\Omega = (0,1)$</p>
+<section id="time-scheme">
+<h2>Time scheme<a class="headerlink" href="#time-scheme" title="Link to this heading">#</a></h2>
+<p>We shall use a <span class="math notranslate nohighlight">\(\theta\)</span>-scheme in this case and consider the following problem</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\frac{u^{n+1}-u^n}{\Delta t} + 
+\theta~ u^{n+1} \partial_x u^{n+1} + (1-\theta)~ u^n \partial_x u^n = \theta~\nu \frac{\partial ^2u^{n+1}}{\partial x^2} + (1-\theta)~\nu \frac{\partial ^2u^{n}}{\partial x^2}
+\end{align}
+\]</div>
+<p>hence</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+u^{n+1} + \Delta t ~ \theta~ u^{n+1} \partial_x u^{n+1} - \Delta t ~ \theta~\nu \frac{\partial ^2u^{n+1}}{\partial x^2} =
+u^{n} - \Delta t ~ (1-\theta)~ u^{n} \partial_x u^{n} + \Delta t ~ (1-\theta)~\nu \frac{\partial ^2u^{n}}{\partial x^2}
+\end{align}
+\]</div>
+<p>from now on, we shall denote by <span class="math notranslate nohighlight">\(f^n\)</span> the right hand side of the previous equation</p>
+<div class="math notranslate nohighlight">
+\[f^n := u^{n} - \Delta t ~ (1-\theta)~ u^{n} \partial_x u^{n} + \Delta t ~ (1-\theta)~\nu \frac{\partial ^2u^{n}}{\partial x^2}\]</div>
+</section>
+<section id="weak-formulation">
+<h2>Weak formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Let <span class="math notranslate nohighlight">\(v \in \mathcal{V}\)</span> be a function test, we have by integrating by parts the highest order term:</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\langle v, u^{n+1}  \rangle + \Delta t ~ \theta~ \langle v, u^{n+1} \partial_x u^{n+1}  \rangle + \Delta t ~ \theta~\nu \langle \frac{\partial v}{\partial x}, \frac{\partial u^{n+1}}{\partial x}  \rangle = \langle v, f^n \rangle
+\end{align}
+\]</div>
+<p>The previous weak formulation is still nonlinear with respect to <span class="math notranslate nohighlight">\(u^{n+1}\)</span>. We shall then follow the same strategy as for the previous chapter on nonlinear Poisson problem.</p>
+<p>The strategy is to define the left hand side as a <strong>LinearForm</strong> with respect to <span class="math notranslate nohighlight">\(v\)</span>, then linearize it around <span class="math notranslate nohighlight">\(u^{n+1}\)</span>. We therefor can use either Picard or Newton method to treat the nonlinearity.</p>
+<p>We consider the following linear form</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+G(v;u,w) := \langle v, u  \rangle + \Delta t ~ \theta~ \langle v, w \partial_x u  \rangle + \Delta t ~ \theta~\nu \langle \frac{\partial v}{\partial x}, \frac{\partial u}{\partial x}  \rangle , \quad \forall u,v,w \in \mathcal{V}
+\end{align}
+\]</div>
+<p>Our problem is then</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\mbox{Find } u^{n+1} \in \mathcal{V}, \mbox{such that}\\
+G(v;u^{n+1},u^{n+1}) = l(v), \quad \forall v \in \mathcal{V}
+\end{align}
+\end{split}\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[
+l(v) := \int_{\Omega} f^n v ~d\Omega, \quad \forall v \in \mathcal{V}
+\]</div>
+</section>
+<section id="sympde-code">
+<h2>SymPDE code<a class="headerlink" href="#sympde-code" title="Link to this heading">#</a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">domain</span> <span class="o">=</span> <span class="n">Line</span><span class="p">()</span>
+
+<span class="n">V</span> <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+<span class="n">u</span>  <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+<span class="n">v</span>  <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;v&#39;</span><span class="p">)</span>
+<span class="n">w</span>  <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;w&#39;</span><span class="p">)</span>
+<span class="n">un</span>  <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;un&#39;</span><span class="p">)</span> <span class="c1"># time iteration</span>
+<span class="n">uk</span>  <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;uk&#39;</span><span class="p">)</span> <span class="c1"># nonlinear solver iteration</span>
+
+<span class="n">x</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">coordinates</span>
+
+<span class="n">nu</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;nu&#39;</span><span class="p">)</span>
+<span class="n">theta</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;theta&#39;</span><span class="p">)</span>
+<span class="n">dt</span> <span class="o">=</span> <span class="n">Constant</span><span class="p">(</span><span class="s1">&#39;dt&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+<section id="defining-the-linear-form-g">
+<h3>Defining the Linear form <span class="math notranslate nohighlight">\(G\)</span><a class="headerlink" href="#defining-the-linear-form-g" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Linear form g: V --&gt; R</span>
+<span class="n">expr</span> <span class="o">=</span> <span class="n">v</span> <span class="o">*</span> <span class="n">u</span> <span class="o">+</span> <span class="n">dt</span><span class="o">*</span><span class="n">theta</span><span class="o">*</span><span class="n">v</span><span class="o">*</span><span class="n">w</span><span class="o">*</span><span class="n">dx</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">+</span> <span class="n">dt</span><span class="o">*</span><span class="n">theta</span><span class="o">*</span><span class="n">nu</span><span class="o">*</span><span class="n">dx</span><span class="p">(</span><span class="n">v</span><span class="p">)</span><span class="o">*</span><span class="n">dx</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
+<span class="n">g</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">expr</span><span class="p">))</span>
+</pre></div>
+</div>
+</section>
+<section id="defining-the-linear-form-l">
+<h3>Defining the Linear form <span class="math notranslate nohighlight">\(l\)</span><a class="headerlink" href="#defining-the-linear-form-l" title="Link to this heading">#</a></h3>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Linear form l: V --&gt; R</span>
+<span class="n">expr</span> <span class="o">=</span> <span class="n">v</span> <span class="o">*</span> <span class="n">un</span> <span class="o">-</span> <span class="n">dt</span><span class="o">*</span><span class="n">theta</span><span class="o">*</span><span class="n">v</span><span class="o">*</span><span class="n">un</span><span class="o">*</span><span class="n">dx</span><span class="p">(</span><span class="n">un</span><span class="p">)</span> <span class="o">-</span> <span class="n">dt</span><span class="o">*</span><span class="n">theta</span><span class="o">*</span><span class="n">nu</span><span class="o">*</span><span class="n">dx</span><span class="p">(</span><span class="n">v</span><span class="p">)</span><span class="o">*</span><span class="n">dx</span><span class="p">(</span><span class="n">un</span><span class="p">)</span>
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">expr</span><span class="p">))</span>
+</pre></div>
+</div>
+</section>
+<section id="picard-method">
+<h3>Picard Method<a class="headerlink" href="#picard-method" title="Link to this heading">#</a></h3>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\mbox{Find } u_{n+1} \in \mathcal{V}\_h, \mbox{such that}\\
+G(v;u_{n+1},u_n) = l(v), \quad \forall v \in \mathcal{V}\_h
+\end{align}
+\end{split}\]</div>
+<section id="picard-iteration">
+<h4>Picard iteration<a class="headerlink" href="#picard-iteration" title="Link to this heading">#</a></h4>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Variational problem</span>
+<span class="n">picard</span> <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">g</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">u</span><span class="o">=</span><span class="n">u</span><span class="p">,</span><span class="n">w</span><span class="o">=</span><span class="n">uk</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
+</pre></div>
+</div>
+</section>
+</section>
+<section id="newton-method">
+<h3>Newton Method<a class="headerlink" href="#newton-method" title="Link to this heading">#</a></h3>
+<p>Let’s define</p>
+<div class="math notranslate nohighlight">
+\[
+F(v;u) := G(v;u,u) -l(v), \quad \forall v \in \mathcal{V}
+\]</div>
+<p>Newton method writes</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\mbox{Find } u_{n+1} \in \mathcal{V}\_h, \mbox{such that}\\
+F^{\prime}(\delta u,v; u_n) = - F(v;u_n), \quad \forall v \in \mathcal{V} \\
+u_{n+1} := u_{n} + \delta u, \quad \delta u \in \mathcal{V}
+\end{align}
+\end{split}\]</div>
+<section id="newton-iteration">
+<h4>Newton iteration<a class="headerlink" href="#newton-iteration" title="Link to this heading">#</a></h4>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">F</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">g</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">w</span><span class="o">=</span><span class="n">u</span><span class="p">)</span><span class="o">-</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
+<span class="n">du</span>  <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;du&#39;</span><span class="p">)</span>
+
+<span class="n">Fprime</span> <span class="o">=</span> <span class="n">linearize</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">trials</span><span class="o">=</span><span class="n">du</span><span class="p">)</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">newton</span> <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">du</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">Fprime</span><span class="p">(</span><span class="n">du</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span><span class="n">u</span><span class="o">=</span><span class="n">uk</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=-</span><span class="n">F</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">u</span><span class="o">=</span><span class="n">uk</span><span class="p">))</span>
+</pre></div>
+</div>
+</section>
+</section>
+</section>
+</section>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#time-scheme">Time scheme</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weak-formulation">Weak formulation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sympde-code">SymPDE code</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-linear-form-g">Defining the Linear form <span class="math notranslate nohighlight">\(G\)</span></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#defining-the-linear-form-l">Defining the Linear form <span class="math notranslate nohighlight">\(l\)</span></a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#picard-method">Picard Method</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#picard-iteration">Picard iteration</a></li>
+</ul>
+</li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="burgers.html">1D Burgers equation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+    <h1>The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#linearization-of-the-streamfunction-velocity-linear-form">Linearization of the streamfunction-velocity linear form</a></li>
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+  <section class="tex2jax_ignore mathjax_ignore" id="the-streamfunction-velocity-formulation-of-the-steady-state-navier-stokes-equations-for-incompressible-fluids">
+<h1>The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids<a class="headerlink" href="#the-streamfunction-velocity-formulation-of-the-steady-state-navier-stokes-equations-for-incompressible-fluids" title="Link to this heading">#</a></h1>
+<p>When <span class="math notranslate nohighlight">\(\Omega\)</span> is a simply connected 2D domain, there exists a unique function <span class="math notranslate nohighlight">\(\psi\)</span> such that <span class="math notranslate nohighlight">\(\mathbf{u} = \boldsymbol{\nabla} \times \psi:= \left( \partial_y \psi, - \partial_x \psi \right)\)</span>;
+substituting this expression for <span class="math notranslate nohighlight">\(\mathbf{u}\)</span> into \eqref{eq:steady-navier-stokes} leads to the so-called ``streamfunction-velocity formulation’’ of the steady-state Navier-Stokes equations for an incompressible fluid.
+For more details we refer the reader to \cite{TAGLIABUE2014277}.
+The variational formulation is then given by:</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \text{find $\psi \in V$ such that} \quad 
+  a(\psi,\phi) + b(\psi,\phi;\psi) = l(\phi)
+  \quad \forall \phi \in V,
+\end{align}
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(V \subset H^2(\Omega)\)</span> and</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align*}
+  a(\psi, \phi) := \int_{\Omega} D\left( \boldsymbol{\nabla} \times \psi \right) : D\left(\boldsymbol{\nabla} \times \phi \right) ~d\Omega,  
+  \quad
+  b(\psi, \phi; \xi) := \int_{\Omega} \left( \left(\boldsymbol{\nabla} \times \xi \cdot \nabla \right) \boldsymbol{\nabla} \times \psi \right) \cdot \boldsymbol{\nabla} \times \phi ~d\Omega,
+  \quad
+  l(\phi) := \int_{\Omega} \mathbf{f} \cdot \boldsymbol{\nabla} \times \phi ~d\Omega.
+\end{align*}
+\]</div>
+<section id="linearization-of-the-streamfunction-velocity-linear-form">
+<h2>Linearization of the streamfunction-velocity linear form<a class="headerlink" href="#linearization-of-the-streamfunction-velocity-linear-form" title="Link to this heading">#</a></h2>
+<p>In the sequel, we use SymPDE to linearize the streamfunction-velocity formulation, starting from a \texttt{LinearForm}.
+As presented in the previous example, we shall write our weak formulation as a linear form where the velocity <span class="math notranslate nohighlight">\(\mathbf{u}\)</span> (\textit{i.e.} \texttt{u}) is considered as a \textit{field}, which is implemented in Python code \ref{code:steady-navier-stokes-nl}.
+Using the function \texttt{linearize} allows us to construct the bilinear form associated to the linearization of the given linear form, around the field \texttt{psi}, see Python code \ref{code:steady-streamfunction-velocity-lin}.
+To get an \texttt{Equation}, the user can add the left-hand-side for the Navier-Stokes equation to the linear form, the appropriate essential boundary conditions, then use the \texttt{NewtonIteration} as presented in the previous example.</p>
+</section>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="burgers.html">1D Burgers equation</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
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+    <h1>the steady-state Navier-Stokes equations for incompressible fluids</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
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+  <section class="tex2jax_ignore mathjax_ignore" id="the-steady-state-navier-stokes-equations-for-incompressible-fluids">
+<h1>the steady-state Navier-Stokes equations for incompressible fluids<a class="headerlink" href="#the-steady-state-navier-stokes-equations-for-incompressible-fluids" title="Link to this heading">#</a></h1>
+<p>The steady-state Navier Stokes problem for an incompressible fluid, with homogeneous Dirichlet boundary conditions (``no slip’’ condition), is defined as</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{equation}
+\left\{
+\begin{aligned}
+        \left( \mathbf{u} \cdot \nabla \right) \mathbf{u} + \nabla p - 2 \nabla \cdot \left( \frac{1}{R_e} D\left( \mathbf{u} \right) \right) &amp;= \mathbf{f} &amp;&amp; \text{in $\Omega$},
+  \\
+        \nabla \cdot \mathbf{u} &amp;= 0 &amp;&amp; \text{in $\Omega$},
+  \\
+        \mathbf{u} &amp;= 0 &amp;&amp; \text{on $\partial\Omega$},
+\end{aligned}
+\right.
+\end{equation}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(R_e\)</span> is the Reynolds number and <span class="math notranslate nohighlight">\(D\left( \mathbf{u} \right) := \frac{1}{2}\left( \nabla \mathbf{u} + \nabla \mathbf{u}^T \right)\)</span> is the strain rate tensor.
+The weak formulation reads</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \text{find $\mathbf{u} \in W$ such that} \quad 
+  a(\mathbf{u},\mathbf{v}) + b(\mathbf{u},\mathbf{v};\mathbf{u}) = l(\mathbf{v}) \quad
+  \forall \mathbf{v} \in W
+\end{align}
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(W \subset \{ \mathbf{v} \in \mathbf{H}_0^1(\Omega): \nabla \cdot \mathbf{v} = 0 \}\)</span> and</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align*}
+  a(\mathbf{u}, \mathbf{v}) := \frac{2}{R_e} \int_{\Omega} D\left( \mathbf{u} \right) : D\left(\mathbf{v} \right) ~d\Omega,
+  \quad
+  b(\mathbf{u}, \mathbf{v}; \mathbf{w}) := \int_{\Omega} \left( \left( \mathbf{w} \cdot \nabla \right) \mathbf{u} \right) \cdot \mathbf{v} ~d\Omega,
+  \quad
+  l(\mathbf{v}) := \int_{\Omega} \mathbf{f} \cdot \mathbf{v} ~d\Omega.
+\end{align*}
+\]</div>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Nonlinear Poisson in 2D</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-topological-domain">1. The topological domain</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#function-space">2. Function space</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-the-linear-form-g">3.1. Define the Linear form <span class="math notranslate nohighlight">\(G\)</span></a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-the-linear-form-l">3.2. Define the linear form <span class="math notranslate nohighlight">\(L\)</span></a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#picard-iteration">4.1. Picard iteration</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#newton-iteration">4.2. Newton iteration</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-f-prime-the-derivative-of-f">Computing <span class="math notranslate nohighlight">\(F^{\prime}\)</span> the derivative of <span class="math notranslate nohighlight">\(F\)</span></a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="nonlinear-poisson-in-2d">
+<h1>Nonlinear Poisson in 2D<a class="headerlink" href="#nonlinear-poisson-in-2d" title="Link to this heading">#</a></h1>
+<p>In this section, we consider the non-linear Poisson problem:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+-\nabla \cdot \left( (1+u^2) \nabla u \right) &amp;= f, \Omega
+\\
+u &amp;= 0, \partial \Omega
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\Omega\)</span> denotes the unit square.</p>
+<p>For testing, we shall take a function <span class="math notranslate nohighlight">\(u\)</span> that fulfills the boundary condition, the compute <span class="math notranslate nohighlight">\(f\)</span> as</p>
+<div class="math notranslate nohighlight">
+\[
+f(x,y) = -\nabla^2 u - F(u)
+\]</div>
+<p>The weak formulation is</p>
+<div class="math notranslate nohighlight">
+\[
+\int_{\Omega} (1+u^2) \nabla u \cdot \nabla v ~ d\Omega = \int_{\Omega} f v ~d\Omega, \quad \forall v \in \mathcal{V}
+\]</div>
+<p>For the sack of generality, we shall consider the linear form</p>
+<div class="math notranslate nohighlight">
+\[
+G(v;u,w) := \int_{\Omega} (1+w^2) \nabla u \cdot \nabla v ~ d\Omega, \quad \forall u,v,w \in \mathcal{V}
+\]</div>
+<p>Our problem is then</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\mbox{Find } u \in \mathcal{V}, \mbox{such that}\\
+G(v;u,u) = l(v), \quad \forall v \in \mathcal{V}
+\end{align}
+\end{split}\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[
+l(v) := \int_{\Omega} f v ~d\Omega, \quad \forall v \in \mathcal{V}
+\]</div>
+<p>As usual, we’ll follow the classical steps,</p>
+<ol class="arabic simple">
+<li><p>Create a domain</p></li>
+<li><p>Create the Function space</p></li>
+<li><p>Define the different linear/bilinear forms</p></li>
+<li><p>Define the Picard/Newton iteration</p></li>
+</ol>
+<section id="the-topological-domain">
+<h2>1. The topological domain<a class="headerlink" href="#the-topological-domain" title="Link to this heading">#</a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">domain</span> <span class="o">=</span> <span class="n">Square</span><span class="p">()</span>
+<span class="n">B_dirichlet_0</span> <span class="o">=</span> <span class="n">domain</span><span class="o">.</span><span class="n">boundary</span>
+</pre></div>
+</div>
+</section>
+<section id="function-space">
+<h2>2. Function space<a class="headerlink" href="#function-space" title="Link to this heading">#</a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">V</span>  <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">domain</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="define-the-linear-form-g">
+<h2>3.1. Define the Linear form <span class="math notranslate nohighlight">\(G\)</span><a class="headerlink" href="#define-the-linear-form-g" title="Link to this heading">#</a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">u</span>  <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">)</span>
+<span class="n">v</span>  <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;v&#39;</span><span class="p">)</span>
+<span class="n">w</span>  <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;w&#39;</span><span class="p">)</span>
+
+<span class="c1"># Linear form g: V --&gt; R</span>
+<span class="n">g</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">w</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">),</span> <span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">))))</span>
+</pre></div>
+</div>
+</section>
+<section id="define-the-linear-form-l">
+<h2>3.2. Define the linear form <span class="math notranslate nohighlight">\(L\)</span><a class="headerlink" href="#define-the-linear-form-l" title="Link to this heading">#</a></h2>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">solution</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span>
+<span class="n">f</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">y</span><span class="p">)</span><span class="o">**</span><span class="mi">3</span><span class="o">*</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
+
+<span class="c1"># Linear form l: V --&gt; R</span>
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">domain</span><span class="p">,</span> <span class="n">f</span> <span class="o">*</span> <span class="n">v</span><span class="p">))</span>
+</pre></div>
+</div>
+</section>
+<section id="picard-iteration">
+<h2>4.1. Picard iteration<a class="headerlink" href="#picard-iteration" title="Link to this heading">#</a></h2>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\mbox{Find } u_{n+1} \in \mathcal{V}\_h, \mbox{such that}\\ 
+G(v;u_{n+1},u_n) = l(v), \quad \forall v \in \mathcal{V}\_h
+\end{align}
+\end{split}\]</div>
+<p>The Picard iteration for our problem can be created this way</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">un</span>  <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;un&#39;</span><span class="p">)</span>
+
+<span class="c1"># Bilinear form a: V x V --&gt; R</span>
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">g</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">u</span><span class="o">=</span><span class="n">u</span><span class="p">,</span><span class="n">w</span><span class="o">=</span><span class="n">un</span><span class="p">))</span>
+
+<span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">B_dirichlet_0</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+
+<span class="c1"># Error norms</span>
+<span class="n">error</span>  <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="n">solution</span>
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">error</span><span class="p">,</span> <span class="n">domain</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="newton-iteration">
+<h2>4.2. Newton iteration<a class="headerlink" href="#newton-iteration" title="Link to this heading">#</a></h2>
+<p>Let’s define</p>
+<div class="math notranslate nohighlight">
+\[
+F(v;u) := G(v;u,u) -l(v), \quad \forall v \in \mathcal{V}
+\]</div>
+<p>Newton method writes</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\mbox{Find } u_{n+1} \in \mathcal{V}\_h, \mbox{such that} \\
+F^{\prime}(\delta u,v; u_n) = - F(v;u_n), \quad \forall v \in \mathcal{V}, \\ 
+u_{n+1} := u_{n} + \delta u, \quad \delta u \in \mathcal{V}
+\end{align}
+\end{split}\]</div>
+<section id="computing-f-prime-the-derivative-of-f">
+<h3>Computing <span class="math notranslate nohighlight">\(F^{\prime}\)</span> the derivative of <span class="math notranslate nohighlight">\(F\)</span><a class="headerlink" href="#computing-f-prime-the-derivative-of-f" title="Link to this heading">#</a></h3>
+<p><strong>SymPDE</strong> allows you to linearize a linear form and get a bilinear form, using the function <strong>linearize</strong></p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">F</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">g</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">w</span><span class="o">=</span><span class="n">u</span><span class="p">)</span><span class="o">-</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
+<span class="n">du</span>  <span class="o">=</span> <span class="n">element_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">name</span><span class="o">=</span><span class="s1">&#39;du&#39;</span><span class="p">)</span>
+
+<span class="n">Fprime</span> <span class="o">=</span> <span class="n">linearize</span><span class="p">(</span><span class="n">F</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">trials</span><span class="o">=</span><span class="n">du</span><span class="p">)</span>
+</pre></div>
+</div>
+<p>The Newton iteration is then defined as follows,</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Dirichlet boundary conditions</span>
+<span class="n">bc</span> <span class="o">=</span> <span class="p">[</span><span class="n">EssentialBC</span><span class="p">(</span><span class="n">du</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">B_dirichlet_0</span><span class="p">)]</span>
+
+<span class="c1"># Variational problem</span>
+<span class="n">equation</span>   <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">du</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">Fprime</span><span class="p">(</span><span class="n">du</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span><span class="n">u</span><span class="o">=</span><span class="n">un</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=-</span><span class="n">F</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="n">u</span><span class="o">=</span><span class="n">un</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+</section>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#function-space">2. Function space</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-the-linear-form-g">3.1. Define the Linear form <span class="math notranslate nohighlight">\(G\)</span></a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#define-the-linear-form-l">3.2. Define the linear form <span class="math notranslate nohighlight">\(L\)</span></a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#computing-f-prime-the-derivative-of-f">Computing <span class="math notranslate nohighlight">\(F^{\prime}\)</span> the derivative of <span class="math notranslate nohighlight">\(F\)</span></a></li>
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+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1 current active has-children"><a class="reference internal" href="subdomains.html">Subdomains</a><input checked="" class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul class="current">
+<li class="toctree-l2 current active"><a class="current reference internal" href="#">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_compiler.html">sympde.exterior.tests.test_compiler</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
+</ul>
+</li>
+</ul>
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+</ul>
+</li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.expr.html">psydac.api.ast.expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.glt.html">psydac.api.ast.glt</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
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+  <section class="tex2jax_ignore mathjax_ignore" id="the-poisson-problem-using-nitsche-method">
+<h1>The Poisson problem using Nitsche method<a class="headerlink" href="#the-poisson-problem-using-nitsche-method" title="Link to this heading">#</a></h1>
+<p>In this section, we consider a domain <span class="math notranslate nohighlight">\(\Omega\)</span> which a union of multiple subdomain, <em>i.e.</em></p>
+<div class="math notranslate nohighlight">
+\[
+\Omega := \bigcup_{i=1}^n \Omega_i, \quad \Omega_i \bigcap \Omega_j = \emptyset, ~ i \neq j  
+\]</div>
+<p>We shall denote by <span class="math notranslate nohighlight">\(\mathcal{I}\)</span> the set of all internal interfaces of <span class="math notranslate nohighlight">\(\Omega\)</span>.</p>
+<p>We shall also need the following operators</p>
+<ul class="simple">
+<li><p>The jump of the function <span class="math notranslate nohighlight">\(u\)</span>, defined as <span class="math notranslate nohighlight">\([\![ u ]\!] := u|\_{\Omega_{i_1}} -  u|\_{\Omega_{i_2}}\)</span> for two adjacent subdomains <span class="math notranslate nohighlight">\(\Omega_{i_1}\)</span> and <span class="math notranslate nohighlight">\(\Omega_{i_2}\)</span></p></li>
+<li><p>The average of the function <span class="math notranslate nohighlight">\(u\)</span>, defined as <span class="math notranslate nohighlight">\(\{u\} := \frac{1}{2} \left( u|\_{\Omega_{i_1}} +  u|\_{\Omega_{i_2}} \right)\)</span> for two adjacent subdomains <span class="math notranslate nohighlight">\(\Omega_{i_1}\)</span> and <span class="math notranslate nohighlight">\(\Omega_{i_2}\)</span></p></li>
+</ul>
+<p>We consider the Poisson equation</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  - \nabla^2 u = f \quad \text{in~$\Omega$}, \quad \quad 
+  u = 0            \quad \text{on~$\partial \Omega$}, \quad \quad 
+   [\![ u ]\!] = 0 \quad \text{on~$\mathcal{I}$}, \quad \quad
+   [\![\partial_n u ]\!] = 0 \quad \text{on~$\mathcal{I}$}.
+\end{align}
+\]</div>
+<p>The variational formulation reads</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+  \text{find $u \in V$ such that} \quad a(u,v) = l(v) \quad \forall v \in V,
+\end{align}
+\]</div>
+<p>where</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(V \subset H^1(\Omega)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(a(u,v) := \int_{\Omega} \nabla u \cdot \nabla v ~ d\Omega + \int_{\mathcal{I}} \left( \kappa [\![ u ]\!] ~ [\![ v ]\!] - [\![ u ]\!] ~ \partial_n v - \partial_n u ~ [\![ v ]\!] \right) ~ d\mathcal{I} \)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(l(v) := \int_{\Omega} f v ~ d\Omega ~ d\Gamma\)</span>.</p></li>
+</ul>
+<section id="examples">
+<h2>Examples<a class="headerlink" href="#examples" title="Link to this heading">#</a></h2>
+<ul class="simple">
+<li><p><a class="reference external" href="https://github.com/pyccel/sympde/blob/devel-documentation/docs/examples/2d_poisson_subdomains_nitsche_dir0.py">Poisson on a square with two subdomains and Homogeneous Boundary Conditions</a></p></li>
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+    <h1>Bingham Plastic Flow in a Pipe</h1>
+    <!-- Table of contents -->
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+                <h2> Contents </h2>
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+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="bingham-plastic-flow-in-a-pipe">
+<h1>Bingham Plastic Flow in a Pipe<a class="headerlink" href="#bingham-plastic-flow-in-a-pipe" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the Bingham plastic model for non-Newtonian fluids with yield stress:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} &amp;= -\frac{1}{\rho}\frac{\partial p}{\partial x} + \frac{\tau_0}{\rho}\frac{\partial^2 u}{\partial y^2} \\
+\frac{\partial^2 u}{\partial y^2} &amp;\geq 0 \quad \text{(Yield stress condition)}
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(u\)</span> is the velocity, <span class="math notranslate nohighlight">\(p\)</span> is the pressure, <span class="math notranslate nohighlight">\(\rho\)</span> is the density, <span class="math notranslate nohighlight">\(\tau_0\)</span> is the yield stress, and <span class="math notranslate nohighlight">\(y\)</span> is the radial coordinate.</p>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet: Specify velocity or pressure conditions.</p></li>
+<li><p>Outlet: Specify outflow conditions.</p></li>
+<li><p>Pipe walls: No-slip conditions.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Specify initial conditions for velocity and pressure.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(u \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \frac{\partial u}{\partial t}v \, d\Omega + \int_{\Omega} u\frac{\partial u}{\partial x}v \, d\Omega + \int_{\Omega} \frac{1}{\rho}\frac{\partial p}{\partial x}v \, d\Omega \\
+&amp;+ \int_{\Omega} \frac{\tau_0}{\rho}\frac{\partial^2 u}{\partial y^2}\frac{\partial v}{\partial y} \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(v \in H_0^1(\Omega)\)</span>.</p>
+</section>
+</section>
+
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+    <h1>Buoyancy-Driven Natural Convection</h1>
+    <!-- Table of contents -->
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+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mathematical-model">Mathematical Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#boundary-conditions">Boundary Conditions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initial-conditions">Initial Conditions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weak-formulation">Weak Formulation</a></li>
+</ul>
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+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="buoyancy-driven-natural-convection">
+<h1>Buoyancy-Driven Natural Convection<a class="headerlink" href="#buoyancy-driven-natural-convection" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the Boussinesq approximation for buoyancy-driven natural convection:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v} &amp;= -\frac{1}{\rho_0}\nabla p + \nu \nabla^2 \mathbf{v} + \beta g (T - T_0)\mathbf{k} \\
+\frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T &amp;= \alpha \nabla^2 T
+\end{align}
+\end{split}\]</div>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Vertical walls: <span class="math notranslate nohighlight">\(\mathbf{v} \cdot \mathbf{t} = 0\)</span> and <span class="math notranslate nohighlight">\(\frac{\partial T}{\partial n} = 0\)</span>.</p></li>
+<li><p>Inlet and outlet: Specify velocity or temperature conditions.</p></li>
+<li><p>Top and bottom: No-slip and adiabatic conditions.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Start with an initial velocity and temperature distribution.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(\mathbf{v} \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(T \in H^1(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \left(\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v}\right) \cdot \mathbf{w} \, d\Omega + \int_{\Omega} \frac{1}{\rho_0}\nabla p \cdot \mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \nu \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega - \beta g (T - T_0)\mathbf{k} \cdot \mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \left(\frac{\partial T}{\partial t} + \mathbf{v} \cdot \nabla T - \alpha \nabla^2 T\right) \psi \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(\mathbf{w}, \psi \in H_0^1(\Omega)\)</span>.</p>
+</section>
+</section>
+
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+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+    <h1>Casson Fluid Flow in a Channel</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mathematical-model">Mathematical Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#boundary-conditions">Boundary Conditions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initial-conditions">Initial Conditions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weak-formulation">Weak Formulation</a></li>
+</ul>
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+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="casson-fluid-flow-in-a-channel">
+<h1>Casson Fluid Flow in a Channel<a class="headerlink" href="#casson-fluid-flow-in-a-channel" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the Casson model for non-Newtonian fluids:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\sigma_{ij} &amp;= -p\delta_{ij} + \sqrt{\frac{\tau_0}{\rho}}\sqrt{\frac{\tau_0}{\rho} + \frac{1}{2}\rho\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)^2} \\
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\sigma_{ij}\)</span> is the stress tensor, <span class="math notranslate nohighlight">\(p\)</span> is the pressure, <span class="math notranslate nohighlight">\(\rho\)</span> is the density, <span class="math notranslate nohighlight">\(u_i\)</span> is the velocity, and <span class="math notranslate nohighlight">\(\tau_0\)</span> is the yield stress.</p>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet and outlet: Specify velocity or pressure conditions.</p></li>
+<li><p>Channel walls: No-slip conditions.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Start with an initial guess for velocity and pressure.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(u_i \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \sigma_{ij} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) \, d\Omega + \int_{\Omega} \frac{\partial p}{\partial x_i}u_i \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(u_i \in H_0^1(\Omega)\)</span>.</p>
+</section>
+</section>
+
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+    <h1>Compressible Flow in a Nozzle</h1>
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+                <h2> Contents </h2>
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+  <section class="tex2jax_ignore mathjax_ignore" id="compressible-flow-in-a-nozzle">
+<h1>Compressible Flow in a Nozzle<a class="headerlink" href="#compressible-flow-in-a-nozzle" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the compressible Euler equations for flow in a converging-diverging nozzle:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) &amp;= 0 \\
+\frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot (\rho \mathbf{v} \otimes \mathbf{v} + p\mathbf{I}) &amp;= 0 \\
+\frac{\partial E}{\partial t} + \nabla \cdot \left[(E + p) \mathbf{v}\right] &amp;= 0
+\end{align}
+\end{split}\]</div>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet: Specify inflow conditions (<span class="math notranslate nohighlight">\(\rho\)</span>, <span class="math notranslate nohighlight">\(\mathbf{v}\)</span>, <span class="math notranslate nohighlight">\(p\)</span>).</p></li>
+<li><p>Outlet: Specify outflow conditions or use characteristic-based boundary conditions.</p></li>
+<li><p>Nozzle walls: <span class="math notranslate nohighlight">\(\mathbf{v} \cdot \mathbf{n} = 0\)</span> (no penetration).</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Specify initial conditions for density, velocity, and pressure.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(\rho, \mathbf{v}, p \in H^1(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \left(\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v})\right) \phi \, d\Omega \\
+&amp;+ \int_{\Omega} \left(\frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot (\rho \mathbf{v} \otimes \mathbf{v} + p\mathbf{I})\right) \cdot \mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \left(\frac{\partial E}{\partial t} + \nabla \cdot \left[(E + p) \mathbf{v}\right]\right) \psi \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(\phi, \mathbf{w}, \psi \in H^1(\Omega)\)</span>.</p>
+</section>
+</section>
+
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+    <h1>Convection-Diffusion Equation in a Channel</h1>
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+  <section class="tex2jax_ignore mathjax_ignore" id="convection-diffusion-equation-in-a-channel">
+<h1>Convection-Diffusion Equation in a Channel<a class="headerlink" href="#convection-diffusion-equation-in-a-channel" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the 2D convection-diffusion equation in a channel:</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\rho u \frac{\partial c}{\partial x} + \rho v \frac{\partial c}{\partial y} - \nabla \cdot (D \nabla c) = f
+\end{align}
+\]</div>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet: <span class="math notranslate nohighlight">\(c = c_{\text{inlet}}\)</span> with $\mathbf{v} \cdot \mathbf{n} &lt; 0$ (inward flow).</p></li>
+<li><p>Outlet: <span class="math notranslate nohighlight">\(\frac{\partial c}{\partial x} = \frac{\partial c}{\partial y} = 0\)</span> (zero gradient).</p></li>
+<li><p>Lateral faces: <span class="math notranslate nohighlight">\(-D \nabla c \cdot \mathbf{n} = 0\)</span> (insulated).</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Assume an initial concentration distribution <span class="math notranslate nohighlight">\(c_0(x, y)\)</span> at <span class="math notranslate nohighlight">\(t = 0\)</span>.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(c \in H_0^1(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \left(\rho u \frac{\partial c}{\partial x} + \rho v \frac{\partial c}{\partial y}\right) \phi \, d\Omega - \int_{\Omega} D \nabla c \cdot \nabla \phi \, d\Omega \\
+&amp;= \int_{\Omega} f \phi \, d\Omega
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(\phi \in H_0^1(\Omega)\)</span>.</p>
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diff --git a/chapter5/cross_power_law_fluid_flow_in_a_channel.html b/chapter5/cross_power_law_fluid_flow_in_a_channel.html
new file mode 100644
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+    <h1>Cross Power Law Fluid Flow in a Channel</h1>
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+  <section class="tex2jax_ignore mathjax_ignore" id="cross-power-law-fluid-flow-in-a-channel">
+<h1>Cross Power Law Fluid Flow in a Channel<a class="headerlink" href="#cross-power-law-fluid-flow-in-a-channel" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the cross power-law model for non-Newtonian fluids:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\sigma_{ij} &amp;= -p\delta_{ij} + \lambda \varepsilon_{kk}\delta_{ij} + 2\mu\left(\varepsilon_{ij} + \alpha\varepsilon_{ij}^2\right) \\
+\varepsilon_{ij} &amp;= \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right) \\
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\sigma_{ij}\)</span> is the stress tensor, <span class="math notranslate nohighlight">\(p\)</span> is the pressure, <span class="math notranslate nohighlight">\(\lambda\)</span> and <span class="math notranslate nohighlight">\(\mu\)</span> are material constants, <span class="math notranslate nohighlight">\(v_i\)</span> is the velocity, <span class="math notranslate nohighlight">\(\varepsilon_{ij}\)</span> is the strain rate, and <span class="math notranslate nohighlight">\(\alpha\)</span> is a power-law index.</p>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet and outlet: Specify velocity or pressure conditions.</p></li>
+<li><p>Channel walls: No-slip conditions.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Start with an initial guess for velocity and pressure.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(v_i \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \sigma_{ij} \varepsilon_{ij} \, d\Omega + \int_{\Omega} \frac{\partial p}{\partial x_i}v_i \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(v_i \in H_0^1(\Omega)\)</span>.</p>
+</section>
+</section>
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+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+    <h1>Electromagnetics</h1>
+    <!-- Table of contents -->
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+                <h2> Contents </h2>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#maxwell-s-equations">Maxwell’s Equations</a></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#finite-element-discretization">Finite Element Discretization</a></li>
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+  <section class="tex2jax_ignore mathjax_ignore" id="electromagnetics">
+<h1>Electromagnetics<a class="headerlink" href="#electromagnetics" title="Link to this heading">#</a></h1>
+<p>Electromagnetic problems are commonly described by Maxwell’s equations, which govern the behavior of electric and magnetic fields. The Finite Element Method (FEM) provides a powerful numerical approach for solving these equations in complex geometries. This section provides a concise overview of the mathematical formulation for electromagnetic problems using finite elements.</p>
+<section id="maxwell-s-equations">
+<h2>Maxwell’s Equations<a class="headerlink" href="#maxwell-s-equations" title="Link to this heading">#</a></h2>
+<p>The time-harmonic Maxwell’s equations in free space are given by:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+    \nabla \times \mathbf{E} &amp;= -j\omega \mu \mathbf{H}, \label{eq:maxwell1} \\
+    \nabla \times \mathbf{H} &amp;= j\omega \varepsilon \mathbf{E}, \label{eq:maxwell2} \\
+    \nabla \cdot \mathbf{B} &amp;= 0, \label{eq:maxwell3} \\
+    \nabla \cdot \mathbf{D} &amp;= 0, \label{eq:maxwell4}
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{E}\)</span> is the electric field, <span class="math notranslate nohighlight">\(\mathbf{H}\)</span> is the magnetic field, <span class="math notranslate nohighlight">\(\mathbf{B}\)</span> is the magnetic flux density, <span class="math notranslate nohighlight">\(\mathbf{D}\)</span> is the electric displacement field, <span class="math notranslate nohighlight">\(\omega\)</span> is the angular frequency, <span class="math notranslate nohighlight">\(\mu\)</span> is the permeability, and <span class="math notranslate nohighlight">\(\varepsilon\)</span> is the permittivity.</p>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>The weak form of Maxwell’s equations is obtained by multiplying each equation with suitable test functions and integrating over the domain <span class="math notranslate nohighlight">\(\Omega\)</span>. For the electric field <span class="math notranslate nohighlight">\(\mathbf{E}\)</span>, the weak form is given by:</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{equation}
+    \int_{\Omega} \nabla \times \mathbf{E} \cdot \nabla \times \mathbf{v} \, d\Omega + \omega^2 \mu \varepsilon \mathbf{E} \cdot \mathbf{v} \, d\Omega = 0,
+\end{equation}
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> is a test function belonging to the function space <span class="math notranslate nohighlight">\(H(\text{curl}; \Omega)\)</span>.</p>
+<p>Similarly, for the magnetic field <span class="math notranslate nohighlight">\(\mathbf{H}\)</span>, the weak form is given by:</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{equation}
+    \int_{\Omega} \nabla \times \mathbf{H} \cdot \nabla \times \mathbf{u} \, d\Omega - \omega^2 \mu \varepsilon \mathbf{H} \cdot \mathbf{u} \, d\Omega = 0,
+\end{equation}
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{u}\)</span> is a test function belonging to the function space <span class="math notranslate nohighlight">\(H(\text{curl}; \Omega)\)</span>.</p>
+</section>
+<section id="finite-element-discretization">
+<h2>Finite Element Discretization<a class="headerlink" href="#finite-element-discretization" title="Link to this heading">#</a></h2>
+<p>To apply the finite element method, the domain <span class="math notranslate nohighlight">\(\Omega\)</span> is discretized into elements, and the electromagnetic fields are approximated using piecewise basis functions. The discretized weak form leads to a system of linear equations, which can be solved numerically to obtain the finite element solution.</p>
+</section>
+<section id="references">
+<h2>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h2>
+<p>The following references provide in-depth coverage of the mathematical background for electromagnetism using finite elements:</p>
+<p>\begin{enumerate}
+\item \cite{jin2015finite}
+\item \cite{bossavit1998whitney}
+\item \cite{monk2003finite}
+\end{enumerate}</p>
+<p>This mathematical background serves as the foundation for implementing finite element simulations of electromagnetic problems. Researchers interested in detailed mathematical derivations and computational techniques are encouraged to explore the referenced works.</p>
+</section>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="current nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+</li>
+</ul>
+</li>
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+    <h1>Free Surface Flow (Navier-Stokes with Free Surface)</h1>
+    <!-- Table of contents -->
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initial-conditions">Initial Conditions</a></li>
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+  <section class="tex2jax_ignore mathjax_ignore" id="free-surface-flow-navier-stokes-with-free-surface">
+<h1>Free Surface Flow (Navier-Stokes with Free Surface)<a class="headerlink" href="#free-surface-flow-navier-stokes-with-free-surface" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the Navier-Stokes equations coupled with a level set or volume-of-fluid method for tracking a free surface:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\rho \frac{\partial \mathbf{v}}{\partial t} + \rho (\mathbf{v} \cdot \nabla) \mathbf{v} &amp;= -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g} \\
+\nabla \cdot \mathbf{v} &amp;= 0 \\
+\frac{\partial \phi}{\partial t} + \mathbf{v} \cdot \nabla \phi &amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\phi\)</span> is the level set or volume fraction representing the free surface.</p>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet and outlet: Specify velocity or pressure conditions.</p></li>
+<li><p>No-slip conditions on solid boundaries.</p></li>
+<li><p>Implement the level set or volume fraction boundary conditions.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Specify initial conditions for velocity, pressure, and the level set or volume fraction.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(\mathbf{v} \in H_0^1(\Omega)\)</span>, <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span>, and <span class="math notranslate nohighlight">\(\phi \in H^1(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \rho \frac{\partial \mathbf{v}}{\partial t} \cdot \mathbf{w} \, d\Omega + \int_{\Omega} \rho (\mathbf{v} \cdot \nabla) \mathbf{v} \cdot \mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \mu \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega - \int_{\Omega} p \nabla \cdot \mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \nabla \cdot \mathbf{v} q \, d\Omega + \int_{\Omega} \frac{\partial \phi}{\partial t} q \, d\Omega + \int_{\Omega} \mathbf{v} \cdot \nabla \phi q \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(\mathbf{w} \in H_0^1(\Omega)\)</span>, <span class="math notranslate nohighlight">\(q \in L^2(\Omega)\)</span>, and <span class="math notranslate nohighlight">\(r \in H^1(\Omega)\)</span>.</p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="current nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
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+</li>
+<li class="toctree-l1 current active"><a class="current reference internal" href="#">Fluid-Structure Interaction</a></li>
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+    <h1>Fluid-Structure Interaction</h1>
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+                <h2> Contents </h2>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#coupled-fluid-structure-equations">Coupled Fluid-Structure Equations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#coupling-conditions">Coupling Conditions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weak-formulation">Weak Formulation</a></li>
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+  <section class="tex2jax_ignore mathjax_ignore" id="fluid-structure-interaction">
+<h1>Fluid-Structure Interaction<a class="headerlink" href="#fluid-structure-interaction" title="Link to this heading">#</a></h1>
+<p>Fluid-Structure Interaction (FSI) involves the coupled interaction between a fluid and a structure, where the motion of one influences the behavior of the other. The Finite Element Method (FEM) is a powerful tool for simulating FSI problems. This section provides an overview of the mathematical formulation for fluid-structure interaction using finite elements.</p>
+<section id="coupled-fluid-structure-equations">
+<h2>Coupled Fluid-Structure Equations<a class="headerlink" href="#coupled-fluid-structure-equations" title="Link to this heading">#</a></h2>
+<p>The governing equations for fluid-structure interaction involve the Navier-Stokes equations for the fluid and the equations of motion for the structure. In a partitioned approach, the coupled system is given by:</p>
+<p>\begin{align}
+\text{Fluid Domain:} \quad \rho_f \left(\frac{\partial \mathbf{u}_f}{\partial t} + (\mathbf{u}_f \cdot \nabla)\mathbf{u}_f\right) &amp;= -\nabla p_f + \mu_f \nabla^2 \mathbf{u}_f + \mathbf{f}_f + \mathbf{F}_s, \label{eq:fsi_fluid_momentum} \
+\nabla \cdot \mathbf{u}_f &amp;= 0, \label{eq:fsi_fluid_continuity} \
+\text{Structure Domain:} \quad \rho_s \frac{\partial^2 \mathbf{u}_s}{\partial t^2} &amp;= \nabla \cdot \mathbf{P}_s + \mathbf{f}_s, \label{eq:fsi_structure_motion}
+\end{align}</p>
+<p>where:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\mathbf{u}_f\)</span> is the fluid velocity,</p></li>
+<li><p><span class="math notranslate nohighlight">\(p_f\)</span> is the fluid pressure,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\rho_f\)</span> is the fluid density,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mu_f\)</span> is the fluid dynamic viscosity,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{f}_f\)</span> is the fluid body force,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{F}_s\)</span> is the fluid-structure interaction force on the fluid by the structure,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{u}_s\)</span> is the structure displacement,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\rho_s\)</span> is the structure density,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{P}_s\)</span> is the structure internal force,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{f}_s\)</span> is the structure external force.</p></li>
+</ul>
+</section>
+<section id="coupling-conditions">
+<h2>Coupling Conditions<a class="headerlink" href="#coupling-conditions" title="Link to this heading">#</a></h2>
+<p>For a fluid-structure interaction problem, coupling conditions at the fluid-structure interface are essential. These conditions ensure continuity of velocities, pressures, and forces between the fluid and structure. A common approach is to enforce kinematic and dynamic conditions:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+    \text{Kinematic Condition:} \quad \mathbf{u}_f &amp;= \mathbf{u}_s, \label{eq:fsi_kinematic_condition} \\
+    \text{Dynamic Condition:} \quad \mathbf{P}_f \cdot \mathbf{n} &amp;= \mathbf{P}_s \cdot \mathbf{n}, \label{eq:fsi_dynamic_condition}
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{n}\)</span> is the unit outward normal vector at the fluid-structure interface, and <span class="math notranslate nohighlight">\(\mathbf{P}_f\)</span> and <span class="math notranslate nohighlight">\(\mathbf{P}_s\)</span> are the fluid and structure stresses, respectively.</p>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>The weak form of the coupled fluid-structure problem involves the variational formulation of the fluid and structure equations, incorporating the coupling conditions. For the fluid domain, the weak form is given by:</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{equation}
+    \int_{\Omega_f} \rho_f \left(\frac{\partial \mathbf{u}_f}{\partial t} + (\mathbf{u}_f \cdot \nabla)\mathbf{u}_f\right) \cdot \mathbf{v}_f \, d\Omega_f + \int_{\Omega_f} \mu_f \nabla \mathbf{u}_f : \nabla \mathbf{v}_f \, d\Omega_f = \int_{\Omega_f} (\mathbf{f}_f + \mathbf{F}_s) \cdot \mathbf{v}_f \, d\Omega_f,
+\end{equation}
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{v}_f\)</span> is a test function belonging to the function space <span class="math notranslate nohighlight">\(H^1(\Omega_f)\)</span>, and <span class="math notranslate nohighlight">\(\Omega_f\)</span> is the fluid domain.</p>
+<p>For the structure domain, the weak form is given by:</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{equation}
+    \int_{\Omega_s} \rho_s \frac{\partial^2 \mathbf{u}_s}{\partial t^2} \cdot \mathbf{v}_s \, d\Omega_s + \int_{\Omega_s} \nabla \cdot \mathbf{P}_s \cdot \mathbf{v}_s \, d\Omega_s = \int_{\Omega_s} \mathbf{f}_s \cdot \mathbf{v}_s \, d\Omega_s,
+\end{equation}
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{v}_s\)</span> is a test function belonging to the function space <span class="math notranslate nohighlight">\(H^1(\Omega_s)\)</span>, and <span class="math notranslate nohighlight">\(\Omega_s\)</span> is the structure domain.</p>
+<p>The coupling conditions, such as \eqref{eq:fsi_kinematic_condition} and \eqref{eq:fsi_dynamic_condition}, should be enforced during the assembly of the global matrices.</p>
+</section>
+<section id="references">
+<h2>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h2>
+<p>The following references provide comprehensive coverage of the mathematical background for fluid-structure interaction using finite elements:</p>
+<p>\begin{enumerate}
+\item \cite{hron2006fluid}<br />
+\item \cite{bathe2014finite}
+\item \cite{quarteroni2017fluid}
+\end{enumerate}</p>
+<p>Researchers interested in detailed mathematical derivations and computational techniques are encouraged to explore the referenced works.</p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="current nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2 current active"><a class="current reference internal" href="#">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+    <h1>Heat Conduction in a Solid</h1>
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+                <h2> Contents </h2>
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+<h1>Heat Conduction in a Solid<a class="headerlink" href="#heat-conduction-in-a-solid" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the heat conduction equation in a solid domain (\Omega):</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\rho C_p \frac{\partial T}{\partial t} - \nabla \cdot (k \nabla T) = Q
+\end{align}
+\]</div>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Dirichlet: <span class="math notranslate nohighlight">\(T = T_{\text{inlet}}\)</span> on the inlet face.</p></li>
+<li><p>Neumann: <span class="math notranslate nohighlight">\(-k \frac{\partial T}{\partial n} = 0\)</span> on the outlet face.</p></li>
+<li><p>Insulated: <span class="math notranslate nohighlight">\(-k \frac{\partial T}{\partial n} = 0\)</span> on the lateral faces.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Assume an initial temperature distribution <span class="math notranslate nohighlight">\(T_0(x, y, z)\)</span> at <span class="math notranslate nohighlight">\(t = 0\)</span>.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(T \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[
+\begin{align}
+\int_{\Omega} \rho C_p \frac{\partial T}{\partial t} \phi \, d\Omega + \int_{\Omega} k \nabla T \cdot \nabla \phi \, d\Omega = \int_{\Omega} Q \phi \, d\Omega
+\end{align}
+\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(\phi \in H_0^1(\Omega)\)</span>.</p>
+</section>
+</section>
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diff --git a/chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html b/chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson.html">The Poisson equation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="current nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
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+<h1>Herschel-Bulkley Fluid Flow in a Pipe<a class="headerlink" href="#herschel-bulkley-fluid-flow-in-a-pipe" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the Herschel-Bulkley model for non-Newtonian fluids with yield stress:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} &amp;= -\frac{1}{\rho}\frac{\partial p}{\partial x} + \frac{\tau_0}{\rho}\left(\frac{\partial u}{\partial x}\right)^{n-1} \\
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(u\)</span> is the velocity, <span class="math notranslate nohighlight">\(p\)</span> is the pressure, <span class="math notranslate nohighlight">\(\rho\)</span> is the density, <span class="math notranslate nohighlight">\(\tau_0\)</span> is the yield stress, and <span class="math notranslate nohighlight">\(n\)</span> is the flow behavior index.</p>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet: Specify velocity or pressure conditions.</p></li>
+<li><p>Outlet: Specify outflow conditions.</p></li>
+<li><p>Pipe walls: No-slip conditions.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Specify initial conditions for velocity and pressure.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(u \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \frac{\partial u}{\partial t}v \, d\Omega + \int_{\Omega} u\frac{\partial u}{\partial x}v \, d\Omega + \int_{\Omega} \frac{1}{\rho}\frac{\partial p}{\partial x}v \, d\Omega \\
+&amp;+ \int_{\Omega} \frac{\tau_0}{\rho}\left(\frac{\partial u}{\partial x}\right)^{n-1}v \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(v \in H_0^1(\Omega)\)</span>.</p>
+</section>
+</section>
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+<ul class="nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="current nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#boundary-conditions">Boundary Conditions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initial-conditions">Initial Conditions</a></li>
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+<h1>Incompressible Flow Past a Cylinder<a class="headerlink" href="#incompressible-flow-past-a-cylinder" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model:<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the steady, incompressible Navier-Stokes equations for fluid flow around a cylinder:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\rho (\mathbf{v} \cdot \nabla) \mathbf{v} &amp;= -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g} \\
+\nabla \cdot \mathbf{v} &amp;= 0
+\end{align}
+\end{split}\]</div>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet: <span class="math notranslate nohighlight">\(\mathbf{v} = (U, 0)\)</span> (uniform velocity).</p></li>
+<li><p>Outlet: <span class="math notranslate nohighlight">\(\frac{\partial \mathbf{v}}{\partial x} = 0\)</span> (zero gradient).</p></li>
+<li><p>Cylinder surface: <span class="math notranslate nohighlight">\(\mathbf{v} \cdot \mathbf{n} = 0\)</span> (no-slip condition).</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Assume a quiescent fluid, i.e., <span class="math notranslate nohighlight">\(\mathbf{v} = (0, 0)\)</span> at <span class="math notranslate nohighlight">\(t = 0\)</span>.</p>
+</section>
+<section id="weak-formulation">
+<h3>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h3>
+<p>Find <span class="math notranslate nohighlight">\(\mathbf{v} \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\int_{\Omega} \rho (\mathbf{v} \cdot \nabla) \mathbf{v} \cdot \mathbf{w} \, d\Omega &amp; = -\int_{\Omega} p \nabla \cdot \mathbf{w} \, d\Omega + \mu \int_{\Omega} \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega + \int_{\Omega} \rho \mathbf{g} \cdot \mathbf{w} \, d\Omega \\
+\int_{\Omega} \nabla \cdot \mathbf{v} q \, d\Omega &amp; = 0
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{w} \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(q \in L^2(\Omega)\)</span>.</p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="current nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
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+    <h1>Magnetohydrodynamics (MHD) Flow</h1>
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+  <section class="tex2jax_ignore mathjax_ignore" id="magnetohydrodynamics-mhd-flow">
+<h1>Magnetohydrodynamics (MHD) Flow<a class="headerlink" href="#magnetohydrodynamics-mhd-flow" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the MHD equations coupling the Navier-Stokes equations with Maxwell’s equations:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\rho \frac{\partial \mathbf{v}}{\partial t} + \rho (\mathbf{v} \cdot \nabla) \mathbf{v} &amp;= -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{J} \times \mathbf{B} \\
+\frac{\partial \mathbf{B}}{\partial t} &amp;= \nabla \times (\mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B}) \\
+\nabla \cdot \mathbf{v} &amp;= 0 \\
+\nabla \cdot \mathbf{B} &amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{J} = \nabla \times \mathbf{B}\)</span> is the current density.</p>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet and outlet: Specify velocity or pressure conditions.</p></li>
+<li><p>Magnetic field conditions depending on the problem.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Specify initial conditions for velocity, pressure, and the magnetic field.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(\mathbf{v} \in H_0^1(\Omega)\)</span>, <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span>, and <span class="math notranslate nohighlight">\(\mathbf{B} \in H(\text{curl}; \Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \rho \frac{\partial \mathbf{v}}{\partial t} \cdot \mathbf{w} \, d\Omega + \int_{\Omega} \rho (\mathbf{v} \cdot \nabla) \mathbf{v} \cdot \mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \mu \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega - \int_{\Omega} p \nabla \cdot \mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \frac{\partial \mathbf{B}}{\partial t} \cdot \boldsymbol{\psi} \, d\Omega - \int_{\Omega} \nabla \times (\mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B}) \cdot \boldsymbol{\psi} \, d\Omega \\
+&amp;+ \int_{\Omega} \nabla \cdot \mathbf{v} q \, d\Omega + \int_{\Omega} \nabla \cdot \mathbf{B} r \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(\mathbf{w} \in H_0^1(\Omega)\)</span>, <span class="math notranslate nohighlight">\(q \in L^2(\Omega)\)</span>, <span class="math notranslate nohighlight">\(\boldsymbol{\psi} \in H(\text{curl}; \Omega)\)</span>, and <span class="math notranslate nohighlight">\(r \in L^2(\Omega)\)</span>.</p>
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+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+</li>
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+    <h1>MHD</h1>
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+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mhd-equations">MHD Equations</a></li>
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+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="mhd">
+<h1>MHD<a class="headerlink" href="#mhd" title="Link to this heading">#</a></h1>
+<p>In magnetohydrodynamics (MHD), finite element analysis can be applied to solve a variety of problems related to the behavior of electrically conducting fluids (plasmas or liquid metals) in the presence of magnetic fields. Here are some common problems in magnetohydrodynamics that can be addressed using finite element methods.</p>
+<p>Finite element analysis provides a powerful and flexible framework for addressing these complex problems in magnetohydrodynamics by discretizing the governing equations and solving them numerically. The specific application will determine the details of the problem setup and the required numerical techniques.</p>
+<section id="mhd-equations">
+<h2>MHD Equations<a class="headerlink" href="#mhd-equations" title="Link to this heading">#</a></h2>
+<p>Magnetohydrodynamics (MHD) is a branch of physics and fluid dynamics that studies the magnetic properties and behavior of electrically conducting fluids, such as plasmas, liquid metals, and saltwater. The MHD equations describe the coupled interactions between the magnetic field, fluid flow, and electric current. Here are the mathematical models for the MHD equations:</p>
+<ol class="arabic simple">
+<li><p>MHD Continuity Equation:
+$<span class="math notranslate nohighlight">\(
+\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0
+\)</span>$
+where:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\rho\)</span> is the fluid density,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{v}\)</span> is the fluid velocity.</p></li>
+</ul>
+</li>
+<li><p>MHD Momentum Equation:
+$<span class="math notranslate nohighlight">\(
+\rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \rho \mathbf{g} + \nabla \cdot \boldsymbol{\tau} + \mathbf{J} \times \mathbf{B}
+\)</span>$
+where:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\frac{D\mathbf{v}}{Dt}\)</span> is the material derivative of velocity,</p></li>
+<li><p><span class="math notranslate nohighlight">\(p\)</span> is the pressure,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{g}\)</span> is the gravitational acceleration,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\boldsymbol{\tau}\)</span> is the stress tensor,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{J}\)</span> is the current density,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{B}\)</span> is the magnetic field.</p></li>
+</ul>
+</li>
+<li><p>MHD Induction Equation:
+$<span class="math notranslate nohighlight">\(
+\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B})
+\)</span>$
+where:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\eta\)</span> is the magnetic diffusivity.</p></li>
+</ul>
+</li>
+<li><p>MHD Ohm’s Law:
+$<span class="math notranslate nohighlight">\(
+\mathbf{J} = \sigma (\mathbf{E} + \mathbf{v} \times \mathbf{B})
+\)</span>$
+where:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\sigma\)</span> is the electrical conductivity,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{E}\)</span> is the electric field.</p></li>
+</ul>
+</li>
+<li><p>MHD Energy Equation:
+$<span class="math notranslate nohighlight">\(
+\rho C_p \frac{DT}{Dt} = -p \nabla \cdot \mathbf{v} + \nabla \cdot (k \nabla T) + \frac{1}{\sigma}(\mathbf{J} \cdot \mathbf{E})
+\)</span>$
+where:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(C_p\)</span> is the specific heat at constant pressure,</p></li>
+<li><p><span class="math notranslate nohighlight">\(T\)</span> is the temperature,</p></li>
+<li><p><span class="math notranslate nohighlight">\(k\)</span> is the thermal conductivity.</p></li>
+</ul>
+</li>
+</ol>
+<p>These equations, when solved together with appropriate boundary conditions, describe the behavior of magnetized fluids in the presence of electric and magnetic fields. Keep in mind that the specific form of these equations may vary based on the assumptions made and the type of fluid being considered (e.g., ideal MHD vs. resistive MHD).</p>
+</section>
+</section>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="current nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 current active has-children"><a class="current reference internal" href="#">Non-Newtonian Fluids</a><input checked="" class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
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+</ul>
+</li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.expr.html">psydac.api.ast.expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
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+    <h1>Oldroyd-B Fluid Flow in a Channel</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mathematical-model">Mathematical Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#boundary-conditions">Boundary Conditions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initial-conditions">Initial Conditions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weak-formulation">Weak Formulation</a></li>
+</ul>
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+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="oldroyd-b-fluid-flow-in-a-channel">
+<h1>Oldroyd-B Fluid Flow in a Channel<a class="headerlink" href="#oldroyd-b-fluid-flow-in-a-channel" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the Oldroyd-B model for viscoelastic fluids:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u} &amp;= -\frac{1}{\rho}\nabla p + \nabla\cdot\boldsymbol{\tau} \\
+\frac{\partial \boldsymbol{\tau}}{\partial t} + \mathbf{u}\cdot\nabla\boldsymbol{\tau} &amp;= \boldsymbol{\tau}\cdot\nabla\mathbf{u} - \frac{\boldsymbol{\tau}}{\lambda} + 2\mu\left(\nabla\mathbf{u} - \frac{\boldsymbol{\tau}}{\lambda}\right) \\
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{u}\)</span> is the velocity, <span class="math notranslate nohighlight">\(p\)</span> is the pressure, <span class="math notranslate nohighlight">\(\rho\)</span> is the density, <span class="math notranslate nohighlight">\(\boldsymbol{\tau}\)</span> is the extra stress tensor, <span class="math notranslate nohighlight">\(\lambda\)</span> is the relaxation time, and <span class="math notranslate nohighlight">\(\mu\)</span> is the viscosity.</p>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet and outlet: Specify velocity or pressure conditions.</p></li>
+<li><p>Channel walls: No-slip conditions.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Specify initial conditions for velocity and pressure.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(\mathbf{u} \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \frac{\partial \mathbf{u}}{\partial t}\cdot\mathbf{w} \, d\Omega + \int_{\Omega} \mathbf{u}\cdot\nabla\mathbf{u}\cdot\mathbf{w} \, d\Omega + \int_{\Omega} \frac{1}{\rho}\nabla p\cdot\mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \frac{\partial \boldsymbol{\tau}}{\partial t}:\mathbf{v} \, d\Omega + \int_{\Omega} \mathbf{u}\cdot\nabla\boldsymbol{\tau}:\mathbf{v} \, d\Omega \\
+&amp;- \int_{\Omega} \boldsymbol{\tau}\cdot\nabla\mathbf{u}:\mathbf{v} \, d\Omega + \int_{\Omega} \frac{\boldsymbol{\tau}}{\lambda}:\mathbf{v} \, d\Omega \\
+&amp;+ \int_{\Omega} 2\mu\left(\nabla\mathbf{u} - \frac{\boldsymbol{\tau}}{\lambda}\right):\nabla\mathbf{w} \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(\mathbf{w} \in H_0^1(\Omega)\)</span>.</p>
+</section>
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+    <h1>Papanastasiou Fluid Flow in a Channel</h1>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initial-conditions">Initial Conditions</a></li>
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+  <section class="tex2jax_ignore mathjax_ignore" id="papanastasiou-fluid-flow-in-a-channel">
+<h1>Papanastasiou Fluid Flow in a Channel<a class="headerlink" href="#papanastasiou-fluid-flow-in-a-channel" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the Papanastasiou model for non-Newtonian fluids:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\sigma_{ij} &amp;= -p\delta_{ij} + 2\mu\left(\varepsilon_{ij} + \frac{\tau_0}{2\mu}\right)\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) \\
+\varepsilon_{ij} &amp;= \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) \\
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\sigma_{ij}\)</span> is the stress tensor, <span class="math notranslate nohighlight">\(p\)</span> is the pressure, <span class="math notranslate nohighlight">\(\mu\)</span> is the viscosity, <span class="math notranslate nohighlight">\(u_i\)</span> is the velocity, <span class="math notranslate nohighlight">\(\varepsilon_{ij}\)</span> is the strain rate, and <span class="math notranslate nohighlight">\(\tau_0\)</span> is the yield stress.</p>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet and outlet: Specify velocity or pressure conditions.</p></li>
+<li><p>Channel walls: No-slip conditions.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Start with an initial guess for velocity and pressure.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(u_i \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \sigma_{ij} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) \, d\Omega + \int_{\Omega} \frac{\partial p}{\partial x_i}u_i \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(u_i \in H_0^1(\Omega)\)</span>.</p>
+</section>
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+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
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+</ul>
+</li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.expr.html">psydac.api.ast.expr</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
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+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.discretization.html">psydac.api.discretization</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.equation.html">psydac.api.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.essential_bc.html">psydac.api.essential_bc</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.expr.html">psydac.api.expr</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.feec.html">psydac.api.feec</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.fem.html">psydac.api.fem</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.glt.html">psydac.api.glt</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.grid.html">psydac.api.grid</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.postprocessing.html">psydac.api.postprocessing</a></li>
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+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.settings.html">psydac.api.settings</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.tests.test_2d_biharmonic.html">psydac.api.tests.test_2d_biharmonic</a></li>
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+    <h1>Particle-Laden Flow (Lagrangian-Eulerian Approach)</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mathematical-model">Mathematical Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#boundary-conditions">Boundary Conditions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initial-conditions">Initial Conditions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weak-formulation">Weak Formulation</a></li>
+</ul>
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+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="particle-laden-flow-lagrangian-eulerian-approach">
+<h1>Particle-Laden Flow (Lagrangian-Eulerian Approach)<a class="headerlink" href="#particle-laden-flow-lagrangian-eulerian-approach" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the incompressible Navier-Stokes equations coupled with the Lagrangian motion of particles:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\rho \frac{\partial \mathbf{v}}{\partial t} + \rho (\mathbf{v} \cdot \nabla) \mathbf{v} &amp;= -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{F}_{\text{drag}} \\
+\nabla \cdot \mathbf{v} &amp;= 0 \\
+\frac{d \mathbf{x}}{dt} &amp;= \mathbf{v}
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{F}_{\text{drag}}\)</span> represents the drag force on the particles.</p>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet and outlet: Specify velocity or pressure conditions.</p></li>
+<li><p>Solid particle boundaries: Implement no-slip conditions.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Specify initial conditions for velocity, pressure, and particle positions.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(\mathbf{v} \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \rho \frac{\partial \mathbf{v}}{\partial t} \cdot \mathbf{w} \, d\Omega + \int_{\Omega} \rho (\mathbf{v} \cdot \nabla) \mathbf{v} \cdot \mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \mu \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega - \int_{\Omega} p \nabla \cdot \mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \nabla \cdot \mathbf{v} q \, d\Omega \\
+&amp;= -\int_{\Omega} \mathbf{F}_{\text{drag}} \cdot \mathbf{w} \, d\Omega
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(\mathbf{w} \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(q \in L^2(\Omega)\)</span>.</p>
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+<li class="toctree-l3 current active"><a class="current reference internal" href="#">Power Law Fluid Flow in a Channel</a></li>
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+</ul>
+</li>
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+</li>
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+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.discretization.html">psydac.api.discretization</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.equation.html">psydac.api.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.essential_bc.html">psydac.api.essential_bc</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.expr.html">psydac.api.expr</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.feec.html">psydac.api.feec</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.fem.html">psydac.api.fem</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.glt.html">psydac.api.glt</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.grid.html">psydac.api.grid</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.postprocessing.html">psydac.api.postprocessing</a></li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Power Law Fluid Flow in a Channel</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mathematical-model">Mathematical Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#boundary-conditions">Boundary Conditions:</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initial-conditions">Initial Conditions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weak-formulation">Weak Formulation</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="power-law-fluid-flow-in-a-channel">
+<h1>Power Law Fluid Flow in a Channel<a class="headerlink" href="#power-law-fluid-flow-in-a-channel" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the steady-state power-law fluid model for non-Newtonian fluids:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\sigma_{ij} &amp;= -p\delta_{ij} + \lambda \varepsilon_{kk}\delta_{ij} + 2\mu \varepsilon_{ij} \\
+\varepsilon_{ij} &amp;= \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right) \\
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\sigma_{ij}\)</span> is the stress tensor, <span class="math notranslate nohighlight">\(p\)</span> is the pressure, <span class="math notranslate nohighlight">\(\lambda\)</span> and <span class="math notranslate nohighlight">\(\mu\)</span> are material constants, <span class="math notranslate nohighlight">\(v_i\)</span> is the velocity, and <span class="math notranslate nohighlight">\(\varepsilon_{ij}\)</span> is the strain rate.</p>
+<section id="boundary-conditions">
+<h3>Boundary Conditions:<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet and outlet: Specify velocity or pressure conditions.</p></li>
+<li><p>Channel walls: No-slip conditions.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Start with an initial guess for velocity and pressure.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(v_i \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \sigma_{ij} \varepsilon_{ij} \, d\Omega + \int_{\Omega} \frac{\partial p}{\partial x_i}v_i \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(v_i \in H_0^1(\Omega)\)</span>.</p>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
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+</ul>
+</li>
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+</ul>
+</li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.expr.html">psydac.api.ast.expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.glt.html">psydac.api.ast.glt</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
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+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.discretization.html">psydac.api.discretization</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.equation.html">psydac.api.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.essential_bc.html">psydac.api.essential_bc</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.expr.html">psydac.api.expr</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.feec.html">psydac.api.feec</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.fem.html">psydac.api.fem</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.glt.html">psydac.api.glt</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.grid.html">psydac.api.grid</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.postprocessing.html">psydac.api.postprocessing</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.printing.pycode.html">psydac.api.printing.pycode</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.settings.html">psydac.api.settings</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.tests.test_2d_biharmonic.html">psydac.api.tests.test_2d_biharmonic</a></li>
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+    <h1>Stokes Flow in a Lid-Driven Cavity</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mathematical-model">Mathematical Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#boundary-conditions">Boundary Conditions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initial-conditions">Initial Conditions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weak-formulation">Weak Formulation</a></li>
+</ul>
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+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="stokes-flow-in-a-lid-driven-cavity">
+<h1>Stokes Flow in a Lid-Driven Cavity<a class="headerlink" href="#stokes-flow-in-a-lid-driven-cavity" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the steady-state Stokes equations for flow in a 2D lid-driven cavity:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+- \nabla p + \mu \nabla^2 \mathbf{v} &amp;= 0 \\
+\nabla \cdot \mathbf{v} &amp;= 0
+\end{align}
+\end{split}\]</div>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Lid: <span class="math notranslate nohighlight">\(\mathbf{v} = (1, 0)\)</span> (lid-driven at constant velocity).</p></li>
+<li><p>Other boundaries: <span class="math notranslate nohighlight">\(\mathbf{v} = (0, 0)\)</span> (no-slip condition).</p></li>
+<li><p>Outlet: <span class="math notranslate nohighlight">\(\frac{\partial p}{\partial n} = 0\)</span> (zero-pressure gradient).</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Start with an initial guess for velocity and pressure.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(\mathbf{v} \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \mu \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega - \int_{\Omega} p \nabla \cdot \mathbf{w} \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(\mathbf{w} \in H_0^1(\Omega)\)</span>.</p>
+</section>
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+    <h1>Two-Phase Flow (Immersed Boundary Method)</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mathematical-model">Mathematical Model</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#boundary-conditions">Boundary Conditions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#initial-conditions">Initial Conditions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#weak-formulation">Weak Formulation</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="two-phase-flow-immersed-boundary-method">
+<h1>Two-Phase Flow (Immersed Boundary Method)<a class="headerlink" href="#two-phase-flow-immersed-boundary-method" title="Link to this heading">#</a></h1>
+<section id="mathematical-model">
+<h2>Mathematical Model<a class="headerlink" href="#mathematical-model" title="Link to this heading">#</a></h2>
+<p>Consider the incompressible Navier-Stokes equations with the Immersed Boundary Method for simulating two-phase flows:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+\rho \frac{\partial \mathbf{v}}{\partial t} + \rho (\mathbf{v} \cdot \nabla) \mathbf{v} &amp;= -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}_{\text{IB}} \\
+\nabla \cdot \mathbf{v} &amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{f}_{\text{IB}}\)</span> is the immersed boundary force.</p>
+<section id="boundary-conditions">
+<h3>Boundary Conditions<a class="headerlink" href="#boundary-conditions" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Inlet and outlet: Specify velocity or pressure conditions.</p></li>
+<li><p>Immersed boundary conditions to model the interface.</p></li>
+</ul>
+</section>
+<section id="initial-conditions">
+<h3>Initial Conditions<a class="headerlink" href="#initial-conditions" title="Link to this heading">#</a></h3>
+<p>Specify initial conditions for velocity, pressure, and the immersed boundary.</p>
+</section>
+</section>
+<section id="weak-formulation">
+<h2>Weak Formulation<a class="headerlink" href="#weak-formulation" title="Link to this heading">#</a></h2>
+<p>Find <span class="math notranslate nohighlight">\(\mathbf{v} \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(p \in L^2(\Omega)\)</span> such that:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{align}
+&amp;\int_{\Omega} \rho \frac{\partial \mathbf{v}}{\partial t} \cdot \mathbf{w} \, d\Omega + \int_{\Omega} \rho (\mathbf{v} \cdot \nabla) \mathbf{v} \cdot \mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \mu \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega - \int_{\Omega} p \nabla \cdot \mathbf{w} \, d\Omega + \int_{\Omega} \mathbf{f}_{\text{IB}} \cdot \mathbf{w} \, d\Omega \\
+&amp;+ \int_{\Omega} \nabla \cdot \mathbf{v} q \, d\Omega \\
+&amp;= 0
+\end{align}
+\end{split}\]</div>
+<p>for all test functions <span class="math notranslate nohighlight">\(\mathbf{w} \in H_0^1(\Omega)\)</span> and <span class="math notranslate nohighlight">\(q \in L^2(\Omega)\)</span>.</p>
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+  <section id="module-psydac.api.ast.expr">
+<span id="psydac-api-ast-expr"></span><h1>psydac.api.ast.expr<a class="headerlink" href="#module-psydac.api.ast.expr" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_atoms_expr</span></code>(atom, basis, indices, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_scalar_field</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_vector_field</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ExprInterface</span></code>(kernel[, name, mapping, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ExprKernel</span></code>(expr, space[, name, mapping, ...])</p></td>
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+  <section id="module-psydac.api.ast.fem">
+<span id="psydac-api-ast-fem"></span><h1>psydac.api.ast.fem<a class="headerlink" href="#module-psydac.api.ast.fem" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_diag_len</span></code>(p, md, mc)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">convert</span></code>(dtype)</p></td>
+<td><p>This function returns the index of a Function Space in a 3D DeRham sequence</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">expand</span></code>(args)</p></td>
+<td><p>This function expands vector functions into indexed functions</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">expand_hdiv_hcurl</span></code>(args)</p></td>
+<td><p>This function expands vector functions of type hdiv and hculr into indexed functions</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_degrees</span></code>(funcs, space)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_multiplicity</span></code>(funcs, space)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_nquads</span></code>(Vh)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">regroup</span></code>(tests)</p></td>
+<td><p>This function regourps the test/trial functions by their Function Space</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">toInteger</span></code>(a)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AST</span></code>(expr, terminal_expr, spaces[, ...])</p></td>
+<td><p>The Ast class transforms a terminal expression returned from sympde into a DefNode</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DefNode</span></code>(name, arguments, local_variables, ...)</p></td>
+<td><p>DefNode represents a function definition where it contains the arguments and the body</p></td>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.api.ast.glt">
+<span id="psydac-api-ast-glt"></span><h1>psydac.api.ast.glt<a class="headerlink" href="#module-psydac.api.ast.glt" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GltInterface</span></code>(kernel[, name, domain, ...])</p></td>
+<td><p></p></td>
+</tr>
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+  <section id="module-psydac.api.ast">
+<span id="psydac-api-ast"></span><h1>psydac.api.ast<a class="headerlink" href="#module-psydac.api.ast" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.ast.basic.html#module-psydac.api.ast.basic" title="psydac.api.ast.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.ast.basic</span></code></a></p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.ast.evaluation.html#module-psydac.api.ast.evaluation" title="psydac.api.ast.evaluation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.ast.evaluation</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.ast.expr.html#module-psydac.api.ast.expr" title="psydac.api.ast.expr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.ast.expr</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.ast.fem.html#module-psydac.api.ast.fem" title="psydac.api.ast.fem"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.ast.fem</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.ast.glt.html#module-psydac.api.ast.glt" title="psydac.api.ast.glt"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.ast.glt</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.ast.linalg.html#module-psydac.api.ast.linalg" title="psydac.api.ast.linalg"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.ast.linalg</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.ast.nodes.html#module-psydac.api.ast.nodes" title="psydac.api.ast.nodes"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.ast.nodes</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.ast.parser.html#module-psydac.api.ast.parser" title="psydac.api.ast.parser"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.ast.parser</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.ast.tests.html#module-psydac.api.ast.tests" title="psydac.api.ast.tests"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.ast.tests</span></code></a></p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.ast.utilities.html#module-psydac.api.ast.utilities" title="psydac.api.ast.utilities"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.ast.utilities</span></code></a></p></td>
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new file mode 100644
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+  <section id="module-psydac.api.ast.linalg">
+<span id="psydac-api-ast-linalg"></span><h1>psydac.api.ast.linalg<a class="headerlink" href="#module-psydac.api.ast.linalg" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Mod</span></code>(a, b)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_diag_len</span></code>(p, md, mc[, return_padding])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">toInteger</span></code>(a)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">variable_to_sympy</span></code>(x)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearOperatorDot</span></code>(ndim, block_shape[, comm])</p></td>
+<td><p>Generate the Matrix Vector Product function for a BlockLinearOperator,StencilMatrix or StencilInterfaceMatrix.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorDot</span></code>(ndim[, backend])</p></td>
+<td><p></p></td>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+  <section id="module-psydac.api.ast.nodes">
+<span id="psydac-api-ast-nodes"></span><h1>psydac.api.ast.nodes<a class="headerlink" href="#module-psydac.api.ast.nodes" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_itergener</span></code>(a, index)</p></td>
+<td><p>Create the generator and the iterator based on a and the index</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_logical_expressions</span></code>(u, nderiv[, lhs])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_number_derivatives</span></code>(expr)</p></td>
+<td><p>returns the number of partial derivatives in expr.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_string</span></code>(n)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">toInteger</span></code>(a)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AddNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Allocate</span></code>(arr, shape)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AndNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ArityType</span></code>()</p></td>
+<td><p>Base class representing a form type: bilinear/linear/functional</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Array</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ArrayNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AtomicNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BaseNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasisAtom</span></code>(expr)</p></td>
+<td><p>Used to describe a temporary for the basis coefficient or in the kernel.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BilinearArity</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Block</span></code>(body)</p></td>
+<td><p>This class represents a Block of statements</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockLinearOperatorNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockScalarLocalBasis</span></code>([trials, tests, expr, ...])</p></td>
+<td><p>This is used to describe a block of scalar dofs over an element</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockStencilMatrixGlobalBasis</span></code>(trials, tests, ...)</p></td>
+<td><p>used to describe local dof over an element as a block stencil matrix</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockStencilMatrixLocalBasis</span></code>(trials, tests, ...)</p></td>
+<td><p>used to describe local dof over an element as a block stencil matrix</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockStencilVectorGlobalBasis</span></code>(tests, pads, ...)</p></td>
+<td><p>used to describe local dof over an element as a block stencil matrix</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockStencilVectorLocalBasis</span></code>(tests, pads, expr)</p></td>
+<td><p>used to describe local dof over an element as a block stencil matrix</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CoefficientBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputeKernelExpr</span></code>(expr[, weights])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputeLogical</span></code>(expr[, weights, lhs])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputeLogicalBasis</span></code>(expr[, lhs])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputeNode</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputePhysical</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputePhysicalBasis</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Element</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ElementOf</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">EqNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">EvalField</span></code>(domain, atoms, q_index, l_index, ...)</p></td>
+<td><p>This function computes atomic expressions needed to evaluate  EvaluteField/VectorField final expression</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">EvalMapping</span></code>(domain, quads, indices_basis, ...)</p></td>
+<td><p>This function computes atomic expressions needed to evaluate  EvalMapping final expression.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Evaluation</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ExprNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Expression</span></code>(*args)</p></td>
+<td><p>The Expression class gives us the possibility to create specific instructions for some dimension, where the generated code is not in a vectorized form.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FieldEvaluation</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FloorDiv</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionalArity</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GeneratorBase</span></code>(target, dummies)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GeometryAtom</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GeometryExpr</span></code>(expr[, dtype])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GeometryExpressions</span></code>(M, nderiv[, dtype])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalElementBasis</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalSpanArray</span></code>(target[, index])</p></td>
+<td><p>This represents the global span array</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalTensorQuadratureBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalTensorQuadratureGrid</span></code>(*args)</p></td>
+<td><p>This class represents the quadrature points and weights in a domain.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalTensorQuadratureTestBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalTensorQuadratureTrialBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalThreadEnds</span></code>([index])</p></td>
+<td><p>This represents the threads ends over the decomposed domain</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalThreadSizes</span></code>([index])</p></td>
+<td><p>This represents the number of elements owned by a thread</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalThreadSpanArray</span></code>(target[, index])</p></td>
+<td><p>This represents the global span array of each thread</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalThreadStarts</span></code>([index])</p></td>
+<td><p>This represents the threads starts over the decomposed domain</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Grid</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IfNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexDerivative</span></code>([length])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexDof</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexDofTest</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexDofTrial</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexElement</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexInnerDofTest</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexNode</span></code>([start, stop, length, index])</p></td>
+<td><p>Base class representing one index of an iterator</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexOuterDofTest</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexQuadrature</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IntDivNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IteratorBase</span></code>(target[, dummies])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthDof</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthDofTest</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthDofTrial</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthElement</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthInnerDofTest</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthNode</span></code>([target, index])</p></td>
+<td><p>Base class representing one length of an iterator</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthOuterDofTest</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthQuadrature</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearArity</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalElementBasis</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalIndexElement</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalSpanArray</span></code>(target[, index])</p></td>
+<td><p>This represents the local span array</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalTensorQuadratureBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalTensorQuadratureGrid</span></code>(*args)</p></td>
+<td><p>This class represents the element wise quadrature points and weights in a domain.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalTensorQuadratureTestBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalTensorQuadratureTrialBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalThreadEnds</span></code>([index])</p></td>
+<td><p>This represents the local threads ends over the decomposed domain</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalThreadStarts</span></code>([index])</p></td>
+<td><p>This represents the local threads starts over the decomposed domain</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalBasisValue</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalGeometryValue</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalValueNode</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Loop</span></code>(iterable, index, *[, stmts, mask, ...])</p></td>
+<td><p>class to describe a dimensionless loop of an iterator over a generator.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MappingEvaluation</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Mask</span></code>(axis, ext)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixCoordsFromRank</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixGlobalBasis</span></code>(target, test[, dtype])</p></td>
+<td><p>used to describe global dof</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixLocalBasis</span></code>(target[, dtype])</p></td>
+<td><p>used to describe local dof over an element</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixQuadrature</span></code>(target[, dtype])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixRankFromCoords</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Max</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Min</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MulNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NeighbourThreadCoordinates</span></code>([start, stop, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NotNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumThreads</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Pads</span></code>(tests[, trials, tests_degree, ...])</p></td>
+<td><p>This represents the global pads</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ParallelBlock</span></code>([default, private, shared, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Pattern</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PhysicalBasisValue</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PhysicalGeometryValue</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PhysicalValueNode</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PlusGlobalTensorQuadratureGrid</span></code>(*args)</p></td>
+<td><p>This class represents the quadrature points and weights in the plus side of an interface.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PlusLocalTensorQuadratureGrid</span></code>(*args)</p></td>
+<td><p>This class represents the element wise quadrature points and weights in the plus side of an interface.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PlusTensorQuadrature</span></code>(*args)</p></td>
+<td><p>This class represents the quadrature point and weight in the plus side of an interface.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ProductGenerator</span></code>(target, dummies)</p></td>
+<td><p>This class represent an element of an array with arbitrary number of dimensions.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ProductIteration</span></code>(iterator, generator)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ProductIterator</span></code>(target[, dummies])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RAT</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RankDimension</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RankNode</span></code>()</p></td>
+<td><p>Base class representing a rank of an iterator</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Reduce</span></code>(op, rhs, lhs, loop)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Reduction</span></code>(op, expr[, lhs])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Reset</span></code>(var[, expr])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ScalarLocalBasis</span></code>([u, v, tag, dtype])</p></td>
+<td><p>This is used to describe scalar dof over an element</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ScalarNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Span</span></code>(target[, index])</p></td>
+<td><p>This represents the span of a basis in an element</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SpanArray</span></code>(target[, index])</p></td>
+<td><p>This represents the global span array</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SplitArray</span></code>(target, positions, lengths)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilMatrixGlobalBasis</span></code>(u, v, pads[, tag, ...])</p></td>
+<td><p>used to describe local dof over an element as a stencil matrix</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilMatrixLocalBasis</span></code>(u, v, pads[, tag, dtype])</p></td>
+<td><p>used to describe local dof over an element as a stencil matrix</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilVectorGlobalBasis</span></code>(v, pads[, tag, dtype])</p></td>
+<td><p>used to describe local dof over an element as a stencil vector</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilVectorLocalBasis</span></code>(v, pads[, tag, dtype])</p></td>
+<td><p>used to describe local dof over an element as a stencil vector</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StrictLessThanNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorAdd</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorAssignExpr</span></code>(lhs, rhs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorExpression</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorGenerator</span></code>(target, dummies)</p></td>
+<td><p>This class represent an array list of array elements with arbitrary number of dimensions.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorIntDiv</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorInteger</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorIteration</span></code>(iterator, generator)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorIterator</span></code>(target[, dummies])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorMax</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorMul</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorQuadrature</span></code>(*args)</p></td>
+<td><p>This class represents the quadrature point and weight in a domain.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorQuadratureBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorQuadratureTestBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorQuadratureTrialBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorTestBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorTrialBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ThreadCoordinates</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ThreadId</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ValueNode</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorAssign</span></code>(lhs, rhs[, op])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">WeightedVolumeQuadrature</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">WhileLoop</span></code>(condition, body)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Zeros</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ZerosLike</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+  <section id="module-psydac.api.ast.parser">
+<span id="psydac-api-ast-parser"></span><h1>psydac.api.ast.parser<a class="headerlink" href="#module-psydac.api.ast.parser" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_scalar_array</span></code>(var)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">parse</span></code>(expr, settings[, backend])</p></td>
+<td><p>This function takes a Psydac Ast and returns a Pyccel Ast</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_string</span></code>(n)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Parser</span></code>(settings[, backend])</p></td>
+<td><p>This class takes a Psyadac Ast and transforms it to a Pyccel Ast by calling the Parser.doit method</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Shape</span></code>(*args)</p></td>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bezier-curves.html">Bézier curves</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
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+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+  <section id="module-psydac.api.ast.tests">
+<span id="psydac-api-ast-tests"></span><h1>psydac.api.ast.tests<a class="headerlink" href="#module-psydac.api.ast.tests" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
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+  <section id="module-psydac.api.ast.tests.test_nodes">
+<span id="psydac-api-ast-tests-test-nodes"></span><h1>psydac.api.ast.tests.test_nodes<a class="headerlink" href="#module-psydac.api.ast.tests.test_nodes" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">assembly_bilinear_form_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">assembly_linear_form_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_atom_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_atom_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_field_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">geometry_atom_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">global_quad_basis_span_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">global_quad_basis_span_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">global_quad_basis_span_2d_matrix_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">global_quad_basis_span_2d_matrix_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">global_quad_basis_span_2d_vector_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">global_quad_basis_span_2d_vector_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">global_quad_span_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">global_span_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">loop_global_local_quad_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">loop_local_dof_quad_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">loop_local_dof_quad_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">loop_local_dof_quad_2d_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">loop_local_dof_quad_2d_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">loop_local_quad_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">loop_local_quad_geometry_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
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+</section>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
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+  <section id="module-psydac.api.ast.utilities">
+<span id="psydac-api-ast-utilities"></span><h1>psydac.api.ast.utilities<a class="headerlink" href="#module-psydac.api.ast.utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">build_pyccel_type_annotations</span></code>(args[, order])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">build_pythran_types_header</span></code>(name, args[, order])</p></td>
+<td><p>builds a types decorator from a list of arguments (of FunctionDef)</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_atoms_expr</span></code>(atomic_exprs, ...)</p></td>
+<td><p>This function computes atomic expressions needed to evaluate  the Kernel final expression</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_atoms_expr_field</span></code>(atomic_exprs, ...)</p></td>
+<td><p>This function computes atomic expressions needed to evaluate  EvaluteField/VectorField final expression</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_atoms_expr_mapping</span></code>(atomic_exprs, ...)</p></td>
+<td><p>This function computes atomic expressions needed to evaluate  EvalMapping final expression</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_boundary_jacobian</span></code>(parent_namespace, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_normal_vector</span></code>(parent_namespace, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_tangent_vector</span></code>(parent_namespace, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">filter_loops</span></code>(indices, ranges, body, boundary)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">filter_product</span></code>(indices, args, boundary)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">fusion_loops</span></code>(loops)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_name</span></code>(lhs)</p></td>
+<td><p>Given a list of variable return the meaningful part of the name of the first variable that has a _name attribute.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_mapping</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">logical2physical</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">math_atoms_as_str</span></code>(expr[, lib])</p></td>
+<td><p>Given a Sympy expression, find all known mathematical atoms (functions and constants) that need to be imported from a math library (e.g. Numpy) when generating Python code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_string</span></code>(n)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">rationalize_eval_mapping</span></code>(mapping, nderiv, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">select_loops</span></code>(indices, ranges, body, boundary)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">variables</span></code>(names, dtype, **args)</p></td>
+<td><p></p></td>
+</tr>
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+</section>
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+<span id="psydac-api-basic"></span><h1>psydac.api.basic<a class="headerlink" href="#module-psydac.api.basic" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicCodeGen</span></code>(expr, *[, folder, comm, root, ...])</p></td>
+<td><p>Basic class for any discrete concept that needs code generation.</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicDiscrete</span></code>(expr, kernel_expr, *[, ...])</p></td>
+<td><p>mapping is the symbolic mapping here.</p></td>
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new file mode 100644
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
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+  <section id="module-psydac.api.discretization">
+<span id="psydac-api-discretization"></span><h1>psydac.api.discretization<a class="headerlink" href="#module-psydac.api.discretization" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">change_dtype</span></code>(V, dtype)</p></td>
+<td><p>Given a FemSpace V, change its underlying vector_space (i.e. the space of its coefficients) so that it matches the required data type.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize</span></code>(a, *args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize_derham</span></code>(derham, domain_h[, ...])</p></td>
+<td><p>Create a discrete De Rham sequence from a symbolic one.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize_domain</span></code>(domain, *[, filename, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize_space</span></code>(V, domain_h, *[, degree, ...])</p></td>
+<td><p>This function creates the discretized space starting from the symbolic space.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">reduce_space_degrees</span></code>(V, Vh, *[, basis, sequence])</p></td>
+<td><p>This function takes a tensor FEM space Vh and reduces some degrees in order to obtain a tensor FEM space Wh that matches the symbolic space V in a certain sequence of spaces.</p></td>
+</tr>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bezier-curves.html">Bézier curves</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
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+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+  <section id="module-psydac.api.equation">
+<span id="psydac-api-equation"></span><h1>psydac.api.equation<a class="headerlink" href="#module-psydac.api.equation" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">l2_boundary_projection</span></code>(equation)</p></td>
+<td><p>Create an auxiliary equation (weak formulation) that solves for the boundary trace of the solution by performing an L2 projection of inhomogeneous essential boundary conditions.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteEquation</span></code>(expr, *args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearSystem</span></code>(lhs, rhs)</p></td>
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+  <section id="module-psydac.api.essential_bc">
+<span id="psydac-api-essential-bc"></span><h1>psydac.api.essential_bc<a class="headerlink" href="#module-psydac.api.essential_bc" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">apply_essential_bc</span></code>(a, *bcs, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">apply_essential_bc_BlockLinearOperator</span></code>(a, bc, *)</p></td>
+<td><p>Apply homogeneous dirichlet boundary conditions in nD.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">apply_essential_bc_BlockVector</span></code>(a, bc, *[, ...])</p></td>
+<td><p>Apply homogeneous dirichlet boundary conditions in nD.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">apply_essential_bc_kronecker_dense_matrix</span></code>(a, ...)</p></td>
+<td><p>This function applies the homogeneous boundary condition to the Kronecker product matrix objects,</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">apply_essential_bc_stencil</span></code>(a, *, axis, ext, ...)</p></td>
+<td><p>This function applies the homogeneous boundary condition to the Stencil objects,</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">check_boundary_type</span></code>(bc)</p></td>
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+  <section id="module-psydac.api.fem">
+<span id="psydac-api-fem"></span><h1>psydac.api.fem<a class="headerlink" href="#module-psydac.api.fem" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">collect_spaces</span></code>(space, *args)</p></td>
+<td><p>This function collect the arguments used in the assembly function</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_diag_len</span></code>(p, md, mc)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_quad_grids_arguments</span></code>(grid[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_test_space_arguments</span></code>(basis_values)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_trial_space_arguments</span></code>(basis_values)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">do_nothing</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">extract_stencil_mats</span></code>(mats)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_nquads</span></code>(Vh)</p></td>
+<td><p>This function is to be deleted in the future, as we intend to make the number of quadrature points a property of DiscreteLinearForms, DiscreteBilinearForms and DiscreteFunctionals rather than a property of spaces.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">reset_arrays</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteBilinearForm</span></code>(expr, kernel_expr, ...)</p></td>
+<td><p>Class that represents the concept of a discrete bi-linear form.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteFunctional</span></code>(expr, kernel_expr, ...[, ...])</p></td>
+<td><p>Class that represents the concept of a discrete functional form.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteLinearForm</span></code>(expr, kernel_expr, ...[, ...])</p></td>
+<td><p>Class that represents the concept of a discrete linear form.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteSesquilinearForm</span></code>(expr, kernel_expr, ...)</p></td>
+<td><p>Class that represents the concept of a discrete sesqui-linear form with the antilinearity on the first variable.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteSumForm</span></code>(a, kernel_expr, *args, **kwargs)</p></td>
+<td><p></p></td>
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+</section>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bezier-curves.html">Bézier curves</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
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+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+  <section id="module-psydac.api.glt">
+<span id="psydac-api-glt"></span><h1>psydac.api.glt<a class="headerlink" href="#module-psydac.api.glt" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteGltExpr</span></code>(expr, *args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GltBasicCodeGen</span></code>(expr, **kwargs)</p></td>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+  <section id="module-psydac.api.grid">
+<span id="psydac-api-grid"></span><h1>psydac.api.grid<a class="headerlink" href="#module-psydac.api.grid" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">create_collocation_basis</span></code>(glob_points, space)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_points_weights</span></code>(spaces, axis, e)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasisValues</span></code>(V, nderiv[, trial, grid])</p></td>
+<td><p>Compute the basis values and the spans.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CollocationBasisValues</span></code>(points, V, nderiv)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">QuadratureGrid</span></code>(V[, axis, ext, trial_space])</p></td>
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+<span id="psydac-api"></span><h1>psydac.api<a class="headerlink" href="#module-psydac.api" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.ast.html#module-psydac.api.ast" title="psydac.api.ast"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.ast</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.basic.html#module-psydac.api.basic" title="psydac.api.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.basic</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.discretization.html#module-psydac.api.discretization" title="psydac.api.discretization"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.discretization</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.equation.html#module-psydac.api.equation" title="psydac.api.equation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.equation</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.essential_bc.html#module-psydac.api.essential_bc" title="psydac.api.essential_bc"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.essential_bc</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.expr.html#module-psydac.api.expr" title="psydac.api.expr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.expr</span></code></a></p></td>
+<td><p></p></td>
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+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.feec.html#module-psydac.api.feec" title="psydac.api.feec"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.feec</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.fem.html#module-psydac.api.fem" title="psydac.api.fem"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.fem</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.glt.html#module-psydac.api.glt" title="psydac.api.glt"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.glt</span></code></a></p></td>
+<td><p></p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.grid.html#module-psydac.api.grid" title="psydac.api.grid"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.grid</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.api.postprocessing.html#module-psydac.api.postprocessing" title="psydac.api.postprocessing"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.postprocessing</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.printing.html#module-psydac.api.printing" title="psydac.api.printing"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.printing</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.settings.html#module-psydac.api.settings" title="psydac.api.settings"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.settings</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.html#module-psydac.api.tests" title="psydac.api.tests"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.utilities.html#module-psydac.api.utilities" title="psydac.api.utilities"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.utilities</span></code></a></p></td>
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new file mode 100644
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+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+    <h1>psydac.api.postprocessing</h1>
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+  <section id="module-psydac.api.postprocessing">
+<span id="psydac-api-postprocessing"></span><h1>psydac.api.postprocessing<a class="headerlink" href="#module-psydac.api.postprocessing" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_grid_lines_2d</span></code>(domain_h, V_h, *[, refine])</p></td>
+<td><p>Get the grid lines (i.e. element boundaries) of a 2D computational domain, which can be easily plotted with Matplotlib.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">OutputManager</span></code>(filename_space, filename_fields)</p></td>
+<td><p>A class meant to streamline the exportation of output data.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PostProcessManager</span></code>([geometry_file, domain, ...])</p></td>
+<td><p>A class to read saved information of a previous simulation and start post-processing from there.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<ul class="nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
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new file mode 100644
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+                    IGA-Python
+                </a>
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+        </ul>
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Introduction</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
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+</ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/data-structure.html">Data Structure</a></li>
+
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/install.html">Installation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
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+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+  <section id="module-psydac.api.printing.pycode">
+<span id="psydac-api-printing-pycode"></span><h1>psydac.api.printing.pycode<a class="headerlink" href="#module-psydac.api.printing.pycode" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pycode</span></code>(expr, **settings)</p></td>
+<td><p>Converts an expr to a string of Python code</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PythonCodePrinter</span></code>([settings])</p></td>
+<td><p></p></td>
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+
+    <script type="text/x-thebe-config">
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Introduction</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
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+<span id="psydac-api-tests-build-domain"></span><h1>psydac.api.tests.build_domain<a class="headerlink" href="#module-psydac.api.tests.build_domain" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">build_pretzel</span></code>([domain_name, r_min, r_max])</p></td>
+<td><p>design pretzel-like domain</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">create_domain</span></code>(patches, interfaces, name)</p></td>
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+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">flip_axis</span></code>([name, c1, c2])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_2D_rotation_mapping</span></code>([name, c1, c2, alpha])</p></td>
+<td><p></p></td>
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+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TransposedPolarMapping</span></code>(name[, dim])</p></td>
+<td><p>Represents a Transposed (x1 &lt;&gt; x2) Polar 2D Mapping object (Annulus).</p></td>
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+  <section id="module-psydac.api.tests">
+<span id="psydac-api-tests"></span><h1>psydac.api.tests<a class="headerlink" href="#module-psydac.api.tests" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.build_domain.html#module-psydac.api.tests.build_domain" title="psydac.api.tests.build_domain"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.build_domain</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_2d_biharmonic.html#module-psydac.api.tests.test_2d_biharmonic" title="psydac.api.tests.test_2d_biharmonic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_2d_biharmonic</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_2d_complex.html#module-psydac.api.tests.test_2d_complex" title="psydac.api.tests.test_2d_complex"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_2d_complex</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_2d_laplace.html#module-psydac.api.tests.test_2d_laplace" title="psydac.api.tests.test_2d_laplace"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_2d_laplace</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_2d_mapping_biharmonic.html#module-psydac.api.tests.test_2d_mapping_biharmonic" title="psydac.api.tests.test_2d_mapping_biharmonic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_2d_mapping_biharmonic</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_2d_mapping_laplace.html#module-psydac.api.tests.test_2d_mapping_laplace" title="psydac.api.tests.test_2d_mapping_laplace"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_2d_mapping_laplace</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_2d_mapping_poisson.html#module-psydac.api.tests.test_2d_mapping_poisson" title="psydac.api.tests.test_2d_mapping_poisson"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_2d_mapping_poisson</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_2d_multipatch_mapping_maxwell.html#module-psydac.api.tests.test_2d_multipatch_mapping_maxwell" title="psydac.api.tests.test_2d_multipatch_mapping_maxwell"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_2d_multipatch_mapping_maxwell</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_2d_multipatch_mapping_poisson.html#module-psydac.api.tests.test_2d_multipatch_mapping_poisson" title="psydac.api.tests.test_2d_multipatch_mapping_poisson"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_2d_multipatch_mapping_poisson</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_2d_multipatch_poisson.html#module-psydac.api.tests.test_2d_multipatch_poisson" title="psydac.api.tests.test_2d_multipatch_poisson"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_2d_multipatch_poisson</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_2d_navier_stokes.html#module-psydac.api.tests.test_2d_navier_stokes" title="psydac.api.tests.test_2d_navier_stokes"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_2d_navier_stokes</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_2d_poisson.html#module-psydac.api.tests.test_2d_poisson" title="psydac.api.tests.test_2d_poisson"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_2d_poisson</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_api_1d_compatible_spaces.html#module-psydac.api.tests.test_api_1d_compatible_spaces" title="psydac.api.tests.test_api_1d_compatible_spaces"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_1d_compatible_spaces</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_api_2d_compatible_spaces.html#module-psydac.api.tests.test_api_2d_compatible_spaces" title="psydac.api.tests.test_api_2d_compatible_spaces"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_2d_compatible_spaces</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_api_2d_fields.html#module-psydac.api.tests.test_api_2d_fields" title="psydac.api.tests.test_api_2d_fields"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_2d_fields</span></code></a></p></td>
+<td><p></p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_api_2d_scalar_analytical_mapping.html#module-psydac.api.tests.test_api_2d_scalar_analytical_mapping" title="psydac.api.tests.test_api_2d_scalar_analytical_mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_2d_scalar_analytical_mapping</span></code></a></p></td>
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+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_api_2d_system.html#module-psydac.api.tests.test_api_2d_system" title="psydac.api.tests.test_api_2d_system"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_2d_system</span></code></a></p></td>
+<td><p></p></td>
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+<td><p></p></td>
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+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_api_2d_vector_mapping.html#module-psydac.api.tests.test_api_2d_vector_mapping" title="psydac.api.tests.test_api_2d_vector_mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_2d_vector_mapping</span></code></a></p></td>
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+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_api_3d_scalar.html#module-psydac.api.tests.test_api_3d_scalar" title="psydac.api.tests.test_api_3d_scalar"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_3d_scalar</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_api_3d_scalar_mapping.html#module-psydac.api.tests.test_api_3d_scalar_mapping" title="psydac.api.tests.test_api_3d_scalar_mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_3d_scalar_mapping</span></code></a></p></td>
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+<td><p></p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_api_3d_vector_mapping.html#module-psydac.api.tests.test_api_3d_vector_mapping" title="psydac.api.tests.test_api_3d_vector_mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_3d_vector_mapping</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_api_expr_2d_scalar.html#module-psydac.api.tests.test_api_expr_2d_scalar" title="psydac.api.tests.test_api_expr_2d_scalar"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_expr_2d_scalar</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_api_feec_1d.html#module-psydac.api.tests.test_api_feec_1d" title="psydac.api.tests.test_api_feec_1d"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_feec_1d</span></code></a></p></td>
+<td><p>1D time-dependent Maxwell simulation using FEEC and time splitting with two operators.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_api_feec_2d.html#module-psydac.api.tests.test_api_feec_2d" title="psydac.api.tests.test_api_feec_2d"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_feec_2d</span></code></a></p></td>
+<td><p>2D time-dependent Maxwell simulation using FEEC and time splitting with two operators.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_api_feec_3d.html#module-psydac.api.tests.test_api_feec_3d" title="psydac.api.tests.test_api_feec_3d"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_feec_3d</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_api_glt_2d_scalar.html#module-psydac.api.tests.test_api_glt_2d_scalar" title="psydac.api.tests.test_api_glt_2d_scalar"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_glt_2d_scalar</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_api_glt_2d_scalar_mapping.html#module-psydac.api.tests.test_api_glt_2d_scalar_mapping" title="psydac.api.tests.test_api_glt_2d_scalar_mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_glt_2d_scalar_mapping</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_api_glt_2d_vector.html#module-psydac.api.tests.test_api_glt_2d_vector" title="psydac.api.tests.test_api_glt_2d_vector"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_api_glt_2d_vector</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_assembly.html#module-psydac.api.tests.test_assembly" title="psydac.api.tests.test_assembly"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_assembly</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_build_derham_mapping.html#module-psydac.api.tests.test_build_derham_mapping" title="psydac.api.tests.test_build_derham_mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_build_derham_mapping</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_equation.html#module-psydac.api.tests.test_equation" title="psydac.api.tests.test_equation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_equation</span></code></a></p></td>
+<td><p></p></td>
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+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.test_postprocessing.html#module-psydac.api.tests.test_postprocessing" title="psydac.api.tests.test_postprocessing"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_postprocessing</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.api.tests.test_quadorder.html#module-psydac.api.tests.test_quadorder" title="psydac.api.tests.test_quadorder"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.test_quadorder</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.api.tests.utils.html#module-psydac.api.tests.utils" title="psydac.api.tests.utils"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.api.tests.utils</span></code></a></p></td>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.api.tests.test_2d_biharmonic">
+<span id="psydac-api-tests-test-2d-biharmonic"></span><h1>psydac.api.tests.test_2d_biharmonic<a class="headerlink" href="#module-psydac.api.tests.test_2d_biharmonic" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_boundaries</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_biharmonic_2d_dir</span></code>(solution, f, ...[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_biharmonic_2d_dir0_1234</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_biharmonic_2d_dir0_123_diri_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_biharmonic_2d_dir0_13_diri_24</span></code>()</p></td>
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+  <section id="module-psydac.api.tests.test_2d_complex">
+<span id="psydac-api-tests-test-2d-complex"></span><h1>psydac.api.tests.test_2d_complex<a class="headerlink" href="#module-psydac.api.tests.test_2d_complex" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_boundaries</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_biharmonic_2d_dir</span></code>(solution, f, ...[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_helmholtz_2d</span></code>(solution, kappa, e_w_0, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_maxwell_2d</span></code>(uex, f, alpha, domain, *[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_2d</span></code>(solution, f, domain[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_complex_biharmonic_2d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_complex_biharmonic_2d_mapping</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_complex_helmholtz_2d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_complex_poisson_2d_multipatch</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_complex_poisson_2d_multipatch_mapping</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_2_patch_dirichlet_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_2_patch_dirichlet_parallel_0</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
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+<td><p></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_laplace_2d</span></code>(solution, f, ...[, backend, comm])</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_laplace_2d_neu0_1234</span></code>()</p></td>
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new file mode 100644
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+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+  <section id="module-psydac.api.tests.test_2d_mapping_biharmonic">
+<span id="psydac-api-tests-test-2d-mapping-biharmonic"></span><h1>psydac.api.tests.test_2d_mapping_biharmonic<a class="headerlink" href="#module-psydac.api.tests.test_2d_mapping_biharmonic" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_boundaries</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_biharmonic_2d_dir</span></code>(filename, solution, f, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_biharmonic_2d_collela_dir0_1234</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_biharmonic_2d_identity_dir0_1234</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_biharmonic_2d_quarter_annulus_dir0_1234</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_laplace_2d</span></code>(filename, solution, f, ...[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
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+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_laplace_2d_collela_neu0_1234</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_laplace_2d_identity_neu0_1234</span></code>()</p></td>
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+    <h1>psydac.api.tests.test_2d_mapping_poisson</h1>
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+                  
+  <section id="module-psydac.api.tests.test_2d_mapping_poisson">
+<span id="psydac-api-tests-test-2d-mapping-poisson"></span><h1>psydac.api.tests.test_2d_mapping_poisson<a class="headerlink" href="#module-psydac.api.tests.test_2d_mapping_poisson" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_boundaries</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_2d</span></code>(filename, solution, f, ...[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_circle_dir0</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_dir0_1234</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_dir0_123_diri_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_dir0_123_neu0_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_dir0_123_neui_4</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_dir0_124_neu0_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_dir0_124_neui_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_dir0_134_neu0_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_dir0_134_neui_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_dir0_13_diri_24</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_dir0_234_neu0_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_dir0_234_neui_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_collela_diri_1234</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_1234</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_1234_parallel</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_123_diri_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_123_neu0_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_123_neui_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_124_neu0_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_124_neui_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_134_neu0_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_134_neui_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_13_diri_24</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_234_neu0_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_234_neui_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_24_neu0_13</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_identity_dir0_4_neu0_123</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_pipe_dir_1234</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_quarter_annulus_dir0_1234</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_quarter_annulus_dir0_12_diri_34</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_quarter_annulus_diri_1234</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_quarter_annulus_diri_12_neui_34</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_quarter_annulus_diri_34_neui_12</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<p class="rubric">Functions</p>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_maxwell_2d</span></code>(uex, f, alpha, domain, *[, ...])</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_2_patch_dirichlet_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_2_patch_dirichlet_2</span></code>()</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_2_patch_dirichlet_parallel_0</span></code>()</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_2_patch_dirichlet_parallel_1</span></code>()</p></td>
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+  <section id="module-psydac.api.tests.test_2d_multipatch_mapping_poisson">
+<span id="psydac-api-tests-test-2d-multipatch-mapping-poisson"></span><h1>psydac.api.tests.test_2d_multipatch_mapping_poisson<a class="headerlink" href="#module-psydac.api.tests.test_2d_multipatch_mapping_poisson" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_2d</span></code>(solution, f, domain[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patches_dirichlet_0</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patches_dirichlet_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patches_dirichlet_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patches_dirichlet_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patches_dirichlet_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patches_dirichlet_5</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patches_dirichlet_6</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patches_dirichlet_parallel_0</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patches_dirichlet_parallel_1</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_4_patch_dirichlet_0</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_4_patches_dirichlet_parallel_0</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.api.tests.test_2d_multipatch_poisson">
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+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_2d</span></code>(solution, f, domain, ncells, ...)</p></td>
+<td><p></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patch_dirichlet_0</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patch_dirichlet_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patch_dirichlet_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patch_dirichlet_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_2_patch_dirichlet_4</span></code>()</p></td>
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+  <section id="module-psydac.api.tests.test_2d_navier_stokes">
+<span id="psydac-api-tests-test-2d-navier-stokes"></span><h1>psydac.api.tests.test_2d_navier_stokes<a class="headerlink" href="#module-psydac.api.tests.test_2d_navier_stokes" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_boundaries</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac_solver</span></code>(M, b)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_steady_state_navier_stokes_2d</span></code>(domain, f, ...)</p></td>
+<td><p>Navier Stokes solver for the 2d steady-state problem.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_time_dependent_navier_stokes_2d</span></code>(...[, ...])</p></td>
+<td><p>Time dependent Navier Stokes solver in a 2d domain.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy_solver</span></code>(M, b)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_navier_stokes_2d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_st_navier_stokes_2d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_st_navier_stokes_2d_parallel</span></code>()</p></td>
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+  <section id="module-psydac.api.tests.test_2d_poisson">
+<span id="psydac-api-tests-test-2d-poisson"></span><h1>psydac.api.tests.test_2d_poisson<a class="headerlink" href="#module-psydac.api.tests.test_2d_poisson" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_boundaries</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_2d</span></code>(solution, f, ...[, backend, comm])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_1234</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_1234_parallel</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_1234_user_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_123_diri_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_123_neu0_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_123_neui_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_124_neu0_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_124_neui_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_134_neu0_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_134_neui_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_13_diri_24</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_13_neu0_24</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_234_neu0_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_234_neui_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_24_neu0_13</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_dir0_4_neu0_123</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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new file mode 100644
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+<ul class="nav bd-sidenav">
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+  <section id="module-psydac.api.tests.test_api_2d_compatible_spaces">
+<span id="psydac-api-tests-test-api-2d-compatible-spaces"></span><h1>psydac.api.tests.test_api_2d_compatible_spaces<a class="headerlink" href="#module-psydac.api.tests.test_api_2d_compatible_spaces" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_maxwell_time_harmonic_2d_dir</span></code>(uex, f, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_mixed_form_2d_dir</span></code>(f0, sol, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_stokes_2d_dir</span></code>(domain, f, ue, pe, *, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_stokes_2d_dir_petsc</span></code>(domain, f, ue, pe, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_time_harmonic_2d_dir_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_mixed_form_2d_dir_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stokes_2d_dir_homogeneous</span></code>(scipy)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stokes_2d_dir_non_homogeneous</span></code>(scipy)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stokes_2d_dir_non_homogeneous_petsc</span></code>()</p></td>
+<td><p></p></td>
+</tr>
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+  <section id="module-psydac.api.tests.test_api_2d_fields">
+<span id="psydac-api-tests-test-api-2d-fields"></span><h1>psydac.api.tests.test_api_2d_fields<a class="headerlink" href="#module-psydac.api.tests.test_api_2d_fields" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_boundary_field_test</span></code>(domain, boundary, f, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_field_test</span></code>(filename, f)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_non_linear_poisson</span></code>(filename[, comm])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_boundary_field_identity</span></code>(n1, axis, ext)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_field_collela</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_field_identity_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_field_identity_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_field_quarter_annulus</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_nonlinear_poisson_circle</span></code>()</p></td>
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new file mode 100644
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+  <section id="module-psydac.api.tests.test_api_2d_scalar_analytical_mapping">
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+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_2d</span></code>(solution, f, domain, ncells, ...)</p></td>
+<td><p></p></td>
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+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_poisson_2d_analytical_mapping_0</span></code>()</p></td>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.api.tests.test_api_2d_system">
+<span id="psydac-api-tests-test-api-2d-system"></span><h1>psydac.api.tests.test_api_2d_system<a class="headerlink" href="#module-psydac.api.tests.test_api_2d_system" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_system_1_2d_dir</span></code>(Fe, Ge, f0, f1, ncells, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_system_1_2d_dir_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_vector_poisson_2d_dir</span></code>(solution, f, ...)</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_vector_poisson_2d_dir_1</span></code>()</p></td>
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+  <section id="module-psydac.api.tests.test_api_2d_vector_mapping">
+<span id="psydac-api-tests-test-api-2d-vector-mapping"></span><h1>psydac.api.tests.test_api_2d_vector_mapping<a class="headerlink" href="#module-psydac.api.tests.test_api_2d_vector_mapping" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_vector_poisson_2d_dir</span></code>(filename, solution, f)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_vector_poisson_2d_dir_collela</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_vector_poisson_2d_dir_identity</span></code>()</p></td>
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+  <section id="module-psydac.api.tests.test_api_2d_vector_multipatch_mapping">
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+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_vector_poisson_2d_dir</span></code>(filename, solution, f)</p></td>
+<td><p></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_vector_poisson_2d_dir_identity</span></code>()</p></td>
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+  <section id="module-psydac.api.tests.test_api_3d_scalar">
+<span id="psydac-api-tests-test-api-3d-scalar"></span><h1>psydac.api.tests.test_api_3d_scalar<a class="headerlink" href="#module-psydac.api.tests.test_api_3d_scalar" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_3d_dir</span></code>(solution, f, ncells, degree)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_3d_dirneu</span></code>(solution, f, boundary, ...)</p></td>
+<td><p></p></td>
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+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_poisson_3d_dir_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_poisson_3d_dir_1_parallel</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_poisson_3d_dirneu_13</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_poisson_3d_dirneu_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_poisson_3d_dirneu_24</span></code>()</p></td>
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+  <section id="module-psydac.api.tests.test_api_3d_scalar_mapping">
+<span id="psydac-api-tests-test-api-3d-scalar-mapping"></span><h1>psydac.api.tests.test_api_3d_scalar_mapping<a class="headerlink" href="#module-psydac.api.tests.test_api_3d_scalar_mapping" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_laplace_3d_neu</span></code>(filename, solution, f[, comm])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_3d_dir</span></code>(filename, solution, f[, comm])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_3d_dirneu</span></code>(filename, solution, f, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_laplace_3d_neu_identity</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_poisson_3d_dir_collela</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_poisson_3d_dirneu_collela_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_poisson_3d_dirneu_collela_24</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_poisson_3d_dirneu_identity_13</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_poisson_3d_dirneu_identity_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_poisson_3d_dirneu_identity_24</span></code>()</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_vector_poisson_3d_dir</span></code>(solution, f, ...)</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_vector_poisson_3d_dir_1</span></code>()</p></td>
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+  <section id="module-psydac.api.tests.test_api_3d_vector_mapping">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_vector_poisson_3d_dir</span></code>(filename, solution, f)</p></td>
+<td><p></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_vector_poisson_3d_dir_collela</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_vector_poisson_3d_dir_identity</span></code>()</p></td>
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+</ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.api.tests.test_api_expr_2d_scalar">
+<span id="psydac-api-tests-test-api-expr-2d-scalar"></span><h1>psydac.api.tests.test_api_expr_2d_scalar<a class="headerlink" href="#module-psydac.api.tests.test_api_expr_2d_scalar" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_2d_dir</span></code>(ncells, degree[, comm])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_expr_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+  <section id="module-psydac.api.tests.test_api_feec_1d">
+<span id="psydac-api-tests-test-api-feec-1d"></span><h1>psydac.api.tests.test_api_feec_1d<a class="headerlink" href="#module-psydac.api.tests.test_api_feec_1d" title="Link to this heading">#</a></h1>
+<p>1D time-dependent Maxwell simulation using FEEC and time splitting with
+two operators. These integrate exactly one of the two equations,
+respectively, over a given amount of time ∆t:</p>
+<ol class="arabic">
+<li><p>Faraday:</p>
+<blockquote>
+<div><p>b_new = b - ∆t D0 e</p>
+</div></blockquote>
+</li>
+<li><p>Amperè-Maxwell:</p>
+<blockquote>
+<div><p>e_new = e + ∆t (M0^{-1} D0^T M1) b</p>
+</div></blockquote>
+</li>
+</ol>
+<p>Given a 1D de Rham sequence with coefficient spaces C0 and C1, the vectors
+e and e_new belong to C0, while the vectors b and b_new belong to C1.
+D0 is the derivative matrix that maps from C0 to C1, while M0 and M1 are
+the mass matrices of the two spaces.</p>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">make_plot</span></code>(ax, t, sol_ex, sol_num, x, xlim, label)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_maxwell_1d</span></code>(*, L, eps, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_1d_dirichlet_penalization</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_1d_dirichlet_penalization_par</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_1d_dirichlet_strong</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_1d_dirichlet_strong_par</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_1d_periodic</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_1d_periodic_mult</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_1d_periodic_par</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">update_plot</span></code>(ax, t, sol_ex, sol_num)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.api.tests.test_api_feec_2d">
+<span id="psydac-api-tests-test-api-feec-2d"></span><h1>psydac.api.tests.test_api_feec_2d<a class="headerlink" href="#module-psydac.api.tests.test_api_feec_2d" title="Link to this heading">#</a></h1>
+<p>2D time-dependent Maxwell simulation using FEEC and time splitting with
+two operators. These integrate exactly one of the two equations,
+respectively, over a given amount of time ∆t:</p>
+<ol class="arabic">
+<li><p>Faraday:</p>
+<blockquote>
+<div><p>b_new = b - ∆t D1 e</p>
+</div></blockquote>
+</li>
+<li><p>Amperè-Maxwell:</p>
+<blockquote>
+<div><p>e_new = e + ∆t (M1^{-1} D1^T M2) b</p>
+</div></blockquote>
+</li>
+</ol>
+<p>Given a 2D de Rham sequence H1-H(curl)-L2 with coefficient spaces
+(C0, C1, C2), the vectors e and e_new belong to C1, while the vectors b
+and b_new belong to C2. D1 is the “scalar curl” matrix that maps from C1
+to C2, while M1 and M2 are the mass matrices of the spaces C1 and C2,
+respectively.</p>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">add_colorbar</span></code>(im, ax, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">plot_field_and_error</span></code>(name, x, y, field_h, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_maxwell_2d_TE</span></code>(*, use_spline_mapping, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_dirichlet</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_dirichlet_par</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_dirichlet_spline_mapping</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_multiplicity</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_periodic</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_periodic_multiplicity</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_periodic_multiplicity_equal_deg</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_2d_periodic_par</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">update_plot</span></code>(fig, t, x, y, field_h, field_ex)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CavitySolution</span></code>(*, a, b, c, nx, ny)</p></td>
+<td><p>Time-harmonic solution of Maxwell's equations in a rectangular cavity with perfectly conducting walls.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.api.tests.test_api_feec_3d">
+<span id="psydac-api-tests-test-api-feec-3d"></span><h1>psydac.api.tests.test_api_feec_3d<a class="headerlink" href="#module-psydac.api.tests.test_api_feec_3d" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluation_all_times</span></code>(fields, x, y, z)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_maxwell_3d_scipy</span></code>(logical_domain, ...[, mult])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_maxwell_3d_stencil</span></code>(logical_domain, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">splitting_integrator_scipy</span></code>(e0, b0, M1, M2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">splitting_integrator_stencil</span></code>(e0, b0, M1, M2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_3d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_3d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_maxwell_3d_2_mult</span></code>()</p></td>
+<td><p></p></td>
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+  <section id="module-psydac.api.tests.test_api_glt_2d_scalar">
+<span id="psydac-api-tests-test-api-glt-2d-scalar"></span><h1>psydac.api.tests.test_api_glt_2d_scalar<a class="headerlink" href="#module-psydac.api.tests.test_api_glt_2d_scalar" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_field_2d_dir</span></code>(ncells, degree[, comm])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_2d_dir</span></code>(ncells, degree[, comm])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_variable_coeff_2d_dir</span></code>(ncells, degree[, comm])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_glt_field_2d_dir_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_glt_poisson_2d_dir_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_glt_variable_coeff_2d_dir_1</span></code>()</p></td>
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new file mode 100644
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+  <section id="module-psydac.api.tests.test_api_glt_2d_scalar_mapping">
+<span id="psydac-api-tests-test-api-glt-2d-scalar-mapping"></span><h1>psydac.api.tests.test_api_glt_2d_scalar_mapping<a class="headerlink" href="#module-psydac.api.tests.test_api_glt_2d_scalar_mapping" title="Link to this heading">#</a></h1>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_poisson_2d_dir</span></code>(filename[, comm])</p></td>
+<td><p></p></td>
+</tr>
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+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_glt_poisson_2d_dir_collela</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_glt_poisson_2d_dir_identity</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_glt_poisson_2d_dir_quarter_annulus</span></code>()</p></td>
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new file mode 100644
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+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.api.tests.test_api_glt_2d_vector">
+<span id="psydac-api-tests-test-api-glt-2d-vector"></span><h1>psydac.api.tests.test_api_glt_2d_vector<a class="headerlink" href="#module-psydac.api.tests.test_api_glt_2d_vector" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_vector_poisson_2d_dir</span></code>(ncells, degree)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_api_glt_vector_poisson_2d_dir_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<span id="psydac-api-tests-test-assembly"></span><h1>psydac.api.tests.test_assembly<a class="headerlink" href="#module-psydac.api.tests.test_assembly" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">backend</span></code>(request)</p></td>
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+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_Norm_complex</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_assemble_complex_parallel</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_assembly_no_synchr_args</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_bilinearForm_complex</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_field_and_constant</span></code>(backend, dtype)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linearForm_complex</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_math_imports</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_multiple_fields</span></code>(backend, dtype)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_non_symmetric_BilinearForm</span></code>(backend)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_non_symmetric_different_space_BilinearForm</span></code>(backend)</p></td>
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new file mode 100644
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+<span id="psydac-api-tests-test-build-derham-mapping"></span><h1>psydac.api.tests.test_build_derham_mapping<a class="headerlink" href="#module-psydac.api.tests.test_build_derham_mapping" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_build_derham_spline_mapping_id_1d</span></code>(...)</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_build_derham_spline_mapping_id_3d</span></code>(...)</p></td>
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+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">build_2_cubes</span></code>()</p></td>
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+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_add_spaces</span></code>(dtype)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_export_fields_parallel</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_export_fields_serial</span></code>(dtype)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_incorrect_arg_export_to_vtk</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_parallel_export_discrete_domain</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_parallel_export_topological_domain</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_reconstruct_DerhamSequence_discrete_domain</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_reconstruct_DerhamSequence_topological_domain</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_reconstruct_multipatch</span></code>(dtype)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_reconstruct_spaces_topological_domain</span></code>(...)</p></td>
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new file mode 100644
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.api.tests.test_quadorder">
+<span id="psydac-api-tests-test-quadorder"></span><h1>psydac.api.tests.test_quadorder<a class="headerlink" href="#module-psydac.api.tests.test_quadorder" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_custom_nquads</span></code>(test_nquads)</p></td>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+  <section id="module-psydac.api.utilities">
+<span id="psydac-api-utilities"></span><h1>psydac.api.utilities<a class="headerlink" href="#module-psydac.api.utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">flatten</span></code>(args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">mkdir_p</span></code>(folder)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_string</span></code>(n)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">touch_init_file</span></code>(path)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">write_code</span></code>(filename, code[, folder])</p></td>
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+
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter1/space.html">Function Space concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+  <section id="module-psydac.cad.cad">
+<span id="psydac-cad-cad"></span><h1>psydac.cad.cad<a class="headerlink" href="#module-psydac.cad.cad" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elevate</span></code>(mapping, axis, times)</p></td>
+<td><p>Elevate the mapping degree times time in the direction axis.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_string</span></code>(n)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">refine</span></code>(mapping, axis, values)</p></td>
+<td><p>Refine the mapping by inserting values in the direction axis.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">translate</span></code>(mapping, displ)</p></td>
+<td><p></p></td>
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+  <section id="module-psydac.cad.gallery">
+<span id="psydac-cad-gallery"></span><h1>psydac.cad.gallery<a class="headerlink" href="#module-psydac.cad.gallery" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">annulus</span></code>([rmin, rmax, center])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">circle</span></code>([radius, center])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">quart_circle</span></code>([rmin, rmax, center])</p></td>
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+  <section id="module-psydac.cad.geometry">
+<span id="psydac-cad-geometry"></span><h1>psydac.cad.geometry<a class="headerlink" href="#module-psydac.cad.geometry" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">export_nurbs_to_hdf5</span></code>(filename, nurbs[, ...])</p></td>
+<td><p>Export a single-patch igakit NURBS object to a Psydac geometry file in HDF5 format</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">import_geopdes_to_nurbs</span></code>(filename)</p></td>
+<td><p>This function reads a geopdes geometry file and convert it to igakit nurbs object</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">refine_knots</span></code>(knots, ncells, degree[, ...])</p></td>
+<td><p>This function refines the knot sequence.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">refine_nurbs</span></code>(nrb[, ncells, degree, ...])</p></td>
+<td><p>This function refines the nurbs object.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Geometry</span></code>([domain, ncells, periodic, ...])</p></td>
+<td><p>Distributed discrete geometry that works for single and multiple patches.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<span id="psydac-cad"></span><h1>psydac.cad<a class="headerlink" href="#module-psydac.cad" title="Link to this heading">#</a></h1>
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+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.cad.cad.html#module-psydac.cad.cad" title="psydac.cad.cad"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.cad.cad</span></code></a></p></td>
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+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.cad.geometry.html#module-psydac.cad.geometry" title="psydac.cad.geometry"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.cad.geometry</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.cad.multipatch.html#module-psydac.cad.multipatch" title="psydac.cad.multipatch"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.cad.multipatch</span></code></a></p></td>
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new file mode 100644
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bezier-curves.html">Bézier curves</a></li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
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+</ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
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+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+  <section id="module-psydac.cad.multipatch">
+<span id="psydac-cad-multipatch"></span><h1>psydac.cad.multipatch<a class="headerlink" href="#module-psydac.cad.multipatch" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">export_multipatch_nurbs_to_hdf5</span></code>(filename, ...)</p></td>
+<td><p>Create the multipatch geometry file using one Igakit NURBS for each patch with the connectivity between the patches.</p></td>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<span id="psydac-cad-tests-test-geometry"></span><h1>psydac.cad.tests.test_geometry<a class="headerlink" href="#module-psydac.cad.tests.test_geometry" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
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+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_export_nurbs_to_hdf5</span></code>(ncells, degree)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_geometry_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_geometry_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_geometry_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_geometry_2d_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_geometry_2d_4</span></code>()</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_import_geopdes_to_nurbs</span></code>(ncells, degree)</p></td>
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+  <section id="module-psydac.cmd.mesh">
+<span id="psydac-cmd-mesh"></span><h1>psydac.cmd.mesh<a class="headerlink" href="#module-psydac.cmd.mesh" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">export_analytical_mapping</span></code>(mapping, ncells, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">export_caid_geometry</span></code>(name, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">main</span></code>()</p></td>
+<td><p>pyccel console command.</p></td>
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diff --git a/documentation/_autosummary/psydac.core.bsplines.html b/documentation/_autosummary/psydac.core.bsplines.html
new file mode 100644
index 0000000..a01bf34
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+    <h1>psydac.core.bsplines</h1>
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+  <section id="module-psydac.core.bsplines">
+<span id="psydac-core-bsplines"></span><h1>psydac.core.bsplines<a class="headerlink" href="#module-psydac.core.bsplines" title="Link to this heading">#</a></h1>
+<p>Basic module that provides the means for evaluating the B-Splines basis
+functions and their derivatives. In order to simplify automatic Fortran code
+generation with Pyccel, no object-oriented features are employed.</p>
+<p>References:</p>
+<blockquote>
+<div><ul class="simple">
+<li><p>[1] L. Piegl and W. Tiller. The NURBS Book, 2nd ed., Springer-Verlag Berlin Heidelberg GmbH, 1997.</p></li>
+<li><p>[2] SELALIB, Semi-Lagrangian Library. <a class="reference external" href="http://selalib.gforge.inria.fr">http://selalib.gforge.inria.fr</a></p></li>
+</ul>
+</div></blockquote>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_ders_on_irregular_grid</span></code>(knots, degree, ...)</p></td>
+<td><p>Evaluate B-Splines and their derivatives on an irregular_grid.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_ders_on_quad_grid</span></code>(knots, degree, ...)</p></td>
+<td><p>Evaluate B-Splines and their derivatives on the quadrature grid.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs</span></code>(knots, degree, x, span[, out])</p></td>
+<td><p>Compute the non-vanishing B-splines at a unique location.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_1st_der</span></code>(knots, degree, x, span[, out])</p></td>
+<td><p>Compute the first derivative of the non-vanishing B-splines at a location.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_all_ders</span></code>(knots, degree, x, span, n)</p></td>
+<td><p>Evaluate value and n derivatives at x of all basis functions with support in interval <span class="math notranslate nohighlight">\([x_{span-1}, x_{span}]\)</span>.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_array</span></code>(knots, degree, span, x[, out])</p></td>
+<td><p>Compute the non-vanishing B-splines at several locations.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_integrals</span></code>(knots, degree[, out])</p></td>
+<td><p>Return the integral of each B-spline basis function over the real line:</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">breakpoints</span></code>(knots, degree[, tol, out])</p></td>
+<td><p>Determine breakpoints' coordinates.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">cell_index</span></code>(breaks, i_grid[, tol, out])</p></td>
+<td><p>Computes in which cells a given array of locations belong.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">collocation_matrix</span></code>(knots, degree, periodic, ...)</p></td>
+<td><p>Computes the collocation matrix</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elements_spans</span></code>(knots, degree[, out])</p></td>
+<td><p>Compute the index of the last non-vanishing spline on each grid element (cell).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elevate_knots</span></code>(knots, degree, periodic[, ...])</p></td>
+<td><p>Given the knot sequence of a spline space S of degree p, compute the knot sequence of a spline space S_0 of degree p+1 such that u' is in S for all u in S_0.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_span</span></code>(knots, degree, x)</p></td>
+<td><p>Determine the knot span index at location x, given the B-Splines' knot sequence and polynomial degree.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_spans</span></code>(knots, degree, x[, out])</p></td>
+<td><p>Determine the knot span index at a set of locations x, given the B-Splines' knot sequence and polynomial degree.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">greville</span></code>(knots, degree, periodic[, out, ...])</p></td>
+<td><p>Compute coordinates of all Greville points.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">histopolation_matrix</span></code>(knots, degree, ...[, ...])</p></td>
+<td><p>Computes the histopolation matrix.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">hrefinement_matrix</span></code>(ts, p, knots)</p></td>
+<td><p>Computes the refinement matrix corresponding to the insertion of a given list of knots.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">make_knots</span></code>(breaks, degree, periodic[, ...])</p></td>
+<td><p>Create spline knots from breakpoints, with appropriate boundary conditions.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">quadrature_grid</span></code>(breaks, quad_rule_x, quad_rule_w)</p></td>
+<td><p>Compute the quadrature points and weights for performing integrals over each element (interval) of the 1D domain, given a certain Gaussian quadrature rule.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.core.bsplines_kernels">
+<span id="psydac-core-bsplines-kernels"></span><h1>psydac.core.bsplines_kernels<a class="headerlink" href="#module-psydac.core.bsplines_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_ders_on_irregular_grid_p</span></code></p></td>
+<td><p>Evaluate B-Splines and their derivatives on an irregular_grid.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_ders_on_quad_grid_p</span></code></p></td>
+<td><p>Evaluate B-Splines and their derivatives on the quadrature grid.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_1st_der_p</span></code></p></td>
+<td><p>Compute the first derivative of the non-vanishing B-splines at a location.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_all_ders_p</span></code></p></td>
+<td><p>Evaluate value and n derivatives at x of all basis functions with support in interval <span class="math notranslate nohighlight">\([x_{span-1}, x_{span}]\)</span>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_array_p</span></code></p></td>
+<td><p>Compute the non-vanishing B-splines at locations in x, given the knot sequence, polynomial degree and knot span.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_p</span></code></p></td>
+<td><p>Compute the non-vanishing B-splines at a unique location.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_integrals_p</span></code></p></td>
+<td><p>Return the integral of each B-spline basis function over the real line:</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">breakpoints_p</span></code></p></td>
+<td><p>Determine breakpoints' coordinates.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">cell_index_p</span></code></p></td>
+<td><p>Computes in which cells a given sorted array of locations belong.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">collocation_matrix_p</span></code></p></td>
+<td><p>Compute the collocation matrix <span class="math notranslate nohighlight">\(C_ij = B_j(x_i)\)</span>, which contains the values of each B-spline basis function <span class="math notranslate nohighlight">\(B_j\)</span> at all locations <span class="math notranslate nohighlight">\(x_i\)</span>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elements_spans_p</span></code></p></td>
+<td><p>Compute the index of the last non-vanishing spline on each grid element (cell).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elevate_knots_p</span></code></p></td>
+<td><p>Given the knot sequence of a spline space S of degree p, compute the knot sequence of a spline space S_0 of degree p+1 such that u' is in S for all u in S_0.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_span_p</span></code></p></td>
+<td><p>Determine the knot span index at location x, given the B-Splines' knot sequence and polynomial degree.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_spans_p</span></code></p></td>
+<td><p>Determine the knot span index at a set of locations x, given the B-Splines' knot sequence and polynomial degree.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">greville_p</span></code></p></td>
+<td><p>Compute coordinates of all Greville points.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">histopolation_matrix_p</span></code></p></td>
+<td><p>Computes the histopolation matrix.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">make_knots_p</span></code></p></td>
+<td><p>Create spline knots from breakpoints, with appropriate boundary conditions.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">merge_sort</span></code></p></td>
+<td><p>Performs a 'in place' merge sort of the input list</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">quadrature_grid_p</span></code></p></td>
+<td><p>Compute the quadrature points and weights for performing integrals over each element (interval) of the 1D domain, given a certain Gaussian quadrature rule.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.core.field_evaluation_kernels">
+<span id="psydac-core-field-evaluation-kernels"></span><h1>psydac.core.field_evaluation_kernels<a class="headerlink" href="#module-psydac.core.field_evaluation_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_2d_irregular_no_weights</span></code></p></td>
+<td><p>Parameters
+
+np1: int
+
+Number of points in the X1 direction
+
+np2: int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_2d_irregular_weighted</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_2d_no_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_2d_weighted</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_3d_irregular_no_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_3d_irregular_weighted</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_3d_no_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_3d_weighted</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_2d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_3d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_irregular_2d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_irregular_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_irregular_3d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_irregular_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_2d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_3d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_2d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_3d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_irregular_2d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_irregular_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_irregular_3d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_irregular_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_irregular_2d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_irregular_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_irregular_3d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_irregular_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_2d_hcurl</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (2, n_x1, n_x2, n_f) where:
+* 2 is the logical dimension of the problem (2 here)
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_f is the number of fields to push-forward in the Hcurl space.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_2d_hdiv</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (2, n_x1, n_x2, n_f) where:
+* 2 is the logical dimension of the problem (2 here)
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_f is the number of fields to push-forward in the Hdiv space.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_2d_l2</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (n_x1, n_x2, n_f) where:
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_f is the number of fields to push-forward in the L2 space.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_3d_hcurl</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (3, n_x1, n_x2, n_x3, n_f) where:
+* 3 is the logical dimension of the problem
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_x3 is the number of points in direction 3 of the implicit grid
+* n_f is the number of fields to push-forward in the Hcurl space.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_3d_hdiv</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (3, n_x1, n_x2, n_x3, n_f) where:
+* 3 is the logical dimension of the problem
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_x3 is the number of points in direction 3 of the implicit grid
+* n_f is the number of fields to push-forward in the Hdiv space.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_3d_l2</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (n_x1, n_x2, n_x3, n_f) where:
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_x3 is the number of points in direction 3 of the implicit grid.
+* n_f is the number of fields to push-forward in the L2 space.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.core.bsplines.html#module-psydac.core.bsplines" title="psydac.core.bsplines"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.core.bsplines</span></code></a></p></td>
+<td><p>Basic module that provides the means for evaluating the B-Splines basis functions and their derivatives.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.core.bsplines_kernels.html#module-psydac.core.bsplines_kernels" title="psydac.core.bsplines_kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.core.bsplines_kernels</span></code></a></p></td>
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+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.core.field_evaluation_kernels.html#module-psydac.core.field_evaluation_kernels" title="psydac.core.field_evaluation_kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.core.field_evaluation_kernels</span></code></a></p></td>
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+  <section id="module-psydac.core.tests.test_bsplines">
+<span id="psydac-core-tests-test-bsplines"></span><h1>psydac.core.tests.test_bsplines<a class="headerlink" href="#module-psydac.core.tests.test_bsplines" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_basis_funs</span></code>(lims, nc, p[, tol])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_basis_funs_1st_der</span></code>(lims, nc, p[, tol])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_basis_funs_all_ders</span></code>(lims, nc, p[, tol])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_cell_index</span></code>(i_grid, expected)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_collocation_matrix</span></code>(lims, nc, p, periodic)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_find_span</span></code>(lims, nc, p[, eps])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_histopolation_matrix</span></code>(lims, nc, p, periodic)</p></td>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
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+</ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
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+  <section id="module-psydac.core.tests.test_bsplines_pyccel">
+<span id="psydac-core-tests-test-bsplines-pyccel"></span><h1>psydac.core.tests.test_bsplines_pyccel<a class="headerlink" href="#module-psydac.core.tests.test_bsplines_pyccel" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_ders_on_quad_grid_true</span></code>(knots, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_1st_der_true</span></code>(knots, degree, x, span)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_all_ders_true</span></code>(knots, degree, x, ...)</p></td>
+<td><p>Evaluate value and n derivatives at x of all basis functions with support in interval [x_{span-1}, x_{span}]. If called with normalization='M', this uses M-splines instead of B-splines. ders[i,j] = (d/dx)^i B_k(x) with k=(span-degree+j),             for 0 &lt;= i &lt;= n and 0 &lt;= j &lt;= degree+1. Parameters ---------- knots : array_like     Knots sequence. degree : int     Polynomial degree of B-splines. x : float     Evaluation point. span : int     Knot span index. n : int     Max derivative of interest. normalization: str     Set to 'B' to get B-Splines and 'M' to get M-Splines Returns ------- ders : numpy.ndarray (n+1,degree+1)     2D array of n+1 (from 0-th to n-th) derivatives at x of all (degree+1)     non-vanishing basis functions in given span. Notes ----- The original Algorithm A2.3 in The NURBS Book [1] is here improved:     - 'left' and 'right' arrays are 1 element shorter;     - inverse of knot differences are saved to avoid unnecessary divisions;     - innermost loops are replaced with vector operations on slices.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_true</span></code>(knots, degree, x, span)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_integrals_true</span></code>(knots, degree)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">breakpoints_true</span></code>(knots, degree[, tol])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">collocation_matrix_true</span></code>(knots, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elements_spans_true</span></code>(knots, degree)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elevate_knots_true</span></code>(knots, degree, periodic)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_span_true</span></code>(knots, degree, x)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">greville_true</span></code>(knots, degree, periodic)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">histopolation_matrix_true</span></code>(knots, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">make_knots_true</span></code>(breaks, degree, periodic[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">quadrature_grid_true</span></code>(breaks, quad_rule_x, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_basis_ders_on_quad_grid</span></code>(knots, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_basis_funs</span></code>(knots, degree, x)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_basis_funs_1st_der</span></code>(knots, degree, x)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_basis_funs_all_ders</span></code>(knots, degree, x, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_basis_integrals</span></code>(knots, degree)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_breakpoints</span></code>(knots, degree)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_collocation_matrix</span></code>(knots, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_elements_spans</span></code>(knots, degree)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_elevate_knots</span></code>(knots, degree, periodic, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_find_span</span></code>(knots, degree, x)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_greville</span></code>(knots, degree, periodic)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_histopolation_matrix</span></code>(knots, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_make_knots</span></code>(breaks, degree, periodic, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_quadrature_grid</span></code>(breaks, nquads)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.core.tests.test_field_evaluation_kernel">
+<span id="psydac-core-tests-test-field-evaluation-kernel"></span><h1>psydac.core.tests.test_field_evaluation_kernel<a class="headerlink" href="#module-psydac.core.tests.test_field_evaluation_kernel" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_irregular_evaluations</span></code>(knots, ldim, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_irregular_jacobians</span></code>(geometry, npts)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_pushforwards_hcurl</span></code>(ldim)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_pushforwards_hdiv</span></code>(ldim)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_pushforwards_l2</span></code>(ldim, jac_det, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_regular_evaluations</span></code>(knots, ldim, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_regular_jacobians</span></code>(geometry, npts_per_cell)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/material-science.html">Material Science</a></li>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CartDataExchanger</span></code>()</p></td>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
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+</ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
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+  <section id="module-psydac.ddm.blocking_data_exchanger">
+<span id="psydac-ddm-blocking-data-exchanger"></span><h1>psydac.ddm.blocking_data_exchanger<a class="headerlink" href="#module-psydac.ddm.blocking_data_exchanger" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockingCartDataExchanger</span></code>(cart, dtype, *[, ...])</p></td>
+<td><p>Type that takes care of updating the ghost regions (padding) of a multi-dimensional array distributed according to the given Cartesian decomposition of a tensor-product grid of coefficients.</p></td>
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+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+  <section id="module-psydac.ddm.cart">
+<span id="psydac-ddm-cart"></span><h1>psydac.ddm.cart<a class="headerlink" href="#module-psydac.ddm.cart" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">create_interfaces_cart</span></code>(domain_decomposition, ...)</p></td>
+<td><p>This function Connects the Cartesian grids when they share an interface.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_mpi_type</span></code>(dtype)</p></td>
+<td><p>Find correct MPI datatype that corresponds to user-provided datatype.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CartDecomposition</span></code>(domain_decomposition, ...)</p></td>
+<td><p>Cartesian decomposition of a tensor-product grid of spline coefficients.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DomainDecomposition</span></code>(ncells, periods[, comm, ...])</p></td>
+<td><p>Cartesian decomposition of an N-Cube grid.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InterfaceCartDecomposition</span></code>(cart_minus, ...)</p></td>
+<td><p>The Cartesian decomposition of an interface constructed from the Cartesian decomposition of the patches that shares an interface.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MultiPatchDomainDecomposition</span></code>(ncells, periods)</p></td>
+<td><p>Cartesian decomposition of multiple N-Cube grids.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.ddm.basic.html#module-psydac.ddm.basic" title="psydac.ddm.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.ddm.basic</span></code></a></p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.ddm.blocking_data_exchanger.html#module-psydac.ddm.blocking_data_exchanger" title="psydac.ddm.blocking_data_exchanger"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.ddm.blocking_data_exchanger</span></code></a></p></td>
+<td><p></p></td>
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+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.ddm.cart.html#module-psydac.ddm.cart" title="psydac.ddm.cart"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.ddm.cart</span></code></a></p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.ddm.interface_data_exchanger.html#module-psydac.ddm.interface_data_exchanger" title="psydac.ddm.interface_data_exchanger"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.ddm.interface_data_exchanger</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.ddm.nonblocking_data_exchanger.html#module-psydac.ddm.nonblocking_data_exchanger" title="psydac.ddm.nonblocking_data_exchanger"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.ddm.nonblocking_data_exchanger</span></code></a></p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.ddm.partition.html#module-psydac.ddm.partition" title="psydac.ddm.partition"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.ddm.partition</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.ddm.petsc.html#module-psydac.ddm.petsc" title="psydac.ddm.petsc"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.ddm.petsc</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.ddm.tests.html#module-psydac.ddm.tests" title="psydac.ddm.tests"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.ddm.tests</span></code></a></p></td>
+<td><p></p></td>
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+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.ddm.utilities.html#module-psydac.ddm.utilities" title="psydac.ddm.utilities"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.ddm.utilities</span></code></a></p></td>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.ddm.interface_data_exchanger">
+<span id="psydac-ddm-interface-data-exchanger"></span><h1>psydac.ddm.interface_data_exchanger<a class="headerlink" href="#module-psydac.ddm.interface_data_exchanger" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InterfaceCartDataExchanger</span></code>(cart, dtype, *[, ...])</p></td>
+<td><p>This takes care of updating the ghost regions between two sides of an interface for a multi-dimensional array distributed according to the given Cartesian decomposition of a tensor-product grid of coefficients.</p></td>
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diff --git a/documentation/_autosummary/psydac.ddm.nonblocking_data_exchanger.html b/documentation/_autosummary/psydac.ddm.nonblocking_data_exchanger.html
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@@ -0,0 +1,958 @@
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter0/cad.html">Computer Aided Design</a><input class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-1"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/install.html">Installation</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
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+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
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+  <section id="module-psydac.ddm.nonblocking_data_exchanger">
+<span id="psydac-ddm-nonblocking-data-exchanger"></span><h1>psydac.ddm.nonblocking_data_exchanger<a class="headerlink" href="#module-psydac.ddm.nonblocking_data_exchanger" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NonBlockingCartDataExchanger</span></code>(cart, dtype, *)</p></td>
+<td><p>Type that takes care of updating the ghost regions (padding) of a multi-dimensional array distributed according to the given Cartesian decomposition of a tensor-product grid of coefficients.</p></td>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.ddm.partition">
+<span id="psydac-ddm-partition"></span><h1>psydac.ddm.partition<a class="headerlink" href="#module-psydac.ddm.partition" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_dims</span></code>(nnodes, gridsizes[, ...])</p></td>
+<td><p>With the aim of distributing a multi-dimensional array on a Cartesian topology, compute the number of processes along each dimension.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_dims_general</span></code>(mpi_size, npts[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_dims_uniform</span></code>(mpi_size, npts)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">partition_procs_per_patch</span></code>(npts, size)</p></td>
+<td><p>Compute the number of processes in each patch and assign to it an ascending range of processes.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
+    <script type="text/x-thebe-config">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<td><p></p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_cart_2d</span></code>(data_exchanger_type[, verbose, ...])</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_cart_2d_reverse_axis_0</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_cart_2d_reverse_axis_1</span></code>()</p></td>
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+</ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<ul class="nav bd-sidenav">
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+  <section id="module-psydac.ddm.tests.test_cart_3d">
+<span id="psydac-ddm-tests-test-cart-3d"></span><h1>psydac.ddm.tests.test_cart_3d<a class="headerlink" href="#module-psydac.ddm.tests.test_cart_3d" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_cart_3d</span></code>(data_exchanger_type[, verbose])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_cart_3d</span></code>(data_exchanger_type)</p></td>
+<td><p></p></td>
+</tr>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_minus_starts_ends</span></code>(plus_starts, ...)</p></td>
+<td><p>Compute the coefficients needed by the minus patch in a given interface.</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_plus_starts_ends</span></code>(minus_starts, ...)</p></td>
+<td><p>Compute the coefficients needed by the plus patch in a given interface.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_carts_2d</span></code>()</p></td>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
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+  <section id="module-psydac.ddm.tests.test_partition">
+<span id="psydac-ddm-tests-test-partition"></span><h1>psydac.ddm.tests.test_partition<a class="headerlink" href="#module-psydac.ddm.tests.test_partition" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_partition_1d_general</span></code>(mpi_size)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_partition_1d_uniform</span></code>(mpi_size)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_partition_2d_dims_mask</span></code>(mpi_size, npts, mask)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_partition_3d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_partition_3d_dims_mask</span></code>(mpi_size, npts, mask)</p></td>
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+  <section id="module-psydac.feec.derivatives">
+<span id="psydac-feec-derivatives"></span><h1>psydac.feec.derivatives<a class="headerlink" href="#module-psydac.feec.derivatives" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">block_tostencil</span></code>(M)</p></td>
+<td><p>Convert a BlockLinearOperator that contains KroneckerStencilMatrix objects to a BlockLinearOperator that contains StencilMatrix objects</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Curl_3D</span></code>(Hcurl, Hdiv)</p></td>
+<td><p>Curl operator in 3D.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Derivative_1D</span></code>(H1, L2)</p></td>
+<td><p>1D derivative.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiffOperator</span></code>(domain, codomain, matrix)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DirectionalDerivativeOperator</span></code>(V, W, diffdir, *)</p></td>
+<td><p>Represents a matrix-free derivative operator in a specific cardinal direction.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Divergence_2D</span></code>(Hdiv, L2)</p></td>
+<td><p>Divergence operator in 2D.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Divergence_3D</span></code>(Hdiv, L2)</p></td>
+<td><p>Divergence operator in 3D.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Gradient_2D</span></code>(H1, Hcurl)</p></td>
+<td><p>Gradient operator in 2D.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Gradient_3D</span></code>(H1, Hcurl)</p></td>
+<td><p>Gradient operator in 3D.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ScalarCurl_2D</span></code>(Hcurl, L2)</p></td>
+<td><p>Scalar curl operator in 2D: computes a scalar field from a vector field.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorCurl_2D</span></code>(H1, Hdiv)</p></td>
+<td><p>Vector curl operator in 2D: computes a vector field from a scalar field.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.feec.global_projectors">
+<span id="psydac-feec-global-projectors"></span><h1>psydac.feec.global_projectors<a class="headerlink" href="#module-psydac.feec.global_projectors" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_1d_0form</span></code>(intp_x1, F, f)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_1d_1form</span></code>(quad_x1, quad_w1, F, f)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_2d_0form</span></code>(intp_x1, intp_x2, F, f)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_2d_1form_hcurl</span></code>(intp_x1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_2d_1form_hdiv</span></code>(intp_x1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_2d_2form</span></code>(quad_x1, quad_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_2d_vec</span></code>(intp_x1, intp_x2, F1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_3d_0form</span></code>(intp_x1, intp_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_3d_1form</span></code>(intp_x1, intp_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_3d_2form</span></code>(intp_x1, intp_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_3d_3form</span></code>(quad_x1, quad_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_3d_vec</span></code>(intp_x1, intp_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalProjector</span></code>(space[, nquads])</p></td>
+<td><p>A global projector to some TensorFemSpace or VectorFemSpace object.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector_H1</span></code>(space[, nquads])</p></td>
+<td><p>Projector from H1 to an H1-conforming finite element space (i.e. a finite dimensional subspace of H1) constructed with tensor-product B-splines in 1, 2 or 3 dimensions.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector_H1vec</span></code>(space[, nquads])</p></td>
+<td><p>Projector from H1^3 = H1 x H1 x H1 to a conforming finite element space, i.e. a finite dimensional subspace of H1^3, constructed with tensor-product B-splines in 2 or 3 dimensions.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector_Hcurl</span></code>(space[, nquads])</p></td>
+<td><p>Projector from H(curl) to an H(curl)-conforming finite element space, i.e. a finite dimensional subspace of H(curl), constructed with tensor-product B- and M-splines in 2 or 3 dimensions.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector_Hdiv</span></code>(space[, nquads])</p></td>
+<td><p>Projector from H(div) to an H(div)-conforming finite element space, i.e. a finite dimensional subspace of H(div), constructed with tensor-product B- and M-splines in 2 or 3 dimensions.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector_L2</span></code>(space[, nquads])</p></td>
+<td><p>Projector from L2 to an L2-conforming finite element space (i.e. a finite dimensional subspace of L2) constructed with tensor-product M-splines in 1, 2 or 3 dimensions.</p></td>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.feec.global_projectors.html#module-psydac.feec.global_projectors" title="psydac.feec.global_projectors"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.feec.global_projectors</span></code></a></p></td>
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+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.feec.multipatch.html#module-psydac.feec.multipatch" title="psydac.feec.multipatch"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.feec.multipatch</span></code></a></p></td>
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new file mode 100644
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+  <section id="module-psydac.feec.multipatch.api">
+<span id="psydac-feec-multipatch-api"></span><h1>psydac.feec.multipatch.api<a class="headerlink" href="#module-psydac.feec.multipatch.api" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize</span></code>(expr, *args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize_derham_multipatch</span></code>(derham, ...)</p></td>
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+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteDerhamMultipatch</span></code>(*, mapping, ...[, ...])</p></td>
+<td><p>Represents the discrete De Rham sequence for multipatch domains.</p></td>
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diff --git a/documentation/_autosummary/psydac.feec.multipatch.examples.h1_source_pbms_conga_2d.html b/documentation/_autosummary/psydac.feec.multipatch.examples.h1_source_pbms_conga_2d.html
new file mode 100644
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--- /dev/null
+++ b/documentation/_autosummary/psydac.feec.multipatch.examples.h1_source_pbms_conga_2d.html
@@ -0,0 +1,958 @@
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+<span id="psydac-feec-multipatch-examples-h1-source-pbms-conga-2d"></span><h1>psydac.feec.multipatch.examples.h1_source_pbms_conga_2d<a class="headerlink" href="#module-psydac.feec.multipatch.examples.h1_source_pbms_conga_2d" title="Link to this heading">#</a></h1>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">solve_h1_source_pbm</span></code>([nc, deg, domain_name, ...])</p></td>
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diff --git a/documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d.html b/documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d.html
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+<span id="psydac-feec-multipatch-examples-hcurl-eigen-pbms-conga-2d"></span><h1>psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d<a class="headerlink" href="#module-psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d" title="Link to this heading">#</a></h1>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_eigenvalues</span></code>(nb_eigs, sigma, A_m, M_m)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">hcurl_solve_eigen_pbm</span></code>([nc, deg, ...])</p></td>
+<td><p>solver for the eigenvalue problem: find lambda in R and u in H0(curl), such that</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">solve_hcurl_source_pbm</span></code>([nc, deg, ...])</p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d" title="psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d</span></code></a></p></td>
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new file mode 100644
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new file mode 100644
index 0000000..bab7e87
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+  <section id="module-psydac.feec.multipatch.examples.ppc_test_cases">
+<span id="psydac-feec-multipatch-examples-ppc-test-cases"></span><h1>psydac.feec.multipatch.examples.ppc_test_cases<a class="headerlink" href="#module-psydac.feec.multipatch.examples.ppc_test_cases" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_sol_ref_V1h</span></code>([source_type, domain, ...])</p></td>
+<td><p>get a reference solution as a V1h FemField</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_source_and_sol_for_magnetostatic_pbm</span></code>([...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_source_and_solution</span></code>([source_type, eta, ...])</p></td>
+<td><p>compute source and reference solution (exact, or reference values) when possible, depending on the source_type</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+                  
+  <section id="module-psydac.feec.multipatch.fem_linear_operators">
+<span id="psydac-feec-multipatch-fem-linear-operators"></span><h1>psydac.feec.multipatch.fem_linear_operators<a class="headerlink" href="#module-psydac.feec.multipatch.fem_linear_operators" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComposedLinearOperator</span></code>(operators)</p></td>
+<td><p>operator L = L_1 .</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FemLinearOperator</span></code>([fem_domain, ...])</p></td>
+<td><p>Linear operators with an additional Fem layer</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IdLinearOperator</span></code>(V)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MultLinearOperator</span></code>(c, A)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SumLinearOperator</span></code>(B, A)</p></td>
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+  <section id="module-psydac.feec.multipatch">
+<span id="psydac-feec-multipatch"></span><h1>psydac.feec.multipatch<a class="headerlink" href="#module-psydac.feec.multipatch" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.api.html#module-psydac.feec.multipatch.api" title="psydac.feec.multipatch.api"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.feec.multipatch.api</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.feec.multipatch.examples.html#module-psydac.feec.multipatch.examples" title="psydac.feec.multipatch.examples"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.feec.multipatch.examples</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.fem_linear_operators.html#module-psydac.feec.multipatch.fem_linear_operators" title="psydac.feec.multipatch.fem_linear_operators"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.feec.multipatch.fem_linear_operators</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.feec.multipatch.multipatch_domain_utilities.html#module-psydac.feec.multipatch.multipatch_domain_utilities" title="psydac.feec.multipatch.multipatch_domain_utilities"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.feec.multipatch.multipatch_domain_utilities</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.operators.html#module-psydac.feec.multipatch.operators" title="psydac.feec.multipatch.operators"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.feec.multipatch.operators</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.feec.multipatch.plotting_utilities.html#module-psydac.feec.multipatch.plotting_utilities" title="psydac.feec.multipatch.plotting_utilities"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.feec.multipatch.plotting_utilities</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.utilities.html#module-psydac.feec.multipatch.utilities" title="psydac.feec.multipatch.utilities"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.feec.multipatch.utilities</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.feec.multipatch.utils_conga_2d.html#module-psydac.feec.multipatch.utils_conga_2d" title="psydac.feec.multipatch.utils_conga_2d"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.feec.multipatch.utils_conga_2d</span></code></a></p></td>
+<td><p></p></td>
+</tr>
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+</section>
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new file mode 100644
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Introduction</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter0/cad.html">Computer Aided Design</a><input class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-1"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
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+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+  <section id="module-psydac.feec.multipatch.multipatch_domain_utilities">
+<span id="psydac-feec-multipatch-multipatch-domain-utilities"></span><h1>psydac.feec.multipatch.multipatch_domain_utilities<a class="headerlink" href="#module-psydac.feec.multipatch.multipatch_domain_utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">build_multipatch_domain</span></code>([domain_name, ...])</p></td>
+<td><p>Create a 2D multipatch domain among the many available.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">create_domain</span></code>(patches, interfaces, name)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">flip_axis</span></code>([name, c1, c2])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_2D_rotation_mapping</span></code>([name, c1, c2, alpha])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_ref_eigenvalues</span></code>(domain_name, operator)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TransposedPolarMapping</span></code>(name[, dim])</p></td>
+<td><p>Represents a Transposed (x1 &lt;&gt; x2) Polar 2D Mapping object (Annulus).</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter0/cad.html">Computer Aided Design</a><input class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-1"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/install.html">Installation</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/material-science.html">Material Science</a></li>
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+  <section id="module-psydac.feec.multipatch.operators">
+<span id="psydac-feec-multipatch-operators"></span><h1>psydac.feec.multipatch.operators<a class="headerlink" href="#module-psydac.feec.multipatch.operators" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">allocate_interface_matrix</span></code>(corners, ...)</p></td>
+<td><p>Allocate the interface matrix for a vertex shared by two patches</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_K0_and_K0_inv</span></code>(V0h[, uniform_patches])</p></td>
+<td><p>Compute the change of basis matrices K0 and K0^{-1} in V0h.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_K1_and_K1_inv</span></code>(V1h[, uniform_patches])</p></td>
+<td><p>Compute the change of basis matrices K1 and K1^{-1} in Hcurl space V1h.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_interface_from_corners</span></code>(corner1, corner2, ...)</p></td>
+<td><p>Return the interface between two corners from two different patches that correspond to a single (physical) vertex.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_patch_index_from_face</span></code>(domain, face)</p></td>
+<td><p>Return the patch index of subdomain/boundary</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_row_col_index</span></code>(corner1, corner2, ...)</p></td>
+<td><p>Return the row and column index of a corner in the StencilInterfaceMatrix</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ortho_proj_Hcurl</span></code>(EE, V1h, domain_h, M1[, ...])</p></td>
+<td><p>return orthogonal projection of E on V1h, given M1 the mass matrix</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BrokenGradient_2D</span></code>(V0h, V1h)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BrokenScalarCurl_2D</span></code>(V1h, V2h)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BrokenTransposedGradient_2D</span></code>(V0h, V1h)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BrokenTransposedScalarCurl_2D</span></code>(V1h, V2h)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ConformingProjection_V0</span></code>(V0h, domain_h[, ...])</p></td>
+<td><p>Conforming projection from global broken V0 space to conforming global V0 space Defined by averaging of interface dofs</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ConformingProjection_V1</span></code>(V1h, domain_h[, ...])</p></td>
+<td><p>Conforming projection from global broken V1 space to conforming V1 global space</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">HodgeOperator</span></code>(Vh, domain_h[, ...])</p></td>
+<td><p>Change of basis operator: dual basis -&gt; primal basis</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Multipatch_Projector_H1</span></code>(V0h)</p></td>
+<td><p>to apply the H1 projection (2D) on every patch</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Multipatch_Projector_Hcurl</span></code>(V1h[, nquads])</p></td>
+<td><p>to apply the Hcurl projection (2D) on every patch</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Multipatch_Projector_L2</span></code>(V2h[, nquads])</p></td>
+<td><p>to apply the L2 projection (2D) on every patch</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.feec.multipatch.plotting_utilities">
+<span id="psydac-feec-multipatch-plotting-utilities"></span><h1>psydac.feec.multipatch.plotting_utilities<a class="headerlink" href="#module-psydac.feec.multipatch.plotting_utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_diag_grid</span></code>(mappings, N)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_grid_quad_weights</span></code>(etas, patch_logvols, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_grid_vals</span></code>(u, etas, mappings_list[, ...])</p></td>
+<td><p>get the physical field values, given the logical field and the logical grid :param u: FemField :param etas: logical grid :param space_kind: specifies the push-forward for the physical values</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_patch_knots_gridlines</span></code>(Vh, N, mappings[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_plotting_grid</span></code>(mappings, N[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_vector_valued</span></code>(u)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">my_small_plot</span></code>(title, vals[, titles, xx, yy, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">my_small_streamplot</span></code>(title, vals_x, vals_y, ...)</p></td>
+<td><p><dl class="field-list simple">
+<dt class="field-odd">param skip<span class="colon">:</span></dt>
+<dd class="field-odd"><p>every skip-th data point will be skipped</p>
+</dd>
+</dl>
+</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">plot_field</span></code>([fem_field, stencil_coeffs, ...])</p></td>
+<td><p>plot a discrete field (given as a FemField or by its coeffs in numpy or stencil format) on the given domain</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.feec.multipatch.utilities">
+<span id="psydac-feec-multipatch-utilities"></span><h1>psydac.feec.multipatch.utilities<a class="headerlink" href="#module-psydac.feec.multipatch.utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">error_fn</span></code>([source_type, method, conf_proj, ...])</p></td>
+<td><p>Get the error filename based on the method used to solve the multpatch problem</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_fem_name</span></code>([method, k, DG_full, ...])</p></td>
+<td><p>Get Fem name used to solve the multipatch problem</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_load_dir</span></code>([method, DG_full, domain_name, ...])</p></td>
+<td><p>get load directory name based on the fem name</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_method_name</span></code>([method, k, conf_proj, ...])</p></td>
+<td><p>Get method name used to solve the multpatch problem</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sol_ref_fn</span></code>(source_type, N_diag[, source_proj])</p></td>
+<td><p>Get the reference solution filename based on the source term type</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">source_name</span></code>([source_type, source_proj])</p></td>
+<td><p>Get the source term name</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">time_count</span></code>([t_stamp, msg])</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<ul class="nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.feec.pull_push">
+<span id="psydac-feec-pull-push"></span><h1>psydac.feec.pull_push<a class="headerlink" href="#module-psydac.feec.pull_push" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_1d_h1</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_1d_l2</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_2d_h1</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_2d_h1vec</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_2d_hcurl</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_2d_hdiv</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_2d_l2</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_3d_h1</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_3d_h1vec</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_3d_hcurl</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_3d_hdiv</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_3d_l2</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_1d_h1</span></code>(f, eta)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_1d_l2</span></code>(f, eta, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_2d_h1</span></code>(f, eta1, eta2)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_2d_hcurl</span></code>(f1, f2, eta1, eta2, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_2d_hdiv</span></code>(f1, f2, eta1, eta2, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_2d_l2</span></code>(f, eta1, eta2, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_3d_h1</span></code>(f, eta1, eta2, eta3)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_3d_hcurl</span></code>(f1, f2, f3, eta1, eta2, eta3, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_3d_hdiv</span></code>(f1, f2, f3, eta1, eta2, eta3, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_3d_l2</span></code>(f, eta1, eta2, eta3, F)</p></td>
+<td><p></p></td>
+</tr>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+  <section id="module-psydac.feec.pushforward">
+<span id="psydac-feec-pushforward"></span><h1>psydac.feec.pushforward<a class="headerlink" href="#module-psydac.feec.pushforward" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Pushforward</span></code>(grid[, mapping, npts_per_cell, ...])</p></td>
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new file mode 100644
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.feec.tests.test_axis_projection">
+<span id="psydac-feec-tests-test-axis-projection"></span><h1>psydac.feec.tests.test_axis_projection<a class="headerlink" href="#module-psydac.feec.tests.test_axis_projection" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
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+  <section id="module-psydac.feec.tests.test_commuting_projections">
+<span id="psydac-feec-tests-test-commuting-projections"></span><h1>psydac.feec.tests.test_commuting_projections<a class="headerlink" href="#module-psydac.feec.tests.test_commuting_projections" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_1d_commuting_pro_1</span></code>(Nel, Nq, p, bc, m)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_2d_commuting_pro_1</span></code>(Nel, Nq, p, bc, m)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_2d_commuting_pro_2</span></code>(Nel, Nq, p, bc, m)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_2d_commuting_pro_3</span></code>(Nel, Nq, p, bc, m)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_2d_commuting_pro_4</span></code>(Nel, Nq, p, bc, m)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_3d_commuting_pro_1</span></code>(Nel, Nq, p, bc, m)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_3d_commuting_pro_2</span></code>(Nel, Nq, p, bc, m)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_3d_commuting_pro_3</span></code>(Nel, Nq, p, bc, m)</p></td>
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+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_transpose_curl_3d</span></code>(Nel, Nq, p, bc, m)</p></td>
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+<td><p></p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_transpose_grad_3d</span></code>(Nel, Nq, p, bc, m)</p></td>
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+    <h1>psydac.feec.tests.test_differentiation_matrices</h1>
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+  <section id="module-psydac.feec.tests.test_differentiation_matrices">
+<span id="psydac-feec-tests-test-differentiation-matrices"></span><h1>psydac.feec.tests.test_differentiation_matrices<a class="headerlink" href="#module-psydac.feec.tests.test_differentiation_matrices" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compare_diff_operators_by_matrixassembly</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_directional_derivative_operator</span></code>(comm, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_Curl_3D</span></code>(domain, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_Derivative_1D</span></code>(domain, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_Divergence_2D</span></code>(domain, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_Divergence_3D</span></code>(domain, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_Gradient_2D</span></code>(domain, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_Gradient_3D</span></code>(domain, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_ScalarCurl_2D</span></code>(domain, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_VectorCurl_2D</span></code>(domain, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_directional_derivative_operator_1d_par</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_directional_derivative_operator_1d_ser</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_directional_derivative_operator_2d_par</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_directional_derivative_operator_2d_ser</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_directional_derivative_operator_3d_par</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_directional_derivative_operator_3d_ser</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_directional_derivative_operator_interface</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_directional_derivative_operator_invalid_wrongsized1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_directional_derivative_operator_invalid_wrongspace2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
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+<span id="psydac-feec-tests-test-global-projectors"></span><h1>psydac.feec.tests.test_global_projectors<a class="headerlink" href="#module-psydac.feec.tests.test_global_projectors" title="Link to this heading">#</a></h1>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_H1_projector_1d</span></code>(domain, ncells, degree, ...)</p></td>
+<td><p></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_L2_projector_1d</span></code>(domain, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_derham_projector_2d_hcurl</span></code>(ncells, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_derham_projector_2d_hdiv</span></code>(ncells, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_derham_projector_2d_hdiv_2</span></code>(ncells, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_derham_projector_3d</span></code>(ncells, degree, ...)</p></td>
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+  <section id="module-psydac.feec.tests.test_projections_parallel">
+<span id="psydac-feec-tests-test-projections-parallel"></span><h1>psydac.feec.tests.test_projections_parallel<a class="headerlink" href="#module-psydac.feec.tests.test_projections_parallel" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">run_projection_comparison</span></code>(domain, ncells, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_projection_parallel_1d</span></code>(domain, ncells, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_projection_parallel_2d</span></code>(domain, ncells, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_projection_parallel_3d</span></code>(domain, ncells, ...)</p></td>
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+  <section id="module-psydac.fem.basic">
+<span id="psydac-fem-basic"></span><h1>psydac.fem.basic<a class="headerlink" href="#module-psydac.fem.basic" title="Link to this heading">#</a></h1>
+<p>In order to avoid multiple inheritence, we define the base objects for Finite
+Elements as abstract classes that contain a topological member. This member can
+be used to specify the used data structure for example.</p>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FemField</span></code>(space[, coeffs])</p></td>
+<td><p>Element of a finite element space V.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FemSpace</span></code>()</p></td>
+<td><p>Generic Finite Element space V.</p></td>
+</tr>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">fem_context</span></code>(filename[, comm])</p></td>
+<td><p>Create tensor-product spline space and mapping from geometry input file in HDF5 format (single-patch only).</p></td>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<span id="psydac-fem-grid"></span><h1>psydac.fem.grid<a class="headerlink" href="#module-psydac.fem.grid" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FemAssemblyGrid</span></code>(space, start, end, *[, ...])</p></td>
+<td><p>Class that collects all 1D information local to process that are necessary for the correct assembly of l.h.s.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
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+  <section id="module-psydac.fem">
+<span id="psydac-fem"></span><h1>psydac.fem<a class="headerlink" href="#module-psydac.fem" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.fem.basic.html#module-psydac.fem.basic" title="psydac.fem.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.basic</span></code></a></p></td>
+<td><p>In order to avoid multiple inheritence, we define the base objects for Finite Elements as abstract classes that contain a topological member.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.fem.context.html#module-psydac.fem.context" title="psydac.fem.context"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.context</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.fem.grid.html#module-psydac.fem.grid" title="psydac.fem.grid"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.grid</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.fem.partitioning.html#module-psydac.fem.partitioning" title="psydac.fem.partitioning"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.partitioning</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.fem.projectors.html#module-psydac.fem.projectors" title="psydac.fem.projectors"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.projectors</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.fem.splines.html#module-psydac.fem.splines" title="psydac.fem.splines"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.splines</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.fem.tensor.html#module-psydac.fem.tensor" title="psydac.fem.tensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.tensor</span></code></a></p></td>
+<td><p>We assume here that a tensor space is the product of fem spaces whom basis are of compact support</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.fem.tests.html#module-psydac.fem.tests" title="psydac.fem.tests"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.tests</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.fem.vector.html#module-psydac.fem.vector" title="psydac.fem.vector"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.vector</span></code></a></p></td>
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+</section>
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+  <section id="module-psydac.fem.partitioning">
+<span id="psydac-fem-partitioning"></span><h1>psydac.fem.partitioning<a class="headerlink" href="#module-psydac.fem.partitioning" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_connectivity</span></code>(domain)</p></td>
+<td><p>Compute the connectivity of the multipatch domain.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_interface_spaces</span></code>(...)</p></td>
+<td><p>Create the fem spaces for each interface in the domain given by the connectivity.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_reduced_interface_spaces</span></code>(spaces, ...)</p></td>
+<td><p>Create the reduced spaces for the interface coefficients.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">create_cart</span></code>(domain_decomposition, spaces)</p></td>
+<td><p>Compute the cartesian decomposition of the coefficient space.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_minus_starts_ends</span></code>(plus_starts, ...)</p></td>
+<td><p>Compute the coefficients needed by the minus patch in a given interface.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_plus_starts_ends</span></code>(minus_starts, ...)</p></td>
+<td><p>Compute the coefficients needed by the plus patch in a given interface.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">partition_coefficients</span></code>(domain_decomposition, ...)</p></td>
+<td><p>Partition the coefficients starting from the grid decomposition.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.fem.projectors">
+<span id="psydac-fem-projectors"></span><h1>psydac.fem.projectors<a class="headerlink" href="#module-psydac.fem.projectors" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">knot_insertion_projection_operator</span></code>(domain, ...)</p></td>
+<td><p>Compute the projection operator based on the knot insertion technique.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">knots_to_insert</span></code>(coarse_grid, fine_grid[, tol])</p></td>
+<td><p>Compute the point difference between the fine grid and coarse grid.</p></td>
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+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SplineSpace</span></code>(degree[, knots, grid, ...])</p></td>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.fem.tensor">
+<span id="psydac-fem-tensor"></span><h1>psydac.fem.tensor<a class="headerlink" href="#module-psydac.fem.tensor" title="Link to this heading">#</a></h1>
+<p>We assume here that a tensor space is the product of fem spaces whom basis are
+of compact support</p>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorFemSpace</span></code>(domain_decomposition, *spaces)</p></td>
+<td><p>Tensor-product Finite Element space V.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
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+  <section id="module-psydac.fem.tests.analytical_profiles_1d">
+<span id="psydac-fem-tests-analytical-profiles-1d"></span><h1>psydac.fem.tests.analytical_profiles_1d<a class="headerlink" href="#module-psydac.fem.tests.analytical_profiles_1d" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AnalyticalProfile1D_Cos</span></code>([n, c])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AnalyticalProfile1D_Poly</span></code>(deg)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AnalyticalProfile1D_Sin</span></code>([n, c])</p></td>
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+<td><p></p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="psydac.fem.tests.test_eval_fields_parallel.html#module-psydac.fem.tests.test_eval_fields_parallel" title="psydac.fem.tests.test_eval_fields_parallel"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.tests.test_eval_fields_parallel</span></code></a></p></td>
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+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.fem.tests.test_splines.html#module-psydac.fem.tests.test_splines" title="psydac.fem.tests.test_splines"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.tests.test_splines</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.fem.tests.test_splines_par.html#module-psydac.fem.tests.test_splines_par" title="psydac.fem.tests.test_splines_par"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.fem.tests.test_splines_par</span></code></a></p></td>
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+  <section id="module-psydac.fem.tests.splines_error_bounds">
+<span id="psydac-fem-tests-splines-error-bounds"></span><h1>psydac.fem.tests.splines_error_bounds<a class="headerlink" href="#module-psydac.fem.tests.splines_error_bounds" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">spline_1d_error_bound</span></code>(profile_1d, dx, deg)</p></td>
+<td><p>Compute error bound for spline approximation of 1D analytical profile.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">spline_1d_error_bound_on_deriv</span></code>(profile_1d, ...)</p></td>
+<td><p>Compute error bound on first derivative, for spline approximation of 1D analytical profile.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">spline_2d_error_bound</span></code>(profile_2d, dx1, dx2, ...)</p></td>
+<td><p>Compute error bound for spline approximation of 2D analytical profile.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">spline_2d_error_bounds_on_grad</span></code>(profile_2d, ...)</p></td>
+<td><p>Compute error bound on gradient, for spline approximation of 2D analytical profile.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">tihomirov_error_bound</span></code>(h, deg, norm_f)</p></td>
+<td><p>Error bound in max norm for spline interpolation of periodic functions from:</p></td>
+</tr>
+</tbody>
+</table>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bezier-curves.html">Bézier curves</a></li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
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+</ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
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+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+  <section id="module-psydac.fem.tests.test_eval_fields_parallel">
+<span id="psydac-fem-tests-test-eval-fields-parallel"></span><h1>psydac.fem.tests.test_eval_fields_parallel<a class="headerlink" href="#module-psydac.fem.tests.test_eval_fields_parallel" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_eval_fields_irregular</span></code>(geometry, ...)</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_eval_fields_regular</span></code>(geometry, npts_per_cell)</p></td>
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+  <section id="module-psydac.fem.tests.test_spline_histopolation">
+<span id="psydac-fem-tests-test-spline-histopolation"></span><h1>psydac.fem.tests.test_spline_histopolation<a class="headerlink" href="#module-psydac.fem.tests.test_spline_histopolation" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compare_and_plot</span></code>(domain, f, fh[, num_pts])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">histopolate_polynomial</span></code>(basis, ncells, degree)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_histopolation_cosine</span></code>(basis, ncells, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_histopolation_exact</span></code>(basis, ncells, degree)</p></td>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.fem.tests.test_splines_par">
+<span id="psydac-fem-tests-test-splines-par"></span><h1>psydac.fem.tests.test_splines_par<a class="headerlink" href="#module-psydac.fem.tests.test_splines_par" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_2d_1</span></code>()</p></td>
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+  <section id="module-psydac.fem.tests.utilities">
+<span id="psydac-fem-tests-utilities"></span><h1>psydac.fem.tests.utilities<a class="headerlink" href="#module-psydac.fem.tests.utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">falling_factorial</span></code>(x, n)</p></td>
+<td><p>Calculate falling factorial of x.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">horner</span></code>(x, *poly_coeffs)</p></td>
+<td><p>Use Horner's Scheme to evaluate a polynomial of coefficients <a href="#id1"><span class="problematic" id="id2">*</span></a>poly_coeffs at location x.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_grid</span></code>(domain, ncells, random_fraction)</p></td>
+<td><p>Create random grid over 1D domain with given number of cells.</p></td>
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+</tbody>
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+  <section id="module-psydac.fem.vector">
+<span id="psydac-fem-vector"></span><h1>psydac.fem.vector<a class="headerlink" href="#module-psydac.fem.vector" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ProductFemSpace</span></code>(*spaces[, connectivity])</p></td>
+<td><p>Product of FEM spaces this class is used to represent FEM spaces on a multi-patch domain.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorFemSpace</span></code>(*spaces)</p></td>
+<td><p>FEM space with a vector basis defined on a single patch this class is used to represent either spaces of vector-valued fem fields, or product spaces involved in systems of equations.</p></td>
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+  <section id="module-psydac.linalg.basic">
+<span id="psydac-linalg-basic"></span><h1>psydac.linalg.basic<a class="headerlink" href="#module-psydac.linalg.basic" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComposedLinearOperator</span></code>(domain, codomain, *args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IdentityOperator</span></code>(domain[, codomain])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InverseLinearOperator</span></code>(A, **kwargs)</p></td>
+<td><p>Abstract base class for the (approximate) inverse A_inv := A^{-1} of a square matrix A.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearOperator</span></code>()</p></td>
+<td><p>Linear operator acting between two (normed) vector spaces V (domain) and W (codomain).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearSolver</span></code>()</p></td>
+<td><p>Solver for square linear system Ax=b, where x and b belong to (normed) vector space V.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PowerLinearOperator</span></code>(domain, codomain, A, n)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ScaledLinearOperator</span></code>(domain, codomain, c, A)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SumLinearOperator</span></code>(domain, codomain, *args)</p></td>
+<td><p>A sum of linear operatos acting between the same (normed) vector spaces V (domain) and W (codomain).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Vector</span></code>()</p></td>
+<td><p>Element of a (normed) vector space V.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorSpace</span></code>()</p></td>
+<td><p>Finite-dimensional vector space V with a scalar (dot) product.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ZeroOperator</span></code>(domain[, codomain])</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.linalg.block">
+<span id="psydac-linalg-block"></span><h1>psydac.linalg.block<a class="headerlink" href="#module-psydac.linalg.block" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockLinearOperator</span></code>(V1, V2[, blocks])</p></td>
+<td><p>Linear operator that can be written as blocks of other Linear Operators.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockVector</span></code>(V[, blocks])</p></td>
+<td><p>Block of Vectors, which is an element of a BlockVectorSpace.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockVectorSpace</span></code>(*spaces[, connectivity])</p></td>
+<td><p>Product Vector Space V of two Vector Spaces (V1,V2) or more.</p></td>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.linalg.direct_solvers">
+<span id="psydac-linalg-direct-solvers"></span><h1>psydac.linalg.direct_solvers<a class="headerlink" href="#module-psydac.linalg.direct_solvers" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BandedSolver</span></code>(u, l, bmat[, transposed])</p></td>
+<td><p>Solve the equation Ax = b for x, assuming A is banded matrix.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SparseSolver</span></code>(spmat[, transposed])</p></td>
+<td><p>Solve the equation Ax = b for x, assuming A is scipy sparse matrix.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
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+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
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+  <section id="module-psydac.linalg.fft">
+<span id="psydac-linalg-fft"></span><h1>psydac.linalg.fft<a class="headerlink" href="#module-psydac.linalg.fft" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedDCT</span></code>(space[, norm, workers, ttype])</p></td>
+<td><p>Equals an n-dimensional DCT operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedDST</span></code>(space[, norm, workers, ttype])</p></td>
+<td><p>Equals an n-dimensional DST operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedFFT</span></code>(space[, norm, workers])</p></td>
+<td><p>Equals an n-dimensional FFT operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedFFTBase</span></code>(space, functions)</p></td>
+<td><p>A base class for the distributed FFT, DCT and DST.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedIDCT</span></code>(space[, norm, workers, ttype])</p></td>
+<td><p>Equals an n-dimensional IDCT operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedIDST</span></code>(space[, norm, workers, ttype])</p></td>
+<td><p>Equals an n-dimensional IDST operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedIFFT</span></code>(space[, norm, workers])</p></td>
+<td><p>Equals an n-dimensional IFFT operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
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+  <section id="module-psydac.linalg">
+<span id="psydac-linalg"></span><h1>psydac.linalg<a class="headerlink" href="#module-psydac.linalg" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.basic.html#module-psydac.linalg.basic" title="psydac.linalg.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.basic</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.block.html#module-psydac.linalg.block" title="psydac.linalg.block"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.block</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.direct_solvers.html#module-psydac.linalg.direct_solvers" title="psydac.linalg.direct_solvers"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.direct_solvers</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.fft.html#module-psydac.linalg.fft" title="psydac.linalg.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.fft</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.html#module-psydac.linalg.kernels" title="psydac.linalg.kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.kernels</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.kron.html#module-psydac.linalg.kron" title="psydac.linalg.kron"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.kron</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.solvers.html#module-psydac.linalg.solvers" title="psydac.linalg.solvers"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.solvers</span></code></a></p></td>
+<td><p>This module provides iterative solvers and preconditioners.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.stencil.html#module-psydac.linalg.stencil" title="psydac.linalg.stencil"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.stencil</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.linalg.tests.html#module-psydac.linalg.tests" title="psydac.linalg.tests"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.tests</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.topetsc.html#module-psydac.linalg.topetsc" title="psydac.linalg.topetsc"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.topetsc</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.utilities.html#module-psydac.linalg.utilities" title="psydac.linalg.utilities"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.utilities</span></code></a></p></td>
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+                  
+  <section id="module-psydac.linalg.kernels.axpy_kernels">
+<span id="psydac-linalg-kernels-axpy-kernels"></span><h1>psydac.linalg.kernels.axpy_kernels<a class="headerlink" href="#module-psydac.linalg.kernels.axpy_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">axpy_1d</span></code></p></td>
+<td><p>Kernel for computing y = alpha * x + y.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">axpy_2d</span></code></p></td>
+<td><p>Kernel for computing y = alpha * x + y.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">axpy_3d</span></code></p></td>
+<td><p>Kernel for computing y = alpha * x + y.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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new file mode 100644
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter0/cad.html">Computer Aided Design</a><input class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-1"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/install.html">Installation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
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+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/multiphysics.html">Multiphysics</a></li>
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+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.axpy_kernels.html#module-psydac.linalg.kernels.axpy_kernels" title="psydac.linalg.kernels.axpy_kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.kernels.axpy_kernels</span></code></a></p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.inner_kernels.html#module-psydac.linalg.kernels.inner_kernels" title="psydac.linalg.kernels.inner_kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.kernels.inner_kernels</span></code></a></p></td>
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+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.stencil2coo_kernels.html#module-psydac.linalg.kernels.stencil2coo_kernels" title="psydac.linalg.kernels.stencil2coo_kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.linalg.kernels.stencil2coo_kernels</span></code></a></p></td>
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+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
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+<span id="psydac-linalg-kernels-inner-kernels"></span><h1>psydac.linalg.kernels.inner_kernels<a class="headerlink" href="#module-psydac.linalg.kernels.inner_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">inner_1d</span></code></p></td>
+<td><p>Kernel for computing the inner product (case of two 1D vectors).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">inner_2d</span></code></p></td>
+<td><p>Kernel for computing the inner product (case of two 2D vectors).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">inner_3d</span></code></p></td>
+<td><p>Kernel for computing the inner product (case of two 3D vectors).</p></td>
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+  <section id="module-psydac.linalg.kernels.stencil2coo_kernels">
+<span id="psydac-linalg-kernels-stencil2coo-kernels"></span><h1>psydac.linalg.kernels.stencil2coo_kernels<a class="headerlink" href="#module-psydac.linalg.kernels.stencil2coo_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_1d_C</span></code></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_1d_F</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_2d_C</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_2d_F</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_3d_C</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_3d_F</span></code></p></td>
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+  <section id="module-psydac.linalg.kernels.transpose_kernels">
+<span id="psydac-linalg-kernels-transpose-kernels"></span><h1>psydac.linalg.kernels.transpose_kernels<a class="headerlink" href="#module-psydac.linalg.kernels.transpose_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">interface_transpose_1d</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">interface_transpose_2d</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">interface_transpose_3d</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">transpose_1d</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">transpose_2d</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">transpose_3d</span></code></p></td>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
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+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
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+  <section id="module-psydac.linalg.kron">
+<span id="psydac-linalg-kron"></span><h1>psydac.linalg.kron<a class="headerlink" href="#module-psydac.linalg.kron" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">kronecker_solve</span></code>(solvers, rhs[, out])</p></td>
+<td><p>Solve linear system Ax=b with A=kron( A_n, A_{n-1}, ..., A_2, A_1 ), given $n$ separate linear solvers $L_n$ for the 1D problems $A_n x_n = b_n$:</p></td>
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+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">KroneckerDenseMatrix</span></code>(V, W, *args[, with_pads])</p></td>
+<td><p>Kronecker product of 1D dense matrices.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">KroneckerLinearSolver</span></code>(V, W, solvers)</p></td>
+<td><p>A solver for Ax=b, where A is a Kronecker matrix from arbirary dimension d, defined by d solvers.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">KroneckerStencilMatrix</span></code>(V, W, *args)</p></td>
+<td><p>Kronecker product of 1D stencil matrices.</p></td>
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+  <section id="module-psydac.linalg.solvers">
+<span id="psydac-linalg-solvers"></span><h1>psydac.linalg.solvers<a class="headerlink" href="#module-psydac.linalg.solvers" title="Link to this heading">#</a></h1>
+<p>This module provides iterative solvers and preconditioners.</p>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">inverse</span></code>(A, solver, **kwargs)</p></td>
+<td><p>A function to create objects of all InverseLinearOperator subclasses.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BiConjugateGradient</span></code>(A, *[, x0, tol, ...])</p></td>
+<td><p>Biconjugate Gradient (BiCG).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BiConjugateGradientStabilized</span></code>(A, *[, x0, ...])</p></td>
+<td><p>Biconjugate Gradient Stabilized (BiCGStab).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ConjugateGradient</span></code>(A, *[, x0, tol, maxiter, ...])</p></td>
+<td><p>Conjugate Gradient (CG).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GMRES</span></code>(A, *[, x0, tol, maxiter, verbose, recycle])</p></td>
+<td><p>Generalized Minimal Residual (GMRES).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LSMR</span></code>(A, *[, x0, tol, atol, btol, maxiter, ...])</p></td>
+<td><p>Least Squares Minimal Residual (LSMR).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MinimumResidual</span></code>(A, *[, x0, tol, maxiter, ...])</p></td>
+<td><p>Minimum Residual (MinRes).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PBiConjugateGradientStabilized</span></code>(A, *[, pc, ...])</p></td>
+<td><p>Preconditioned Biconjugate Gradient Stabilized (PBiCGStab).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PConjugateGradient</span></code>(A, *[, pc, x0, tol, ...])</p></td>
+<td><p>Preconditioned Conjugate Gradient (PCG).</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+                  
+  <section id="module-psydac.linalg.stencil">
+<span id="psydac-linalg-stencil"></span><h1>psydac.linalg.stencil<a class="headerlink" href="#module-psydac.linalg.stencil" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_diag_len</span></code>(pads, shifts_domain, ...[, ...])</p></td>
+<td><p>Compute the diagonal length and the padding of the stencil matrix for each direction, using the shifts of the domain and the codomain.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">flip_axis</span></code>(index, n)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilDiagonalMatrix</span></code>(V, W, data)</p></td>
+<td><p>Linear operator which operates between stencil vector spaces, and which can be represented by a matrix with non-zero entries only on its main diagonal.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilInterfaceMatrix</span></code>(V, W, s_d, s_c, ...)</p></td>
+<td><p>Matrix in n-dimensional stencil format for an interface.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilMatrix</span></code>(V, W[, pads, backend])</p></td>
+<td><p>Matrix in n-dimensional stencil format.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilVector</span></code>(V)</p></td>
+<td><p>Vector in n-dimensional stencil format.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilVectorSpace</span></code>(cart[, dtype])</p></td>
+<td><p>Vector space for n-dimensional stencil format.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<span id="psydac-linalg-tests-test-block"></span><h1>psydac.linalg.tests.test_block<a class="headerlink" href="#module-psydac.linalg.tests.test_block" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_global_starts_ends</span></code>(...)</p></td>
+<td><p></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_2D_block_linear_operator_serial_init</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_2D_block_serial_math</span></code>(dtype, npts, p, P1, P2)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_2D_block_vector_serial_init</span></code>(dtype, n1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_2D_block_vector_space_serial_init</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_3D_block_serial_basic_operator</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_block_2d_array_to_psydac_1</span></code>(dtype, n1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_block_2d_array_to_psydac_2</span></code>(dtype, n1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_block_linear_operator_dot_backend</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_block_linear_operator_parallel_dot</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_block_linear_operator_serial_dot</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_block_matrix_operator_parallel_dot_backend</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_block_serial_dimension</span></code>(ndim, p, P1, P2, ...)</p></td>
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+<span id="psydac-linalg-tests-test-fft"></span><h1>psydac.linalg.tests.test_fft<a class="headerlink" href="#module-psydac.linalg.tests.test_fft" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_global_starts_ends</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">decode_fft_type</span></code>(ffttype)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">method_test</span></code>(seed, comm, config, dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_fft_par</span></code>(seed, params, ffttype)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_fft_ser</span></code>(seed, params, ffttype)</p></td>
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new file mode 100644
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+  <section id="module-psydac.linalg.tests.test_kron_direct_solver">
+<span id="psydac-linalg-tests-test-kron-direct-solver"></span><h1>psydac.linalg.tests.test_kron_direct_solver<a class="headerlink" href="#module-psydac.linalg.tests.test_kron_direct_solver" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compare_solve</span></code>(seed, comm, npts, pads, ...[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_global_starts_ends</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_M1_block_kron_solver</span></code>(V1, ncells, degree, ...)</p></td>
+<td><p>Given a 3D DeRham sequenece (V0 = H(grad) --grad--&gt; V1 = H(curl) --curl--&gt; V2 = H(div) --div--&gt; V3 = L2) discretized using ncells, degree and periodic,</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_inverse_mass_matrices</span></code>(derham_h, domain_h)</p></td>
+<td><p>Given a 2D DeRham sequence (V0 = H(grad) --curl--&gt; V1 = H(div) --div--&gt; V2 = L2) and its discrete domain, which shall be rectangular, returns the inverse of the mass matrices for all three spaces using kronecker solvers.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">kron_solve_seq_ref</span></code>(Y, A, transposed)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">matrix_to_bandsolver</span></code>(A)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">matrix_to_sparse</span></code>(A)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_matrix</span></code>(seed, domain, codomain)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_vectordata</span></code>(seed, npts, dtype)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_2d_mass_solver</span></code>(ncells, degree, bounds, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_3d_m1_solver</span></code>(ncells, degree, periodic)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_direct_solvers</span></code>(dtype, seed, n, p, P, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_solver_1d_par</span></code>(seed, n1, p1, P1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_solver_1d_ser</span></code>(dtype, seed, n1, p1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_solver_2d_par</span></code>(seed, n1, n2, p1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_solver_2d_ser</span></code>(dtype, seed, n1, n2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_solver_2d_transposed_par</span></code>(seed, n1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_solver_2d_transposed_ser</span></code>(seed, n1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_solver_3d_par</span></code>(seed, n1, n2, n3, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_solver_3d_ser</span></code>(seed, n1, n2, n3, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_solver_nd_par</span></code>(seed, dim, dtype)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_solver_nd_ser</span></code>(seed, dim, dtype)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_kron_solver_nompi</span></code>(seed, params, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">to_bnd</span></code>(A)</p></td>
+<td><p></p></td>
+</tr>
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+  <section id="module-psydac.linalg.tests.test_linalg">
+<span id="psydac-linalg-tests-test-linalg"></span><h1>psydac.linalg.tests.test_linalg<a class="headerlink" href="#module-psydac.linalg.tests.test_linalg" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">array_equal</span></code>(a, b)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_global_starts_ends</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_StencilVectorSpace</span></code>(n1, n2, p1, p2, P1, P2)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_positive_definite_stencilmatrix</span></code>(V)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_pos_def</span></code>(A)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sparse_equal</span></code>(a, b)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_in_place_operations</span></code>(n1, n2, p1, p2[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_internal_storage</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_inverse_transpose_interaction</span></code>(n1, n2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_operator_evaluation</span></code>(n1, n2, p1, p2)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_positive_definite_matrix</span></code>(n1, n2, p1, p2)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_square_block_basic</span></code>(n1, n2, p1, p2[, P1, P2])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_square_stencil_basic</span></code>(n1, n2, p1, p2[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_x0update</span></code>(solver)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">define_data</span></code>(n, p, matrix_data[, dtype])</p></td>
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+<span id="psydac-linalg-tests-test-stencil-interface-matrix"></span><h1>psydac.linalg.tests.test_stencil_interface_matrix<a class="headerlink" href="#module-psydac.linalg.tests.test_stencil_interface_matrix" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_global_starts_ends</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_minus_starts_ends</span></code>(plus_starts, ...)</p></td>
+<td><p>Compute the coefficients needed by the minus patch in a given interface.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_plus_starts_ends</span></code>(minus_starts, ...)</p></td>
+<td><p>Compute the coefficients needed by the plus patch in a given interface.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_interface_matrix_1d_serial_init</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_interface_matrix_2d_parallel_dot</span></code>(n1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_interface_matrix_2d_serial_init</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_interface_matrix_3d_serial_init</span></code>(...)</p></td>
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+  <section id="module-psydac.linalg.tests.test_stencil_matrix">
+<span id="psydac-linalg-tests-test-stencil-matrix"></span><h1>psydac.linalg.tests.test_stencil_matrix<a class="headerlink" href="#module-psydac.linalg.tests.test_stencil_matrix" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_global_starts_ends</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_1d_parallel_backend_dot</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_1d_parallel_dot</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_1d_parallel_sync</span></code>(n1, p1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_1d_parallel_toarray</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_1d_parallel_transpose</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_1d_serial_dot</span></code>(dtype, n1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_1d_serial_init</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_1d_serial_spurious_entries</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_1d_serial_toarray</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_1d_serial_transpose</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_basic_ops</span></code>(dtype, n1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_copy</span></code>(dtype, n1, n2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_math</span></code>(dtype, n1, n2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_parallel_backend_dot</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_parallel_dot</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_parallel_sync</span></code>(n1, n2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_parallel_transpose</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_backend_dot_1</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_backend_dot_2</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_backend_dot_3</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_backend_switch</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_dot_1</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_dot_2</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_dot_3</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_dot_4</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_dot_5</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_dot_6</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
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+<td><p></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_spurious_entries</span></code>(...)</p></td>
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+<td><p></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_tosparse</span></code>(...)</p></td>
+<td><p></p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_transpose_1</span></code>(...)</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_transpose_2</span></code>(...)</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_transpose_3</span></code>(...)</p></td>
+<td><p></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_transpose_4</span></code>(...)</p></td>
+<td><p></p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_transpose_5</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_transpose_6</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_transpose_7</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_transpose_8</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_transpose_9</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_vdot</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_3d_serial_init</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_3d_serial_transpose_1</span></code>(...)</p></td>
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+  <section id="module-psydac.linalg.tests.test_stencil_vector">
+<span id="psydac-linalg-tests-test-stencil-vector"></span><h1>psydac.linalg.tests.test_stencil_vector<a class="headerlink" href="#module-psydac.linalg.tests.test_stencil_vector" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_global_starts_ends</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_matrix_2d_serial_toarray</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_1d_parallel_init</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_2d_array_to_psydac</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_2d_basic_ops</span></code>(dtype, n1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_2d_parallel_dot</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_2d_parallel_init</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_2d_parallel_toarray</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_2d_serial_conjugate</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_2d_serial_copy</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_2d_serial_dot</span></code>(dtype, n1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_2d_serial_init</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_2d_serial_math</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_2d_serial_update_ghost_region_interior</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_3d_parallel_dot</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_3d_parallel_init</span></code>(dtype, ...)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_space_1d_serial_init</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_space_2D_parallel_parent</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_space_2D_serial_set_interface</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_space_2D_serial_zeros</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_space_2d_parallel_init</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_space_2d_serial_init</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_space_3D_serial_parent</span></code>(...)</p></td>
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+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_space_3d_parallel_init</span></code>(...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stencil_vector_space_3d_serial_init</span></code>(...)</p></td>
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+  <section id="module-psydac.linalg.topetsc">
+<span id="psydac-linalg-topetsc"></span><h1>psydac.linalg.topetsc<a class="headerlink" href="#module-psydac.linalg.topetsc" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">flatten_vec</span></code>(vec)</p></td>
+<td><p>Return the flattened 1D array values and indices owned by the process of the given vector.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">mat_topetsc</span></code>(mat)</p></td>
+<td><p>Convert Psydac Matrix to PETSc data structure.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">vec_topetsc</span></code>(vec)</p></td>
+<td><p>Convert Psydac Vector to PETSc data structure.</p></td>
+</tr>
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+  <section id="module-psydac.linalg.utilities">
+<span id="psydac-linalg-utilities"></span><h1>psydac.linalg.utilities<a class="headerlink" href="#module-psydac.linalg.utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">array_to_psydac</span></code>(x, Xh)</p></td>
+<td><p>converts a numpy array to StencilVector or BlockVector format</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">petsc_to_psydac</span></code>(vec, Xh)</p></td>
+<td><p>converts a petsc Vec object to a StencilVector or a BlockVector format.</p></td>
+</tr>
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+</section>
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+  <section id="module-psydac.mapping.discrete">
+<span id="psydac-mapping-discrete"></span><h1>psydac.mapping.discrete<a class="headerlink" href="#module-psydac.mapping.discrete" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_string</span></code>(n)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NurbsMapping</span></code>(*components[, name])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SplineMapping</span></code>(*components[, name])</p></td>
+<td><p></p></td>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.mapping.discrete_gallery">
+<span id="psydac-mapping-discrete-gallery"></span><h1>psydac.mapping.discrete_gallery<a class="headerlink" href="#module-psydac.mapping.discrete_gallery" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discrete_mapping</span></code>(mapping, ncells, degree, ...)</p></td>
+<td><p></p></td>
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+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Collela3D</span></code>(name[, dim])</p></td>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/install.html">Installation</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.mapping.tests">
+<span id="psydac-mapping-tests"></span><h1>psydac.mapping.tests<a class="headerlink" href="#module-psydac.mapping.tests" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.mapping.tests.test_discrete_mapping.html#module-psydac.mapping.tests.test_discrete_mapping" title="psydac.mapping.tests.test_discrete_mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.mapping.tests.test_discrete_mapping</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.mapping.tests.visual_test_discrete_mapping_2d.html#module-psydac.mapping.tests.visual_test_discrete_mapping_2d" title="psydac.mapping.tests.visual_test_discrete_mapping_2d"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.mapping.tests.visual_test_discrete_mapping_2d</span></code></a></p></td>
+<td><p></p></td>
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+  <section id="module-psydac.mapping.tests.test_discrete_mapping">
+<span id="psydac-mapping-tests-test-discrete-mapping"></span><h1>psydac.mapping.tests.test_discrete_mapping<a class="headerlink" href="#module-psydac.mapping.tests.test_discrete_mapping" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
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+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_build_mesh_i</span></code>(geometry_file, npts_i)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_build_mesh_reg</span></code>(geometry_file, npts_per_cell)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_nurbs_circle</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_parallel_jacobians_irregular</span></code>(geometry, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_parallel_jacobians_regular</span></code>(geometry, ...)</p></td>
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new file mode 100644
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+  <section id="module-psydac.mapping.tests.visual_test_discrete_mapping_3d_surface">
+<span id="psydac-mapping-tests-visual-test-discrete-mapping-3d-surface"></span><h1>psydac.mapping.tests.visual_test_discrete_mapping_3d_surface<a class="headerlink" href="#module-psydac.mapping.tests.visual_test_discrete_mapping_3d_surface" title="Link to this heading">#</a></h1>
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+  <section id="module-psydac.polar.c1_cart">
+<span id="psydac-polar-c1-cart"></span><h1>psydac.polar.c1_cart<a class="headerlink" href="#module-psydac.polar.c1_cart" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">C1_Cart</span></code>(cart[, radial_dim])</p></td>
+<td><p>A CartDecomposition subclass whose objects are generated from an existing CartDecomposition object by removing all degrees of freedom corresponding to i_d = 0, 1 with d being the radial dimension.</p></td>
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+<ul class="nav bd-sidenav">
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+<span id="psydac-polar-c1-linops"></span><h1>psydac.polar.c1_linops<a class="headerlink" href="#module-psydac.polar.c1_linops" title="Link to this heading">#</a></h1>
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+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearOperator_StencilToDense</span></code>(V, W, data)</p></td>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
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+</ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
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+  <section id="module-psydac.polar.c1_spaces">
+<span id="psydac-polar-c1-spaces"></span><h1>psydac.polar.c1_spaces<a class="headerlink" href="#module-psydac.polar.c1_spaces" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">new_c1_vector_space</span></code>(V[, radial_dim, angle_dim])</p></td>
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+<span id="psydac-polar-dense"></span><h1>psydac.polar.dense<a class="headerlink" href="#module-psydac.polar.dense" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DenseMatrix</span></code>(V, W, data)</p></td>
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+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DenseVector</span></code>(V, data)</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DenseVectorSpace</span></code>(n, *[, dtype, cart, ...])</p></td>
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+<span id="psydac-polar"></span><h1>psydac.polar<a class="headerlink" href="#module-psydac.polar" title="Link to this heading">#</a></h1>
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+</ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<ul class="nav bd-sidenav">
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+  <section id="module-psydac.polar.tests.test_c1_projections">
+<span id="psydac-polar-tests-test-c1-projections"></span><h1>psydac.polar.tests.test_c1_projections<a class="headerlink" href="#module-psydac.polar.tests.test_c1_projections" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_c1_projections</span></code>(degrees, ncells[, verbose])</p></td>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
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+    <h1>psydac.pyccel.ast.builtins</h1>
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+  <section id="module-psydac.pyccel.ast.builtins">
+<span id="psydac-pyccel-ast-builtins"></span><h1>psydac.pyccel.ast.builtins<a class="headerlink" href="#module-psydac.pyccel.ast.builtins" title="Link to this heading">#</a></h1>
+<p>The Python interpreter has a number of built-in functions and types that are
+always available.</p>
+<p>In this module we implement some of them in alphabetical order.</p>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">python_builtin_datatype</span></code>(name)</p></td>
+<td><p>Given a symbol name, return the corresponding datatype.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Enumerate</span></code>(arg)</p></td>
+<td><p>Represents the enumerate stmt</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Len</span></code>(*args)</p></td>
+<td><p>Represents a 'len' expression in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">List</span></code>(*args, **kwargs)</p></td>
+<td><p>Represent lists in the code with dynamic memory management.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Map</span></code>(*args)</p></td>
+<td><p>Represents the map stmt</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Print</span></code>(expr)</p></td>
+<td><p>Represents a print function in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PythonBool</span></code>(arg)</p></td>
+<td><p>Represents a call to Python's native bool() function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PythonComplex</span></code>(arg0[, arg1])</p></td>
+<td><p>Represents a call to Python's native complex() function.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PythonFloat</span></code>(arg)</p></td>
+<td><p>Represents a call to Python's native float() function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PythonInt</span></code>(arg)</p></td>
+<td><p>Represents a call to Python's native int() function.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PythonTuple</span></code>(*args)</p></td>
+<td><p>Represents a call to Python's native tuple() function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Range</span></code>(*args)</p></td>
+<td><p>Represents a range.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Zip</span></code>(*args)</p></td>
+<td><p>Represents a zip stmt.</p></td>
+</tr>
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+</table>
+</section>
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+  <section id="module-psydac.pyccel.ast.core">
+<span id="psydac-pyccel-ast-core"></span><h1>psydac.pyccel.ast.core<a class="headerlink" href="#module-psydac.pyccel.ast.core" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">allocatable_like</span></code>(expr[, verbose])</p></td>
+<td><p>finds attributes of an expression</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">broadcast</span></code>(shape_1, shape_2)</p></td>
+<td><p>This function broadcast two shapes using numpy broadcasting rules</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">create_incremented_string</span></code>(forbidden_exprs[, ...])</p></td>
+<td><p>This function takes a prefix and a counter and uses them to construct a new name of the form:         prefix_counter Where counter is formatted to fill 4 characters The new name is checked against a list of forbidden expressions. If the constructed name is forbidden then the counter is incremented until a valid name is found.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">create_variable</span></code>(forbidden_names[, prefix, ...])</p></td>
+<td><p>This function takes a prefix and a counter and uses them to construct a Symbol with a name of the form:         prefix_counter Where counter is formatted to fill 4 characters The new name is checked against a list of forbidden expressions. If the constructed name is forbidden then the counter is incremented until a valid name is found.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">extract_subexpressions</span></code>(expr)</p></td>
+<td><p>this function takes an expression and returns a list of statements if this expression contains sub expressions that need to be evaluated outside of the expression</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">float2int</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_assigned_symbols</span></code>(expr)</p></td>
+<td><p>Returns all assigned symbols (as sympy Symbol) in the AST.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_initial_value</span></code>(expr, var)</p></td>
+<td><p>Returns the first assigned value to var in the Expression expr.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_iterable_ranges</span></code>(it[, var_name])</p></td>
+<td><p>Returns ranges of an iterable object.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">handle_precedence</span></code>(args, my_precedence)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">inline</span></code>(func, args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">int2float</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_simple_assign</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">operator</span></code>(op)</p></td>
+<td><p>Returns the operator singleton for the given operator</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">process_shape</span></code>(shape)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">subs</span></code>(expr, new_elements)</p></td>
+<td><p>Substitutes old for new in an expression after sympifying args.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AddOp</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AliasAssign</span></code>(lhs, rhs)</p></td>
+<td><p>Represents aliasing for code generation.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AnnotatedArgument</span></code>(name, annotation)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AnnotatedComment</span></code>(accel, txt)</p></td>
+<td><p>Represents a Annotated Comment in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Argument</span></code>(name, *[, kwonly])</p></td>
+<td><p>An abstract Argument data structure.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AsName</span></code>(name, target)</p></td>
+<td><p>Represents a renaming of a variable, used with Import.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Assert</span></code>(test)</p></td>
+<td><p>Represents a assert statement in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Assign</span></code>(lhs, rhs[, strict, status, like])</p></td>
+<td><p>Represents variable assignment for code generation.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AugAssign</span></code>(lhs, op, rhs[, strict, status, like])</p></td>
+<td><p>Represents augmented variable assignment for code generation.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Block</span></code>(name, variables, body)</p></td>
+<td><p>Represents a block in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Break</span></code>(*args)</p></td>
+<td><p>Represents a break in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ClassDef</span></code>(name[, attributes, methods, ...])</p></td>
+<td><p>Represents a class definition.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CodeBlock</span></code>(body)</p></td>
+<td><p>Represents a list of stmt for code generation.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Comment</span></code>(text)</p></td>
+<td><p>Represents a Comment in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CommentBlock</span></code>(txt)</p></td>
+<td><p>Represents a Block of Comments</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Concatenate</span></code>(args, is_list)</p></td>
+<td><p>Represents the String concatination operation.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Constant</span></code>(*args, **kwargs)</p></td>
+<td><p>Examples</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ConstructorCall</span></code>(func, arguments[, ...])</p></td>
+<td><p>It  serves as a constructor for undefined function classes.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Continue</span></code>(*args)</p></td>
+<td><p>Represents a continue in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Declare</span></code>(dtype, variable[, intent, value, static])</p></td>
+<td><p>Represents a variable declaration in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Del</span></code>(expr)</p></td>
+<td><p>Represents a memory deallocation in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DivOp</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Dlist</span></code>(val, length)</p></td>
+<td><p>this is equivalent to the zeros function of numpy arrays for the python list.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DoConcurrent</span></code>(target, iter, body[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DottedName</span></code>(*args)</p></td>
+<td><p>Represents a dotted variable.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DottedVariable</span></code>(lhs, rhs)</p></td>
+<td><p>Represents a dotted variable.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">EmptyNode</span></code>()</p></td>
+<td><p>Represents an empty node in the abstract syntax tree (AST).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ErrorExit</span></code>(*args)</p></td>
+<td><p>Exist with error.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Eval</span></code>(*args)</p></td>
+<td><p>Basic class for eval instruction.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Exit</span></code>(*args)</p></td>
+<td><p>Basic class for exists.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">F2PYFunctionDef</span></code>(name, arguments, results, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">For</span></code>(target, iter, body[, local_vars, strict])</p></td>
+<td><p>Represents a 'for-loop' in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ForAll</span></code>(iter, target, mask, body)</p></td>
+<td><p>class that represents the forall statement in fortran</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ForIterator</span></code>(target, iter, body[, strict])</p></td>
+<td><p>Class that describes iterable classes defined by the user.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionCall</span></code>(func, args[, current_function])</p></td>
+<td><p>Represents a function call in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionDef</span></code>(name, arguments, results, body, ...)</p></td>
+<td><p>Represents a function definition.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GetDefaultFunctionArg</span></code>(arg, func)</p></td>
+<td><p>Creates a FunctionDef for handling optional arguments in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">If</span></code>(*args)</p></td>
+<td><p>Represents a if statement in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IfTernaryOperator</span></code>(*args)</p></td>
+<td><p>class for the Ternery operator</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Import</span></code>(source[, target])</p></td>
+<td><p>Represents inclusion of dependencies in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexedElement</span></code>(base, *args, **kw_args)</p></td>
+<td><p>Represents a mathematical object with indices.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexedVariable</span></code>(label[, shape, dtype, prec, ...])</p></td>
+<td><p>Represents an indexed variable, like x in x[i], in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Interface</span></code>(name, functions[, hide])</p></td>
+<td><p>Represent an Interface</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Is</span></code>(*args)</p></td>
+<td><p>Represents a is expression in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IsNot</span></code>(*args)</p></td>
+<td><p>Represents a is expression in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Load</span></code>(module[, funcs, as_lambda, nargs])</p></td>
+<td><p>Similar to 'importlib' in python.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ModOp</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Module</span></code>(*args, **kwargs)</p></td>
+<td><p>Represents a module in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MulOp</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeOp</span></code>(*args, **kwargs)</p></td>
+<td><p>Base type for native operands.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NewLine</span></code>()</p></td>
+<td><p>Represents a NewLine in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Nil</span></code>(*args)</p></td>
+<td><p>class for None object in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ParallelBlock</span></code>(clauses, variables, body)</p></td>
+<td><p>Represents a parallel block in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ParallelRange</span></code>(*args)</p></td>
+<td><p>Representes a parallel range using OpenMP/OpenACC.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ParserResult</span></code>([program, module, mod_name, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Pass</span></code>(*args)</p></td>
+<td><p>Basic class for pass instruction.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Product</span></code>(*args)</p></td>
+<td><p>Represents a Product stmt.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Program</span></code>(name, variables, body[, imports])</p></td>
+<td><p>Represents a Program in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelAdd</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelAnd</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelArraySize</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelAssociativeParenthesis</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelBitAnd</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelBitOperator</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelBitOr</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelBitXor</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelBooleanOperator</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelDiv</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelEq</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelFloorDiv</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelGe</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelGt</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelInvert</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelLShift</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelLe</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelLt</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelMinus</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelMod</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelMul</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelNe</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelNot</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelOperator</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelOr</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelPow</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelRShift</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelUnary</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelUnarySub</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PythonFunction</span></code>(name, arguments, results, ...)</p></td>
+<td><p>Represents a Python-Function definition.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Raise</span></code>(*args)</p></td>
+<td><p>Represents a raise in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Random</span></code>(*args)</p></td>
+<td><p>Represents a 'random' number in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Return</span></code>(expr[, stmt])</p></td>
+<td><p>Represents a function return in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SeparatorComment</span></code>(n)</p></td>
+<td><p>Represents a Separator Comment in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Slice</span></code>(start, end)</p></td>
+<td><p>Represents a slice in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StarredArguments</span></code>(args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">String</span></code>(arg)</p></td>
+<td><p>Represents the String</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SubOp</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Subroutine</span></code>(name[, bases, __dict__])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SumFunction</span></code>(body, iterator[, stmts])</p></td>
+<td><p>Represents a Sympy Sum Function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SymbolicAssign</span></code>(lhs, rhs)</p></td>
+<td><p>Represents symbolic aliasing for code generation.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SymbolicPrint</span></code>(expr)</p></td>
+<td><p>Represents a print function of symbolic expressions in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SympyFunction</span></code>(name, arguments, results, ...)</p></td>
+<td><p>Represents a function definition.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Tensor</span></code>(*args, **kwargs)</p></td>
+<td><p>Base class for tensor.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Tile</span></code>(start, stop)</p></td>
+<td><p>Representes a tile.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TupleImport</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TupleVariable</span></code>(arg_vars, dtype, name, *args, ...)</p></td>
+<td><p>Represents a tuple variable in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ValuedArgument</span></code>(*args, **kwargs)</p></td>
+<td><p>Represents a valued argument in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ValuedVariable</span></code>(*args, **kwargs)</p></td>
+<td><p>Represents a valued variable in the code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Variable</span></code>(*args, **kwargs)</p></td>
+<td><p>Represents a typed variable.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VariableAddress</span></code>(*args)</p></td>
+<td><p>Represents the address of a variable.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Void</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VoidFunction</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">While</span></code>(test, body[, local_vars])</p></td>
+<td><p>Represents a 'while' statement in the code.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">With</span></code>(test, body, settings)</p></td>
+<td><p>Represents a 'with' statement in the code.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+                  
+  <section id="module-psydac.pyccel.ast.datatypes">
+<span id="psydac-pyccel-ast-datatypes"></span><h1>psydac.pyccel.ast.datatypes<a class="headerlink" href="#module-psydac.pyccel.ast.datatypes" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DataTypeFactory</span></code>(name[, argnames, BaseClass, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">datatype</span></code>(arg)</p></td>
+<td><p>Returns the datatype singleton for the given dtype.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_default_value</span></code>(dtype)</p></td>
+<td><p>Returns the default value of a native datatype.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_iterable_datatype</span></code>(dtype)</p></td>
+<td><p>Returns True if dtype is an iterable class.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_pyccel_datatype</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_with_construct_datatype</span></code>(dtype)</p></td>
+<td><p>Returns True if dtype is an with_construct class.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CustomDataType</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DataType</span></code>(*args, **kwargs)</p></td>
+<td><p>Base class representing native datatypes</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionType</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeBool</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeComplex</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeGeneric</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeInteger</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeNil</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeParallelRange</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeRange</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeReal</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeString</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeSymbol</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeTensor</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeTuple</span></code>(*args, **kwargs)</p></td>
+<td><p>Base class representing native datatypes</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NativeVoid</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">UnionType</span></code>(args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VariableType</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.pyccel.ast.functionalexpr">
+<span id="psydac-pyccel-ast-functionalexpr"></span><h1>psydac.pyccel.ast.functionalexpr<a class="headerlink" href="#module-psydac.pyccel.ast.functionalexpr" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionalFor</span></code>(loops[, expr, lhs, indices, index])</p></td>
+<td><p><p>.</p>
+</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionalMap</span></code>(loops[, expr, lhs, indices, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionalMax</span></code>(loops[, expr, lhs, indices, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionalMin</span></code>(loops[, expr, lhs, indices, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionalSum</span></code>(loops[, expr, lhs, indices, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GeneratorComprehension</span></code>(*args)</p></td>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<ul class="nav bd-sidenav">
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+  <section id="module-psydac.pyccel.ast">
+<span id="psydac-pyccel-ast"></span><h1>psydac.pyccel.ast<a class="headerlink" href="#module-psydac.pyccel.ast" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.pyccel.ast.basic.html#module-psydac.pyccel.ast.basic" title="psydac.pyccel.ast.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.pyccel.ast.basic</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.pyccel.ast.builtins.html#module-psydac.pyccel.ast.builtins" title="psydac.pyccel.ast.builtins"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.pyccel.ast.builtins</span></code></a></p></td>
+<td><p>The Python interpreter has a number of built-in functions and types that are always available.</p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.pyccel.ast.core.html#module-psydac.pyccel.ast.core" title="psydac.pyccel.ast.core"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.pyccel.ast.core</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.pyccel.ast.datatypes.html#module-psydac.pyccel.ast.datatypes" title="psydac.pyccel.ast.datatypes"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.pyccel.ast.datatypes</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="psydac.pyccel.ast.functionalexpr.html#module-psydac.pyccel.ast.functionalexpr" title="psydac.pyccel.ast.functionalexpr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.pyccel.ast.functionalexpr</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="psydac.pyccel.ast.mathext.html#module-psydac.pyccel.ast.mathext" title="psydac.pyccel.ast.mathext"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.pyccel.ast.mathext</span></code></a></p></td>
+<td><p></p></td>
+</tr>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+  <section id="module-psydac.pyccel.ast.mathext">
+<span id="psydac-pyccel-ast-mathext"></span><h1>psydac.pyccel.ast.mathext<a class="headerlink" href="#module-psydac.pyccel.ast.mathext" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathAcos</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathAcosh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathAsin</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathAsinh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathAtan</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathAtan2</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathAtanh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathCeil</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathCopysign</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathCos</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathCosh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathDegrees</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathErf</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathErfc</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathExp</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathExpm1</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathFabs</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathFactorial</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathFloor</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathFmod</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathFrexp</span></code>(*args)</p></td>
+<td><p>frexp(x)</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathFsum</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathFunctionBase</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathFunctionBool</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathFunctionFloat</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathFunctionInt</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathGamma</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathGcd</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathHypot</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathIsclose</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathIsfinite</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathIsinf</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathIsnan</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathLdexp</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathLgamma</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathLog</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathLog10</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathLog1p</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathLog2</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathModf</span></code>(*args)</p></td>
+<td><p>modf(x)</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathPow</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathRadians</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathRemainder</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathSin</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathSinh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathSqrt</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathTan</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathTanh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MathTrunc</span></code>(*args)</p></td>
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+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ImaginaryUnit</span></code>(*args)</p></td>
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+    <h1>psydac.pyccel.ast.numpyext</h1>
+    <!-- Table of contents -->
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+  <section id="module-psydac.pyccel.ast.numpyext">
+<span id="psydac-pyccel-ast-numpyext"></span><h1>psydac.pyccel.ast.numpyext<a class="headerlink" href="#module-psydac.pyccel.ast.numpyext" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Shape</span></code>(arg)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">process_dtype</span></code>(dtype)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Array</span></code>(*args)</p></td>
+<td><p>Represents a call to  numpy.array for code generation.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Complex128</span></code>(arg0[, arg1])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Complex64</span></code>(arg0[, arg1])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Cross</span></code>(*args)</p></td>
+<td><p>Represents numpy.cross.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Diag</span></code>(*args)</p></td>
+<td><p>Represents numpy.diag.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Empty</span></code>(*args)</p></td>
+<td><p>Represents a call to numpy.empty for code generation.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">EmptyLike</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Float32</span></code>(arg)</p></td>
+<td><p>Represents a call to numpy.float32() function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Float64</span></code>(arg)</p></td>
+<td><p>Represents a call to numpy.float64() function.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Full</span></code>(*args)</p></td>
+<td><p>Represents a call to numpy.full for code generation.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FullLike</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Imag</span></code>(*args)</p></td>
+<td><p>Represents a call to  numpy.imag for code generation.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Int32</span></code>([arg, base])</p></td>
+<td><p>Represents a call to numpy.int32() function.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Int64</span></code>([arg, base])</p></td>
+<td><p>Represents a call to numpy.int64() function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Linspace</span></code>(*args)</p></td>
+<td><p>Represents numpy.linspace.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Matmul</span></code>(*args)</p></td>
+<td><p>Represents a call to numpy.matmul for code generation.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Norm</span></code>(*args)</p></td>
+<td><p>Represents call to numpy.norm</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyAbs</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyArccos</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyArccosh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyArcsin</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyArcsinh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyArctan</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyArctan2</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyArctanh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyComplex</span></code>(arg0[, arg1])</p></td>
+<td><p>Represents a call to numpy.complex() function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyCos</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyCosh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyExp</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyFloat</span></code>(arg)</p></td>
+<td><p>Represents a call to numpy.float() function.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyFloor</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyHypot</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyInt</span></code>([arg, base])</p></td>
+<td><p>Represents a call to numpy.int() function.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyLog</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyMax</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyMin</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyMod</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyNewArray</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyRandint</span></code>(*args)</p></td>
+<td><p>Represents a call to  numpy.random.random or numpy.random.rand for code generation.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpySin</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpySinh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpySqrt</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpySum</span></code>(*args)</p></td>
+<td><p>Represents a call to  numpy.sum for code generation.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyTan</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyTanh</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyUfuncBase</span></code>(*args)</p></td>
+<td><p>Base class for Numpy's universal functions.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyUfuncBinary</span></code>(*args)</p></td>
+<td><p>Numpy's universal function with two arguments.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumpyUfuncUnary</span></code>(*args)</p></td>
+<td><p>Numpy's universal function with one argument.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Ones</span></code>(*args)</p></td>
+<td><p>Represents a call to numpy.ones for code generation.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">OnesLike</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Product</span></code>(*args)</p></td>
+<td><p>Represents a call to  numpy.prod for code generation.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Rand</span></code>(*args)</p></td>
+<td><p>Represents a call to  numpy.random.random or numpy.random.rand for code generation.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Real</span></code>(*args)</p></td>
+<td><p>Represents a call to  numpy.real for code generation.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Sqrt</span></code>(base)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Where</span></code>(*args)</p></td>
+<td><p>Represents a call to  numpy.where</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Zeros</span></code>(*args)</p></td>
+<td><p>Represents a call to numpy.zeros for code generation.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ZerosLike</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Singleton</span></code></p></td>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.pyccel.ast.type_inference">
+<span id="psydac-pyccel-ast-type-inference"></span><h1>psydac.pyccel.ast.type_inference<a class="headerlink" href="#module-psydac.pyccel.ast.type_inference" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">str_dtype</span></code>(dtype)</p></td>
+<td><p>This function takes a datatype and returns a sympy datatype as a string</p></td>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/install.html">Installation</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
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+  <section id="module-psydac.pyccel.ast.utilities">
+<span id="psydac-pyccel-ast-utilities"></span><h1>psydac.pyccel.ast.utilities<a class="headerlink" href="#module-psydac.pyccel.ast.utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">build_types_decorator</span></code>(args[, order])</p></td>
+<td><p>builds a types decorator from a list of arguments (of FunctionDef)</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">builtin_function</span></code>(expr[, args])</p></td>
+<td><p>Returns a builtin-function call applied to given arguments.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">builtin_import</span></code>(expr)</p></td>
+<td><p>Returns a builtin pyccel-extension function/object from an import.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">collect_relevant_imports</span></code>(func_dictionary, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_function_from_ast</span></code>(ast, func_name)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">split_positional_keyword_arguments</span></code>(*args)</p></td>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.pyccel.codegen">
+<span id="psydac-pyccel-codegen"></span><h1>psydac.pyccel.codegen<a class="headerlink" href="#module-psydac.pyccel.codegen" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
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+<span id="psydac-pyccel-codegen-printing-pycode"></span><h1>psydac.pyccel.codegen.printing.pycode<a class="headerlink" href="#module-psydac.pyccel.codegen.printing.pycode" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pycode</span></code>(expr, **settings)</p></td>
+<td><p>Converts an expr to a string of Python code Parameters ========== expr : Expr     A SymPy expression. fully_qualified_modules : bool     Whether or not to write out full module names of functions     (<code class="docutils literal notranslate"><span class="pre">math.sin</span></code> vs. <code class="docutils literal notranslate"><span class="pre">sin</span></code>). default: <code class="docutils literal notranslate"><span class="pre">True</span></code>. Examples ======== &gt;&gt;&gt; from sympy import tan, Symbol &gt;&gt;&gt; from sympy.printing.pycode import pycode &gt;&gt;&gt; pycode(tan(Symbol('x')) + 1) 'math.tan(x) + 1'.</p></td>
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+  <section id="module-psydac.pyccel.errors.errors">
+<span id="psydac-pyccel-errors-errors"></span><h1>psydac.pyccel.errors.errors<a class="headerlink" href="#module-psydac.pyccel.errors.errors" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Errors</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ErrorsMode</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">make_symbol</span></code>(s)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ErrorInfo</span></code>(filename[, line, column, ...])</p></td>
+<td><p>Representation of a single error message.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Exceptions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelCodegenError</span></code>(message[, errors])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelError</span></code>(message[, errors])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelSemanticError</span></code>(message[, errors])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PyccelSyntaxError</span></code>(message[, errors])</p></td>
+<td><p></p></td>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
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new file mode 100644
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+</ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.pyccel.symbolic.lambdify">
+<span id="psydac-pyccel-symbolic-lambdify"></span><h1>psydac.pyccel.symbolic.lambdify<a class="headerlink" href="#module-psydac.pyccel.symbolic.lambdify" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">cse</span></code>(expr)</p></td>
+<td><p>symplify a complicated sympy expression into a list of expression using the cse sympy function</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">lambdify</span></code>(expr, args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pyccel_sum</span></code>(expr)</p></td>
+<td><p>convert the sympy sum to the pyccel node FunctionalSum</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">set_fst</span></code>(expr, fst)</p></td>
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+  <section id="module-psydac.utilities">
+<span id="psydac-utilities"></span><h1>psydac.utilities<a class="headerlink" href="#module-psydac.utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.utilities.quadratures.html#module-psydac.utilities.quadratures" title="psydac.utilities.quadratures"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.utilities.quadratures</span></code></a></p></td>
+<td><p>This module contains some routines to generate quadrature points in 1D it has also a routine uniform, which generates uniform points with weights equal to 1</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.utilities.utils.html#module-psydac.utilities.utils" title="psydac.utilities.utils"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.utilities.utils</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="../../psydac/docs/source/modules/STUBDIR/psydac.utilities.vtk.html#module-psydac.utilities.vtk" title="psydac.utilities.vtk"><code class="xref py py-obj docutils literal notranslate"><span class="pre">psydac.utilities.vtk</span></code></a></p></td>
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+  <section id="module-psydac.utilities.quadratures">
+<span id="psydac-utilities-quadratures"></span><h1>psydac.utilities.quadratures<a class="headerlink" href="#module-psydac.utilities.quadratures" title="Link to this heading">#</a></h1>
+<p>This module contains some routines to generate quadrature points in 1D
+it has also a routine uniform, which generates uniform points
+with weights equal to 1</p>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">gauss_legendre</span></code>(ordergl[, tol])</p></td>
+<td><p>Returns nodal abscissas {x} and weights {A} of Gauss-Legendre m-point quadrature.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">gauss_lobatto</span></code>(k)</p></td>
+<td><p>Returns nodal abscissas {x} and weights {A} of Gauss-Legendre m-point quadrature.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">quadrature</span></code>(a, k[, method])</p></td>
+<td><p>this routine generates a quad pts on the grid linspace(a,b,N)</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
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+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/data-structure.html">Data Structure</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/install.html">Installation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
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+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/material-science.html">Material Science</a></li>
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+  <section id="module-psydac.utilities.utils">
+<span id="psydac-utilities-utils"></span><h1>psydac.utilities.utils<a class="headerlink" href="#module-psydac.utilities.utils" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">animate_field</span></code>(fields, domain, mapping[, ...])</p></td>
+<td><p>Animate a sequence of scalar fields over a geometry.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_real</span></code>(x)</p></td>
+<td><p>Determine whether the given input represents a real number.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">refine_array_1d</span></code>(x, n[, remove_duplicates])</p></td>
+<td><p>Refines a 1D array by subdividing each interval (x[i], x[i+1]) into n identical parts.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">roll_edges</span></code>(domain, points)</p></td>
+<td><p>If necessary, &quot;roll&quot; back intervals that cross boundary of periodic domain.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">split_field</span></code>(uh, spaces[, out])</p></td>
+<td><p>Split a field into a list of fields that corresponds to the spaces.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">split_space</span></code>(Xh)</p></td>
+<td><p>Split the flattened fem spaces into</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">unroll_edges</span></code>(domain, xgrid)</p></td>
+<td><p>If necessary, &quot;unroll&quot; intervals that cross boundary of periodic domain.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.utilities.vtk">
+<span id="psydac-utilities-vtk"></span><h1>psydac.utilities.vtk<a class="headerlink" href="#module-psydac.utilities.vtk" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">writeParallelVTKUnstructuredGrid</span></code>(path, ...)</p></td>
+<td><p>Writes a parallel vtk file from grid-like data: VTKStructuredGrid or VTKRectilinearGrid</p></td>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
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+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
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+  <section id="module-sympde.core.algebra">
+<span id="sympde-core-algebra"></span><h1>sympde.core.algebra<a class="headerlink" href="#module-sympde.core.algebra" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CrossBasic</span></code>(*args)</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InnerBasic</span></code>(*args)</p></td>
+<td><p></p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Inner_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Inner_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearOperator</span></code>(expr)</p></td>
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+  <section id="module-sympde.core.basic">
+<span id="sympde-core-basic"></span><h1>sympde.core.basic<a class="headerlink" href="#module-sympde.core.basic" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicDerivable</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicMapping</span></code>(label[, shape, offset, strides])</p></td>
+<td><p>Represents a basic class for mapping.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CalculusFunction</span></code>(*args)</p></td>
+<td><p>this class is needed to distinguish between functions and calculus functions when manipulating our expressions</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Constant</span></code>(*args, **kwargs)</p></td>
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+        </ul>
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Introduction</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter0/howto.html">What to expect from IGA-Python</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/install.html">Installation</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
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+  <section id="module-sympde.core.utils">
+<span id="sympde-core-utils"></span><h1>sympde.core.utils<a class="headerlink" href="#module-sympde.core.utils" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">expand_name_patterns</span></code>(names, *[, seq])</p></td>
+<td><p>Expand names according to the given pattern or list of patterns, following exactly the same rules of Sympy's &quot;symbols&quot; function.</p></td>
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+  <section id="module-sympde.expr.equation">
+<span id="sympde-expr-equation"></span><h1>sympde.expr.equation<a class="headerlink" href="#module-sympde.expr.equation" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find</span></code>(trials, *, forall, lhs, rhs[, bc, ...])</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicBoundaryCondition</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicConstraint</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Equation</span></code>(lhs, rhs, trials, tests[, bc, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">EssentialBC</span></code>(lhs, rhs, boundary[, position, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Mean</span></code>(lhs, rhs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NewtonIteration</span></code>(form, fields[, bc, trials])</p></td>
+<td><p></p></td>
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+  <section id="module-sympde.expr.errors">
+<span id="sympde-expr-errors"></span><h1>sympde.expr.errors<a class="headerlink" href="#module-sympde.expr.errors" title="Link to this heading">#</a></h1>
+<p class="rubric">Exceptions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">UnconsistentArgumentsError</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">UnconsistentBCError</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">UnconsistentError</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">UnconsistentLhsError</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">UnconsistentLinearExpressionError</span></code>(*args, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">UnconsistentRhsError</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
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+  <section id="module-sympde.expr.evaluation">
+<span id="sympde-expr-evaluation"></span><h1>sympde.expr.evaluation<a class="headerlink" href="#module-sympde.expr.evaluation" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_sequence</span></code>(a)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Advection</span></code>(axis[, weight])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AdvectionT</span></code>(axis[, weight])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Basic1dForm</span></code>(name, axis[, weight])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Bilaplacian</span></code>(axis[, weight])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BoundaryExpression</span></code>(target, expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DomainExpression</span></code>(target, expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InterfaceExpression</span></code>(target, u, v, expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">KernelExpression</span></code>(target, expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Mass</span></code>(axis[, weight])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Stiffness</span></code>(axis[, weight])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorExpr</span></code>(*_args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TerminalExpr</span></code>(expr, domain)</p></td>
+<td><p>This class takes a SymPDE expression and transforms its vector operators into atomic ones for a specified domain.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-sympde.expr.expr">
+<span id="sympde-expr-expr"></span><h1>sympde.expr.expr<a class="headerlink" href="#module-sympde.expr.expr" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">add_int</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">expand</span></code>(expr)</p></td>
+<td><p>Expand an expression by making sure that Mul objects are evaluated by using the * operator (which might be overloaded by the types in the expression).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">integral</span></code>(domain, expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_linear_expression</span></code>(expr, args[, integral, ...])</p></td>
+<td><p>checks if an expression is linear with respect to the given arguments.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">linearize</span></code>(form, fields[, trials])</p></td>
+<td><p>Parameters
+
+form
+
+LinearForm
+
+Linear form F(v; u) to be linearized. F is nonlinear in the parameter u</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">mul_add</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">mul_integral</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BilinearForm</span></code>(arguments, expr, **options)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Functional</span></code>(expr, domain[, evaluate])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IntAdd</span></code>(*args, **options)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Integral</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearExpr</span></code>(arguments, expr, **options)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearForm</span></code>(arguments, expr, **options)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Norm</span></code>(expr, domain[, kind, evaluate])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SemiNorm</span></code>(expr, domain[, kind, evaluate])</p></td>
+<td><p></p></td>
+</tr>
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+</table>
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+<ul class="nav bd-sidenav">
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+  <section id="module-sympde.expr.tests.test_equation_2d">
+<span id="sympde-expr-tests-test-equation-2d"></span><h1>sympde.expr.tests.test_equation_2d<a class="headerlink" href="#module-sympde.expr.tests.test_equation_2d" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_equation_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_equation_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_equation_2d_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_equation_2d_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_equation_2d_5</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_equation_2d_6</span></code>()</p></td>
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+  <section id="module-sympde.expr.tests.test_expr">
+<span id="sympde-expr-tests-test-expr"></span><h1>sympde.expr.tests.test_expr<a class="headerlink" href="#module-sympde.expr.tests.test_expr" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_area_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_bilinear_form_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_bilinear_form_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_bilinear_form_2d_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_bilinear_form_2d_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_functional_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_interface_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_interface_integral_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_interface_integral_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_interface_integral_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_interface_integral_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linear_expr_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linear_expr_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linear_form_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linear_form_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linearity_bilinear_form_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linearity_linear_form_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linearize_expr_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linearize_expr_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linearize_form_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linearize_form_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linearize_form_2d_3</span></code>()</p></td>
+<td><p>steady Euler equation.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_linearize_form_2d_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_norm_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_norm_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_stabilization_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_terminal_expr_bilinear_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_terminal_expr_bilinear_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_terminal_expr_bilinear_2d_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_terminal_expr_bilinear_2d_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_terminal_expr_bilinear_3d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_terminal_expr_linear_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_terminal_expr_linear_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_terminal_expr_linear_2d_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_terminal_expr_linear_2d_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_terminal_expr_linear_2d_5</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_user_function_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_tensorize_2d_1_mapping</span></code>()</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_tensorize_2d_2</span></code>()</p></td>
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+    <h1>sympde.exterior.calculus</h1>
+    <!-- Table of contents -->
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+        <div id="jb-print-toc">
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+                  
+  <section id="module-sympde.exterior.calculus">
+<span id="sympde-exterior-calculus"></span><h1>sympde.exterior.calculus<a class="headerlink" href="#module-sympde.exterior.calculus" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Adjoint</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AdjointExteriorDerivative</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AdjointInteriorProduct</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AdjointLieDerivative</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicOperator</span></code>(*args)</p></td>
+<td><p>Basic class for calculus operators.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ExteriorDerivative</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ExteriorProduct</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Hodge</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InteriorProduct</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Ld</span></code></p></td>
+<td><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">AdjointLieDerivative</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LieDerivative</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PullBack</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">d</span></code></p></td>
+<td><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">ExteriorDerivative</span></code></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">delta</span></code></p></td>
+<td><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">AdjointExteriorDerivative</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">hodge</span></code></p></td>
+<td><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">Hodge</span></code></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ip</span></code></p></td>
+<td><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">InteriorProduct</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">jp</span></code></p></td>
+<td><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">AdjointInteriorProduct</span></code></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ld</span></code></p></td>
+<td><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">LieDerivative</span></code></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">wedge</span></code></p></td>
+<td><p>alias of <code class="xref py py-class docutils literal notranslate"><span class="pre">ExteriorProduct</span></code></p></td>
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+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+  <section id="module-sympde.exterior.compiler">
+<span id="sympde-exterior-compiler"></span><h1>sympde.exterior.compiler<a class="headerlink" href="#module-sympde.exterior.compiler" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">augmented_expression</span></code>(expr, tests, atoms[, weak])</p></td>
+<td><p></p></td>
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+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ExteriorCalculusExpr</span></code>(*_args, **kwargs)</p></td>
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+    <h1>sympde.exterior.datatype</h1>
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+  <section id="module-sympde.exterior.datatype">
+<span id="sympde-exterior-datatype"></span><h1>sympde.exterior.datatype<a class="headerlink" href="#module-sympde.exterior.datatype" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FormTypeFactory</span></code>([argnames, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_index_form</span></code>(dtype)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FiveFormType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FormType</span></code>()</p></td>
+<td><p>Base class representing differential form types</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FourFormType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">OneFormType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SixFormType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ThreeFormType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TwoFormType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ZeroFormType</span></code>()</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DifferentialForm</span></code>(name, index, dim)</p></td>
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+<span id="sympde-exterior"></span><h1>sympde.exterior<a class="headerlink" href="#module-sympde.exterior" title="Link to this heading">#</a></h1>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-sympde.exterior.inference">
+<span id="sympde-exterior-inference"></span><h1>sympde.exterior.inference<a class="headerlink" href="#module-sympde.exterior.inference" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">infere_type</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<span id="sympde-exterior-tests-test-compiler"></span><h1>sympde.exterior.tests.test_compiler<a class="headerlink" href="#module-sympde.exterior.tests.test_compiler" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_compiler_3d_1</span></code>()</p></td>
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+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_compiler_3d_2</span></code>()</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_compiler_3d_poisson</span></code>()</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_compiler_3d_stokes</span></code>()</p></td>
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+<p class="rubric">Functions</p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
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+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+                <h2> Contents </h2>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sympde.printing.latex"><code class="docutils literal notranslate"><span class="pre">latex()</span></code></a></li>
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+  <section id="sympde-printing-latex">
+<h1>sympde.printing.latex<a class="headerlink" href="#sympde-printing-latex" title="Link to this heading">#</a></h1>
+<dl class="py function">
+<dt class="sig sig-object py" id="sympde.printing.latex">
+<span class="sig-prename descclassname"><span class="pre">sympde.printing.</span></span><span class="sig-name descname"><span class="pre">latex</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">expr</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">**</span></span><span class="n"><span class="pre">settings</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#sympde.printing.latex" title="Link to this definition">#</a></dt>
+<dd></dd></dl>
+
+</section>
+
+    <script type="text/x-thebe-config">
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_latex_2d_3</span></code>()</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_latex_2d_4</span></code>()</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_latex_2d_5</span></code>()</p></td>
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+  <section id="module-sympde.topology.analytical_mapping">
+<span id="sympde-topology-analytical-mapping"></span><h1>sympde.topology.analytical_mapping<a class="headerlink" href="#module-sympde.topology.analytical_mapping" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AffineMapping</span></code>(name[, dim])</p></td>
+<td><p>Represents a 1D/2D/3D Affine Mapping object.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CollelaMapping2D</span></code>(name[, dim])</p></td>
+<td><p>Represents a Collela 2D Mapping object.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CzarnyMapping</span></code>(name[, dim])</p></td>
+<td><p>Represents a Czarny 2D Mapping object.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IdentityMapping</span></code>(name[, dim])</p></td>
+<td><p>Represents an identity 1D/2D/3D Mapping object.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PolarMapping</span></code>(name[, dim])</p></td>
+<td><p>Represents a Polar 2D Mapping object (Annulus).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SphericalMapping</span></code>(name[, dim])</p></td>
+<td><p>Parametrization of a sphere (or a portion of it) using spherical coordinates (x1, x2, x3) = (r, theta, phi), where:</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TargetMapping</span></code>(name[, dim])</p></td>
+<td><p>Represents a Target 2D Mapping object.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TorusMapping</span></code>(name[, dim])</p></td>
+<td><p>Parametrization of a torus (or a portion of it) of major radius R0, using toroidal coordinates (x1, x2, x3) = (r, theta, phi), where:</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TorusSurfaceMapping</span></code>(name[, dim])</p></td>
+<td><p>3D surface obtained by &quot;slicing&quot; the torus above at r = a.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TwistedTargetMapping</span></code>(name[, dim])</p></td>
+<td><p>3D volume obtained by &quot;extruding&quot; the TwistedTargetSurfaceMapping along z.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TwistedTargetSurfaceMapping</span></code>(name[, dim])</p></td>
+<td><p>3D surface obtained by &quot;twisting&quot; the TargetMapping out of the (x, y) plane</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
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+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+  <section id="module-sympde.topology.basic">
+<span id="sympde-topology-basic"></span><h1>sympde.topology.basic<a class="headerlink" href="#module-sympde.topology.basic" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicDomain</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Boundary</span></code>(name, domain[, axis, ext, mapping, ...])</p></td>
+<td><p>Represents an undefined boundary over a domain.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Connectivity</span></code>([data])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CornerBoundary</span></code>(*boundaries)</p></td>
+<td><p>Represents an undefined corner over a domain in 2D.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CornerInterface</span></code>(*corners)</p></td>
+<td><p>Represents a shared corner over multiple patches in 2D.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Edge</span></code>(name)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Interface</span></code>(name, bnd_minus, bnd_plus[, ...])</p></td>
+<td><p>Represents an interface between two subdomains through two boundaries.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InteriorDomain</span></code>(name[, dim, dtype, mapping, ...])</p></td>
+<td><p>Represents an undefined interior domain.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Interval</span></code>([name, coordinate, bounds])</p></td>
+<td><p>Represents a 1D interval.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ProductDomain</span></code>(*args[, name])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Union</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<span id="sympde-topology-callable-mapping"></span><h1>sympde.topology.callable_mapping<a class="headerlink" href="#module-sympde.topology.callable_mapping" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CallableMapping</span></code>(mapping, **kwargs)</p></td>
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+                  
+  <section id="module-sympde.topology.datatype">
+<span id="sympde-topology-datatype"></span><h1>sympde.topology.datatype<a class="headerlink" href="#module-sympde.topology.datatype" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">H1RegularityType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">H1SpaceType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">HcurlRegularityType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">HcurlSpaceType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">HdivRegularityType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">HdivSpaceType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">L2RegularityType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">L2SpaceType</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RegularityType</span></code>()</p></td>
+<td><p>Base class representing the regularity of a space of functions</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SpaceType</span></code>()</p></td>
+<td><p>Base class representing function space types</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">UndefinedSpaceType</span></code>()</p></td>
+<td><p></p></td>
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+    <h1>sympde.topology.derivatives</h1>
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+  <section id="module-sympde.topology.derivatives">
+<span id="sympde-topology-derivatives"></span><h1>sympde.topology.derivatives<a class="headerlink" href="#module-sympde.topology.derivatives" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_partial_derivatives</span></code>(expr)</p></td>
+<td><p>returns all partial derivative expressions</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_atom_derivatives</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_atom_logical_derivatives</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_index_derivatives</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_index_derivatives_atom</span></code>(expr, atom[, verbose])</p></td>
+<td><p>This function return a dictionary of partial derivative indices for a given atom.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_index_logical_derivatives</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_index_logical_derivatives_atom</span></code>(expr, atom)</p></td>
+<td><p>This function return a dictionary of partial derivative indices for a given atom.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_max_logical_partial_derivatives</span></code>(expr[, F])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_max_partial_derivatives</span></code>(expr[, F])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_number_derivatives</span></code>(expr)</p></td>
+<td><p>returns the number of partial derivatives in expr.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sort_partial_derivatives</span></code>(expr)</p></td>
+<td><p>returns the partial derivatives of an expression, sorted.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BracketBasic</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Bracket_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CrossBasic</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Cross_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Cross_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CurlBasic</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Curl_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Curl_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DifferentialOperator</span></code>(expr)</p></td>
+<td><p>This class is a linear operator that applies the Leibniz formula</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DivBasic</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Div_1d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Div_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Div_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DotBasic</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Dot_1d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Dot_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Dot_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GradBasic</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Grad_1d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Grad_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Grad_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">HessianBasic</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Hessian_1d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Hessian_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Hessian_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LaplaceBasic</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Laplace_1d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Laplace_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Laplace_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalBracket_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalCurl_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalCurl_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalDiv_1d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalDiv_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalDiv_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalGrad_1d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalGrad_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalGrad_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalHessian_1d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalHessian_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalHessian_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalLaplace_1d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalLaplace_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalLaplace_3d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalRot_2d</span></code>(*_args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Rot_2d</span></code>(*_args)</p></td>
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+            
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+                  
+  <section id="module-sympde.topology.domain">
+<span id="sympde-topology-domain"></span><h1>sympde.topology.domain<a class="headerlink" href="#module-sympde.topology.domain" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">split</span></code>(domain, value)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Area</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicArea</span></code>(domain)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicGeometryOperator</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BoundaryVector</span></code>(label[, shape, offset, strides])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Cube</span></code>([name, bounds1, bounds2, bounds3])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Domain</span></code>(name, *[, interiors, boundaries, ...])</p></td>
+<td><p>Represents an undefined domain.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DomainArea</span></code>(domain)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ElementArea</span></code>(domain)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ElementDomain</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Line</span></code>([name, bounds])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MinusNormalVector</span></code>(label[, shape, offset, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NCube</span></code>(name, dim, min_coords, max_coords)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NCubeInterior</span></code>(name[, dim, dtype, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NormalVector</span></code>(label[, shape, offset, strides])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PeriodicDomain</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PlusNormalVector</span></code>(label[, shape, offset, strides])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Square</span></code>([name, bounds1, bounds2])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TangentVector</span></code>(label[, shape, offset, strides])</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<span id="sympde-topology"></span><h1>sympde.topology<a class="headerlink" href="#module-sympde.topology" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="sympde.topology.analytical_mapping.html#module-sympde.topology.analytical_mapping" title="sympde.topology.analytical_mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.analytical_mapping</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="sympde.topology.basic.html#module-sympde.topology.basic" title="sympde.topology.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.basic</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="sympde.topology.callable_mapping.html#module-sympde.topology.callable_mapping" title="sympde.topology.callable_mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.callable_mapping</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="sympde.topology.datatype.html#module-sympde.topology.datatype" title="sympde.topology.datatype"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.datatype</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="sympde.topology.derivatives.html#module-sympde.topology.derivatives" title="sympde.topology.derivatives"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.derivatives</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="sympde.topology.domain.html#module-sympde.topology.domain" title="sympde.topology.domain"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.domain</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="sympde.topology.mapping.html#module-sympde.topology.mapping" title="sympde.topology.mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.mapping</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="sympde.topology.measure.html#module-sympde.topology.measure" title="sympde.topology.measure"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.measure</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="sympde.topology.space.html#module-sympde.topology.space" title="sympde.topology.space"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.space</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="sympde.topology.tests.html#module-sympde.topology.tests" title="sympde.topology.tests"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.tests</span></code></a></p></td>
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+  <section id="module-sympde.topology.mapping">
+<span id="sympde-topology-mapping"></span><h1>sympde.topology.mapping<a class="headerlink" href="#module-sympde.topology.mapping" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">cancel</span></code>(f)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_logical_test_function</span></code>(u)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicCallableMapping</span></code>()</p></td>
+<td><p>Transformation of coordinates, which can be evaluated.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Contravariant</span></code>(F, v)</p></td>
+<td><p>Examples</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Covariant</span></code>(F, v)</p></td>
+<td><p>Examples</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InterfaceMapping</span></code>(minus, plus)</p></td>
+<td><p>InterfaceMapping is used to represent a mapping in the interface.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InverseMapping</span></code>(mapping)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Jacobian</span></code>(F)</p></td>
+<td><p>This class calculates the Jacobian of a mapping F where [J_{F}]_{i,j} =  frac{partial F_{i}}{partial x_{j}} or simply J_{F} =  (nabla F)^T</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">JacobianInverseSymbol</span></code>(mapping[, axis])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">JacobianSymbol</span></code>(mapping[, axis])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalExpr</span></code>(expr, domain, **options)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MappedDomain</span></code>(mapping, logical_domain)</p></td>
+<td><p><p>.</p>
+</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Mapping</span></code>(name[, dim])</p></td>
+<td><p>Represents a Mapping object.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MappingApplication</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MultiPatchMapping</span></code>(dic)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PullBack</span></code>(u[, mapping])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SymbolicExpr</span></code>(*_args, **kwargs)</p></td>
+<td><p>returns a sympy expression where partial derivatives are converted into sympy Symbols.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SymbolicWeightedVolume</span></code>(*args)</p></td>
+<td><p>This class represents the symbolic weighted volume of a quadrature rule</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+  <section id="module-sympde.topology.measure">
+<span id="sympde-topology-measure"></span><h1>sympde.topology.measure<a class="headerlink" href="#module-sympde.topology.measure" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicMeasure</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CanonicalMeasure</span></code>(ldim)</p></td>
+<td><p>Represents a canonical measure.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CartesianMeasure</span></code>(ldim)</p></td>
+<td><p>Represents a cartesian measure.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Measure</span></code>(coordinates)</p></td>
+<td><p>Represents a measure from coordinates.</p></td>
+</tr>
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+</section>
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+<ul class="nav bd-sidenav">
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+  <section id="module-sympde.topology.space">
+<span id="sympde-topology-space"></span><h1>sympde.topology.space<a class="headerlink" href="#module-sympde.topology.space" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TestFunction</span></code>(space[, name])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">element_of</span></code>(space, name)</p></td>
+<td><p>Create a single element of a given space (possibly a ProductSpace).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elements_of</span></code>(space, names)</p></td>
+<td><p>Create multiple elements of same space (possibly a ProductSpace).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">trace_0</span></code>(x, B)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">trace_1</span></code>(x, B)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicFunctionSpace</span></code>(name, domain, shape, kind)</p></td>
+<td><p>Represents a basic continuous Function space.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Derham</span></code>(domain[, sequence])</p></td>
+<td><p><p>.</p>
+</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexedVectorFunction</span></code>(base, *args, **kw_args)</p></td>
+<td><p>Represents a mathematical object with indices.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ProductSpace</span></code>(*spaces)</p></td>
+<td><p>Represents a product of continuous Sobolev spaces.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projection</span></code>(projector, expr)</p></td>
+<td><p>Represents a projection</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector</span></code>(space[, kind])</p></td>
+<td><p>Represents a Projector over a function space.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ScalarFunction</span></code>(space, name)</p></td>
+<td><p>Represents a test function as an element of a fem space.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ScalarFunctionSpace</span></code>(name, domain[, kind])</p></td>
+<td><p>Represents a basic continuous scalar Function space.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Trace</span></code>(expr, boundary[, order])</p></td>
+<td><p>Represents the trace over a boundary and a space function</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorFunction</span></code>(space, name)</p></td>
+<td><p>Represents a vector test function as an element of a fem space.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorFunctionSpace</span></code>(name, domain[, kind])</p></td>
+<td><p>Represents a basic continuous vector Function space.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+  <section id="module-sympde.topology.tests">
+<span id="sympde-topology-tests"></span><h1>sympde.topology.tests<a class="headerlink" href="#module-sympde.topology.tests" title="Link to this heading">#</a></h1>
+<p class="rubric">Modules</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="sympde.topology.tests.test_callable_mapping.html#module-sympde.topology.tests.test_callable_mapping" title="sympde.topology.tests.test_callable_mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.tests.test_callable_mapping</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="sympde.topology.tests.test_derivatives.html#module-sympde.topology.tests.test_derivatives" title="sympde.topology.tests.test_derivatives"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.tests.test_derivatives</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="sympde.topology.tests.test_field_2d.html#module-sympde.topology.tests.test_field_2d" title="sympde.topology.tests.test_field_2d"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.tests.test_field_2d</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="sympde.topology.tests.test_gallery.html#module-sympde.topology.tests.test_gallery" title="sympde.topology.tests.test_gallery"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.tests.test_gallery</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="sympde.topology.tests.test_logical_expr.html#module-sympde.topology.tests.test_logical_expr" title="sympde.topology.tests.test_logical_expr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.tests.test_logical_expr</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="sympde.topology.tests.test_mapping.html#module-sympde.topology.tests.test_mapping" title="sympde.topology.tests.test_mapping"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.tests.test_mapping</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="sympde.topology.tests.test_space.html#module-sympde.topology.tests.test_space" title="sympde.topology.tests.test_space"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.tests.test_space</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="sympde.topology.tests.test_symbolic_expr.html#module-sympde.topology.tests.test_symbolic_expr" title="sympde.topology.tests.test_symbolic_expr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.tests.test_symbolic_expr</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="sympde.topology.tests.test_topology.html#module-sympde.topology.tests.test_topology" title="sympde.topology.tests.test_topology"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sympde.topology.tests.test_topology</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-sympde.topology.tests.test_callable_mapping">
+<span id="sympde-topology-tests-test-callable-mapping"></span><h1>sympde.topology.tests.test_callable_mapping<a class="headerlink" href="#module-sympde.topology.tests.test_callable_mapping" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_affine_mapping_1d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_affine_mapping_2d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_affine_mapping_3d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_affine_mapping_array_1d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_affine_mapping_array_2d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_affine_mapping_array_3d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_identity_mapping_1d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_identity_mapping_2d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_identity_mapping_3d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_identity_mapping_array_1d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_identity_mapping_array_2d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_identity_mapping_array_3d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_polar_mapping</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_polar_mapping_array</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_user_defined_callable_mapping</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-sympde.topology.tests.test_derivatives">
+<span id="sympde-topology-tests-test-derivatives"></span><h1>sympde.topology.tests.test_derivatives<a class="headerlink" href="#module-sympde.topology.tests.test_derivatives" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">indices_as_str</span></code>(a)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
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+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_partial_derivatives_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_partial_derivatives_2</span></code>()</p></td>
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+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
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+<td><p></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_interval</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_orthogonal_hexahedron</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_rectangle</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_unit_cube</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_unit_line</span></code>()</p></td>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-sympde.topology.tests.test_logical_expr">
+<span id="sympde-topology-tests-test-logical-expr"></span><h1>sympde.topology.tests.test_logical_expr<a class="headerlink" href="#module-sympde.topology.tests.test_logical_expr" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_collela_mapping_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_czarny_mapping_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_identity_mapping_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_identity_mapping_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_logical_expr_1d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_logical_expr_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_logical_expr_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_logical_expr_2d_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_logical_expr_3d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_logical_expr_3d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_logical_expr_3d_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_logical_expr_3d_4</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_logical_expr_3d_5</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_polar_mapping_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_symbolic_expr_1d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_symbolic_expr_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_symbolic_expr_3d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_target_mapping_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_torus_mapping_3d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_twisted_target_mapping_3d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-sympde.topology.tests.test_mapping">
+<span id="sympde-topology-tests-test-mapping"></span><h1>sympde.topology.tests.test_mapping<a class="headerlink" href="#module-sympde.topology.tests.test_mapping" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
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+</tr>
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+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_mapping_1d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_mapping_2d</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_mapping_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_mapping_3d</span></code>()</p></td>
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+<p class="rubric">Functions</p>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_projector_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_space_1d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_space_2d_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_space_2d_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_space_3d_1</span></code>()</p></td>
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+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
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+</ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-sympde.topology.tests.test_symbolic_expr">
+<span id="sympde-topology-tests-test-symbolic-expr"></span><h1>sympde.topology.tests.test_symbolic_expr<a class="headerlink" href="#module-sympde.topology.tests.test_symbolic_expr" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_derivatives_2d_with_mapping</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_derivatives_2d_without_mapping</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
+    <script type="text/x-thebe-config">
+    {
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+  <section id="module-sympde.topology.tests.test_topology">
+<span id="sympde-topology-tests-test-topology"></span><h1>sympde.topology.tests.test_topology<a class="headerlink" href="#module-sympde.topology.tests.test_topology" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_function</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">teardown_module</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_2d_domain_without_bnd</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_3d_domain_without_bnd</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_boundary_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_boundary_2</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_boundary_3</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_domain_1</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_domain_join_line</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_domain_join_square</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_element</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_get_subdomain</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_hash</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_interior_domain</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">test_topology_1</span></code>()</p></td>
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+</tr>
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+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+                  
+  <section id="module-sympde.utilities.utils">
+<span id="sympde-utilities-utils"></span><h1>sympde.utilities.utils<a class="headerlink" href="#module-sympde.utilities.utils" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">lambdify_sympde</span></code>(variables, expr)</p></td>
+<td><p>Custom lambify function that covers the shortcomings of sympy's lambdify.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">plot_2d</span></code>(domain[, draw, isolines, refinement])</p></td>
+<td><p>Plot a 2D domain</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">plot_2d_single_patch</span></code>(patch, mapping, ax[, ...])</p></td>
+<td><p>Plots a singe patch in a 2D domain</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">plot_3d</span></code>(domain[, draw, refinement])</p></td>
+<td><p>Plot a 3D domain</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">plot_3d_single_patch</span></code>(patch, mapping, ax[, ...])</p></td>
+<td><p>Plot a singe patch in a 3D domain</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">plot_domain</span></code>(domain[, draw, isolines, refinement])</p></td>
+<td><p>Plots a 2D  or 3D domain using matplotlib</p></td>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter0/howto.html">What to expect from IGA-Python</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+                    IGA-Python
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+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Introduction</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter0/howto.html">What to expect from IGA-Python</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter0/cad.html">Computer Aided Design</a><input class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-1"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter0/bsplines-surfaces.html">B-Splines surfaces</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter0/iga.html">Isogeometric Analysis</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/data-structure.html">Data Structure</a></li>
+
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/install.html">Installation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter1/sympde.html">SymPDE concepts and their mathematical meaning</a><input class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-5"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/topology.html">Topology concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/space.html">Function Space concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="current nav bd-sidenav">
+<li class="toctree-l1 current active has-children"><a class="current reference internal" href="#">SymPDE API Reference</a><input checked="" class="toctree-checkbox" id="toctree-checkbox-13" name="toctree-checkbox-13" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-13"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="chapter2/poisson.html">The Poisson equation</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter3/burgers.html">1D Burgers equation</a></li>
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+</li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+        <li><a href="documentation/_autosummary/psydac.api.ast.basic.html#module-psydac.api.ast.basic">psydac.api.ast.basic</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.basic.html#module-psydac.api.ast.basic">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.ast.evaluation.html#module-psydac.api.ast.evaluation">psydac.api.ast.evaluation</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.evaluation.html#module-psydac.api.ast.evaluation">[1]</a>
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+        <li><a href="documentation/_autosummary/psydac.api.ast.expr.html#module-psydac.api.ast.expr">psydac.api.ast.expr</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.expr.html#module-psydac.api.ast.expr">[1]</a>
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+        <li><a href="documentation/_autosummary/psydac.api.ast.fem.html#module-psydac.api.ast.fem">psydac.api.ast.fem</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.fem.html#module-psydac.api.ast.fem">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.ast.glt.html#module-psydac.api.ast.glt">psydac.api.ast.glt</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.glt.html#module-psydac.api.ast.glt">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.ast.linalg.html#module-psydac.api.ast.linalg">psydac.api.ast.linalg</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.linalg.html#module-psydac.api.ast.linalg">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.ast.nodes.html#module-psydac.api.ast.nodes">psydac.api.ast.nodes</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.nodes.html#module-psydac.api.ast.nodes">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.ast.parser.html#module-psydac.api.ast.parser">psydac.api.ast.parser</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.parser.html#module-psydac.api.ast.parser">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.html#module-psydac.api.ast.tests">psydac.api.ast.tests</a>
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+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.boundary.html#module-psydac.api.ast.tests.boundary">psydac.api.ast.tests.boundary</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.system_1.html#module-psydac.api.ast.tests.system_1">psydac.api.ast.tests.system_1</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.system_2.html#module-psydac.api.ast.tests.system_2">psydac.api.ast.tests.system_2</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.system_3.html#module-psydac.api.ast.tests.system_3">psydac.api.ast.tests.system_3</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.test_nodes.html#module-psydac.api.ast.tests.test_nodes">psydac.api.ast.tests.test_nodes</a>
+</li>
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+        <li><a href="documentation/_autosummary/psydac.api.equation.html#module-psydac.api.equation">psydac.api.equation</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.equation.html#module-psydac.api.equation">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.essential_bc.html#module-psydac.api.essential_bc">psydac.api.essential_bc</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.essential_bc.html#module-psydac.api.essential_bc">[1]</a>
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+        <li><a href="documentation/_autosummary/psydac.api.expr.html#module-psydac.api.expr">psydac.api.expr</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.expr.html#module-psydac.api.expr">[1]</a>
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+</li>
+        <li><a href="documentation/_autosummary/psydac.api.grid.html#module-psydac.api.grid">psydac.api.grid</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.grid.html#module-psydac.api.grid">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.postprocessing.html#module-psydac.api.postprocessing">psydac.api.postprocessing</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.postprocessing.html#module-psydac.api.postprocessing">[1]</a>
+</li>
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+</li>
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+</li>
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+</li>
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+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.build_domain.html#module-psydac.api.tests.build_domain">psydac.api.tests.build_domain</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_biharmonic.html#module-psydac.api.tests.test_2d_biharmonic">psydac.api.tests.test_2d_biharmonic</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_complex.html#module-psydac.api.tests.test_2d_complex">psydac.api.tests.test_2d_complex</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_laplace.html#module-psydac.api.tests.test_2d_laplace">psydac.api.tests.test_2d_laplace</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_mapping_biharmonic.html#module-psydac.api.tests.test_2d_mapping_biharmonic">psydac.api.tests.test_2d_mapping_biharmonic</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_mapping_laplace.html#module-psydac.api.tests.test_2d_mapping_laplace">psydac.api.tests.test_2d_mapping_laplace</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_mapping_poisson.html#module-psydac.api.tests.test_2d_mapping_poisson">psydac.api.tests.test_2d_mapping_poisson</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_multipatch_mapping_maxwell.html#module-psydac.api.tests.test_2d_multipatch_mapping_maxwell">psydac.api.tests.test_2d_multipatch_mapping_maxwell</a>
+</li>
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+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_navier_stokes.html#module-psydac.api.tests.test_2d_navier_stokes">psydac.api.tests.test_2d_navier_stokes</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_poisson.html#module-psydac.api.tests.test_2d_poisson">psydac.api.tests.test_2d_poisson</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_1d_compatible_spaces.html#module-psydac.api.tests.test_api_1d_compatible_spaces">psydac.api.tests.test_api_1d_compatible_spaces</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_compatible_spaces.html#module-psydac.api.tests.test_api_2d_compatible_spaces">psydac.api.tests.test_api_2d_compatible_spaces</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_fields.html#module-psydac.api.tests.test_api_2d_fields">psydac.api.tests.test_api_2d_fields</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_scalar_analytical_mapping.html#module-psydac.api.tests.test_api_2d_scalar_analytical_mapping">psydac.api.tests.test_api_2d_scalar_analytical_mapping</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_system.html#module-psydac.api.tests.test_api_2d_system">psydac.api.tests.test_api_2d_system</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_vector.html#module-psydac.api.tests.test_api_2d_vector">psydac.api.tests.test_api_2d_vector</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_vector_mapping.html#module-psydac.api.tests.test_api_2d_vector_mapping">psydac.api.tests.test_api_2d_vector_mapping</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_vector_multipatch_mapping.html#module-psydac.api.tests.test_api_2d_vector_multipatch_mapping">psydac.api.tests.test_api_2d_vector_multipatch_mapping</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_3d_scalar.html#module-psydac.api.tests.test_api_3d_scalar">psydac.api.tests.test_api_3d_scalar</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_3d_scalar_mapping.html#module-psydac.api.tests.test_api_3d_scalar_mapping">psydac.api.tests.test_api_3d_scalar_mapping</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_3d_vector.html#module-psydac.api.tests.test_api_3d_vector">psydac.api.tests.test_api_3d_vector</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_3d_vector_mapping.html#module-psydac.api.tests.test_api_3d_vector_mapping">psydac.api.tests.test_api_3d_vector_mapping</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_expr_2d_scalar.html#module-psydac.api.tests.test_api_expr_2d_scalar">psydac.api.tests.test_api_expr_2d_scalar</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_feec_1d.html#module-psydac.api.tests.test_api_feec_1d">psydac.api.tests.test_api_feec_1d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_feec_2d.html#module-psydac.api.tests.test_api_feec_2d">psydac.api.tests.test_api_feec_2d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_feec_3d.html#module-psydac.api.tests.test_api_feec_3d">psydac.api.tests.test_api_feec_3d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_glt_2d_scalar.html#module-psydac.api.tests.test_api_glt_2d_scalar">psydac.api.tests.test_api_glt_2d_scalar</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_glt_2d_scalar_mapping.html#module-psydac.api.tests.test_api_glt_2d_scalar_mapping">psydac.api.tests.test_api_glt_2d_scalar_mapping</a>
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+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_glt_2d_vector.html#module-psydac.api.tests.test_api_glt_2d_vector">psydac.api.tests.test_api_glt_2d_vector</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_assembly.html#module-psydac.api.tests.test_assembly">psydac.api.tests.test_assembly</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_build_derham_mapping.html#module-psydac.api.tests.test_build_derham_mapping">psydac.api.tests.test_build_derham_mapping</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_equation.html#module-psydac.api.tests.test_equation">psydac.api.tests.test_equation</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_postprocessing.html#module-psydac.api.tests.test_postprocessing">psydac.api.tests.test_postprocessing</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_quadorder.html#module-psydac.api.tests.test_quadorder">psydac.api.tests.test_quadorder</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.tests.utils.html#module-psydac.api.tests.utils">psydac.api.tests.utils</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.api.utilities.html#module-psydac.api.utilities">psydac.api.utilities</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.cad.html#module-psydac.cad">psydac.cad</a>
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+        <li><a href="documentation/_autosummary/psydac.cad.cad.html#module-psydac.cad.cad">psydac.cad.cad</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cad.cad.html#module-psydac.cad.cad">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.cad.gallery.html#module-psydac.cad.gallery">psydac.cad.gallery</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cad.gallery.html#module-psydac.cad.gallery">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.cad.geometry.html#module-psydac.cad.geometry">psydac.cad.geometry</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cad.geometry.html#module-psydac.cad.geometry">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.cad.multipatch.html#module-psydac.cad.multipatch">psydac.cad.multipatch</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cad.multipatch.html#module-psydac.cad.multipatch">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.cad.tests.html#module-psydac.cad.tests">psydac.cad.tests</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.cad.tests.test_geometry.html#module-psydac.cad.tests.test_geometry">psydac.cad.tests.test_geometry</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.cad.utils.html#module-psydac.cad.utils">psydac.cad.utils</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cad.utils.html#module-psydac.cad.utils">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.cmd.html#module-psydac.cmd">psydac.cmd</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.cmd.mesh.html#module-psydac.cmd.mesh">psydac.cmd.mesh</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cmd.mesh.html#module-psydac.cmd.mesh">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.core.html#module-psydac.core">psydac.core</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.core.bsplines.html#module-psydac.core.bsplines">psydac.core.bsplines</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.core.bsplines.html#module-psydac.core.bsplines">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.core.bsplines_kernels.html#module-psydac.core.bsplines_kernels">psydac.core.bsplines_kernels</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.core.bsplines_kernels.html#module-psydac.core.bsplines_kernels">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.core.field_evaluation_kernels.html#module-psydac.core.field_evaluation_kernels">psydac.core.field_evaluation_kernels</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.core.field_evaluation_kernels.html#module-psydac.core.field_evaluation_kernels">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.core.tests.html#module-psydac.core.tests">psydac.core.tests</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.core.tests.test_bsplines.html#module-psydac.core.tests.test_bsplines">psydac.core.tests.test_bsplines</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.core.tests.test_bsplines_pyccel.html#module-psydac.core.tests.test_bsplines_pyccel">psydac.core.tests.test_bsplines_pyccel</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.core.tests.test_field_evaluation_kernel.html#module-psydac.core.tests.test_field_evaluation_kernel">psydac.core.tests.test_field_evaluation_kernel</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.html#module-psydac.ddm">psydac.ddm</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.basic.html#module-psydac.ddm.basic">psydac.ddm.basic</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.basic.html#module-psydac.ddm.basic">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.blocking_data_exchanger.html#module-psydac.ddm.blocking_data_exchanger">psydac.ddm.blocking_data_exchanger</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.blocking_data_exchanger.html#module-psydac.ddm.blocking_data_exchanger">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.cart.html#module-psydac.ddm.cart">psydac.ddm.cart</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.cart.html#module-psydac.ddm.cart">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.interface_data_exchanger.html#module-psydac.ddm.interface_data_exchanger">psydac.ddm.interface_data_exchanger</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.interface_data_exchanger.html#module-psydac.ddm.interface_data_exchanger">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.nonblocking_data_exchanger.html#module-psydac.ddm.nonblocking_data_exchanger">psydac.ddm.nonblocking_data_exchanger</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.nonblocking_data_exchanger.html#module-psydac.ddm.nonblocking_data_exchanger">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.partition.html#module-psydac.ddm.partition">psydac.ddm.partition</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.partition.html#module-psydac.ddm.partition">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.petsc.html#module-psydac.ddm.petsc">psydac.ddm.petsc</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.petsc.html#module-psydac.ddm.petsc">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.html#module-psydac.ddm.tests">psydac.ddm.tests</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.test_cart_1d.html#module-psydac.ddm.tests.test_cart_1d">psydac.ddm.tests.test_cart_1d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.test_cart_2d.html#module-psydac.ddm.tests.test_cart_2d">psydac.ddm.tests.test_cart_2d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.test_cart_3d.html#module-psydac.ddm.tests.test_cart_3d">psydac.ddm.tests.test_cart_3d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.test_multicart_2d.html#module-psydac.ddm.tests.test_multicart_2d">psydac.ddm.tests.test_multicart_2d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.test_partition.html#module-psydac.ddm.tests.test_partition">psydac.ddm.tests.test_partition</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.ddm.utilities.html#module-psydac.ddm.utilities">psydac.ddm.utilities</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.utilities.html#module-psydac.ddm.utilities">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.html#module-psydac.feec">psydac.feec</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.derivatives.html#module-psydac.feec.derivatives">psydac.feec.derivatives</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.derivatives.html#module-psydac.feec.derivatives">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.global_projectors.html#module-psydac.feec.global_projectors">psydac.feec.global_projectors</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.global_projectors.html#module-psydac.feec.global_projectors">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.html#module-psydac.feec.multipatch">psydac.feec.multipatch</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.api.html#module-psydac.feec.multipatch.api">psydac.feec.multipatch.api</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.api.html#module-psydac.feec.multipatch.api">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.html#module-psydac.feec.multipatch.examples">psydac.feec.multipatch.examples</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.h1_source_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.h1_source_pbms_conga_2d">psydac.feec.multipatch.examples.h1_source_pbms_conga_2d</a>
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+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d">psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d">psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d">psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.ppc_test_cases.html#module-psydac.feec.multipatch.examples.ppc_test_cases">psydac.feec.multipatch.examples.ppc_test_cases</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.fem_linear_operators.html#module-psydac.feec.multipatch.fem_linear_operators">psydac.feec.multipatch.fem_linear_operators</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.fem_linear_operators.html#module-psydac.feec.multipatch.fem_linear_operators">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.multipatch_domain_utilities.html#module-psydac.feec.multipatch.multipatch_domain_utilities">psydac.feec.multipatch.multipatch_domain_utilities</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.operators.html#module-psydac.feec.multipatch.operators">psydac.feec.multipatch.operators</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.operators.html#module-psydac.feec.multipatch.operators">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.plotting_utilities.html#module-psydac.feec.multipatch.plotting_utilities">psydac.feec.multipatch.plotting_utilities</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.utilities.html#module-psydac.feec.multipatch.utilities">psydac.feec.multipatch.utilities</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.utilities.html#module-psydac.feec.multipatch.utilities">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.utils_conga_2d.html#module-psydac.feec.multipatch.utils_conga_2d">psydac.feec.multipatch.utils_conga_2d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.pull_push.html#module-psydac.feec.pull_push">psydac.feec.pull_push</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.pull_push.html#module-psydac.feec.pull_push">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.pushforward.html#module-psydac.feec.pushforward">psydac.feec.pushforward</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.pushforward.html#module-psydac.feec.pushforward">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.html#module-psydac.feec.tests">psydac.feec.tests</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_axis_projection.html#module-psydac.feec.tests.test_axis_projection">psydac.feec.tests.test_axis_projection</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_commuting_projections.html#module-psydac.feec.tests.test_commuting_projections">psydac.feec.tests.test_commuting_projections</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_commuting_projections_dual.html#module-psydac.feec.tests.test_commuting_projections_dual">psydac.feec.tests.test_commuting_projections_dual</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_differentiation_matrices.html#module-psydac.feec.tests.test_differentiation_matrices">psydac.feec.tests.test_differentiation_matrices</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_global_projectors.html#module-psydac.feec.tests.test_global_projectors">psydac.feec.tests.test_global_projectors</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_projections_parallel.html#module-psydac.feec.tests.test_projections_parallel">psydac.feec.tests.test_projections_parallel</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.html#module-psydac.fem">psydac.fem</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.basic.html#module-psydac.fem.basic">psydac.fem.basic</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.basic.html#module-psydac.fem.basic">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.context.html#module-psydac.fem.context">psydac.fem.context</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.context.html#module-psydac.fem.context">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.grid.html#module-psydac.fem.grid">psydac.fem.grid</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.grid.html#module-psydac.fem.grid">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.partitioning.html#module-psydac.fem.partitioning">psydac.fem.partitioning</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.partitioning.html#module-psydac.fem.partitioning">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.projectors.html#module-psydac.fem.projectors">psydac.fem.projectors</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.projectors.html#module-psydac.fem.projectors">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.splines.html#module-psydac.fem.splines">psydac.fem.splines</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.splines.html#module-psydac.fem.splines">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.tensor.html#module-psydac.fem.tensor">psydac.fem.tensor</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.tensor.html#module-psydac.fem.tensor">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.html#module-psydac.fem.tests">psydac.fem.tests</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.analytical_profiles_1d.html#module-psydac.fem.tests.analytical_profiles_1d">psydac.fem.tests.analytical_profiles_1d</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.analytical_profiles_base.html#module-psydac.fem.tests.analytical_profiles_base">psydac.fem.tests.analytical_profiles_base</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.splines_error_bounds.html#module-psydac.fem.tests.splines_error_bounds">psydac.fem.tests.splines_error_bounds</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.test_eval_fields_parallel.html#module-psydac.fem.tests.test_eval_fields_parallel">psydac.fem.tests.test_eval_fields_parallel</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.test_product.html#module-psydac.fem.tests.test_product">psydac.fem.tests.test_product</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.test_spline_histopolation.html#module-psydac.fem.tests.test_spline_histopolation">psydac.fem.tests.test_spline_histopolation</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.test_spline_interpolation.html#module-psydac.fem.tests.test_spline_interpolation">psydac.fem.tests.test_spline_interpolation</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.test_splines.html#module-psydac.fem.tests.test_splines">psydac.fem.tests.test_splines</a>
+</li>
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+</li>
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+</li>
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+</li>
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+</li>
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+</li>
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+</li>
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+</li>
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+</li>
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+</li>
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+</li>
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+</li>
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+        <li><a href="documentation/_autosummary/psydac.polar.tests.test_c1_linops.html#module-psydac.polar.tests.test_c1_linops">psydac.polar.tests.test_c1_linops</a>
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+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.html#module-psydac.pyccel.ast">psydac.pyccel.ast</a>
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+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.core.html#module-psydac.pyccel.ast.core">psydac.pyccel.ast.core</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.datatypes.html#module-psydac.pyccel.ast.datatypes">psydac.pyccel.ast.datatypes</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.functionalexpr.html#module-psydac.pyccel.ast.functionalexpr">psydac.pyccel.ast.functionalexpr</a>
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+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.mathext.html#module-psydac.pyccel.ast.mathext">psydac.pyccel.ast.mathext</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.numbers.html#module-psydac.pyccel.ast.numbers">psydac.pyccel.ast.numbers</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.numpyext.html#module-psydac.pyccel.ast.numpyext">psydac.pyccel.ast.numpyext</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.singleton.html#module-psydac.pyccel.ast.singleton">psydac.pyccel.ast.singleton</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.type_inference.html#module-psydac.pyccel.ast.type_inference">psydac.pyccel.ast.type_inference</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.utilities.html#module-psydac.pyccel.ast.utilities">psydac.pyccel.ast.utilities</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.codegen.html#module-psydac.pyccel.codegen">psydac.pyccel.codegen</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.codegen.printing.html#module-psydac.pyccel.codegen.printing">psydac.pyccel.codegen.printing</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.codegen.printing.pycode.html#module-psydac.pyccel.codegen.printing.pycode">psydac.pyccel.codegen.printing.pycode</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.errors.html#module-psydac.pyccel.errors">psydac.pyccel.errors</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.errors.errors.html#module-psydac.pyccel.errors.errors">psydac.pyccel.errors.errors</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.errors.messages.html#module-psydac.pyccel.errors.messages">psydac.pyccel.errors.messages</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.symbolic.html#module-psydac.pyccel.symbolic">psydac.pyccel.symbolic</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.pyccel.symbolic.lambdify.html#module-psydac.pyccel.symbolic.lambdify">psydac.pyccel.symbolic.lambdify</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.utilities.html#module-psydac.utilities">psydac.utilities</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.utilities.quadratures.html#module-psydac.utilities.quadratures">psydac.utilities.quadratures</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.utilities.quadratures.html#module-psydac.utilities.quadratures">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.utilities.utils.html#module-psydac.utilities.utils">psydac.utilities.utils</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.utilities.utils.html#module-psydac.utilities.utils">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/psydac.utilities.vtk.html#module-psydac.utilities.vtk">psydac.utilities.vtk</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.utilities.vtk.html#module-psydac.utilities.vtk">[1]</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.calculus.html#module-sympde.calculus">sympde.calculus</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.core.html#module-sympde.core">sympde.core</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.core.algebra.html#module-sympde.core.algebra">sympde.core.algebra</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.core.basic.html#module-sympde.core.basic">sympde.core.basic</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.core.tests.html#module-sympde.core.tests">sympde.core.tests</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.core.tests.test_utils.html#module-sympde.core.tests.test_utils">sympde.core.tests.test_utils</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.core.utils.html#module-sympde.core.utils">sympde.core.utils</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.expr.html#module-sympde.expr">sympde.expr</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.expr.basic.html#module-sympde.expr.basic">sympde.expr.basic</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.expr.equation.html#module-sympde.expr.equation">sympde.expr.equation</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.expr.errors.html#module-sympde.expr.errors">sympde.expr.errors</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.expr.evaluation.html#module-sympde.expr.evaluation">sympde.expr.evaluation</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.expr.expr.html#module-sympde.expr.expr">sympde.expr.expr</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.expr.tests.html#module-sympde.expr.tests">sympde.expr.tests</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.expr.tests.test_equation_2d.html#module-sympde.expr.tests.test_equation_2d">sympde.expr.tests.test_equation_2d</a>
+</li>
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+</li>
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+</li>
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+</li>
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+</li>
+        <li><a href="documentation/_autosummary/sympde.exterior.html#module-sympde.exterior">sympde.exterior</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.exterior.calculus.html#module-sympde.exterior.calculus">sympde.exterior.calculus</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.exterior.compiler.html#module-sympde.exterior.compiler">sympde.exterior.compiler</a>
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+        <li><a href="documentation/_autosummary/sympde.exterior.datatype.html#module-sympde.exterior.datatype">sympde.exterior.datatype</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.exterior.form.html#module-sympde.exterior.form">sympde.exterior.form</a>
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+        <li><a href="documentation/_autosummary/sympde.exterior.inference.html#module-sympde.exterior.inference">sympde.exterior.inference</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.exterior.tests.html#module-sympde.exterior.tests">sympde.exterior.tests</a>
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+        <li><a href="documentation/_autosummary/sympde.exterior.tests.test_compiler.html#module-sympde.exterior.tests.test_compiler">sympde.exterior.tests.test_compiler</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.exterior.tests.test_datatype.html#module-sympde.exterior.tests.test_datatype">sympde.exterior.tests.test_datatype</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.exterior.tests.test_exterior.html#module-sympde.exterior.tests.test_exterior">sympde.exterior.tests.test_exterior</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.exterior.tests.test_inference.html#module-sympde.exterior.tests.test_inference">sympde.exterior.tests.test_inference</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.printing.html#module-sympde.printing">sympde.printing</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.printing.tests.html#module-sympde.printing.tests">sympde.printing.tests</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.printing.tests.test_latex.html#module-sympde.printing.tests.test_latex">sympde.printing.tests.test_latex</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.html#module-sympde.topology">sympde.topology</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.analytical_mapping.html#module-sympde.topology.analytical_mapping">sympde.topology.analytical_mapping</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.basic.html#module-sympde.topology.basic">sympde.topology.basic</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.callable_mapping.html#module-sympde.topology.callable_mapping">sympde.topology.callable_mapping</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.datatype.html#module-sympde.topology.datatype">sympde.topology.datatype</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.derivatives.html#module-sympde.topology.derivatives">sympde.topology.derivatives</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.domain.html#module-sympde.topology.domain">sympde.topology.domain</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.mapping.html#module-sympde.topology.mapping">sympde.topology.mapping</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.measure.html#module-sympde.topology.measure">sympde.topology.measure</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.space.html#module-sympde.topology.space">sympde.topology.space</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.html#module-sympde.topology.tests">sympde.topology.tests</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_callable_mapping.html#module-sympde.topology.tests.test_callable_mapping">sympde.topology.tests.test_callable_mapping</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_derivatives.html#module-sympde.topology.tests.test_derivatives">sympde.topology.tests.test_derivatives</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_field_2d.html#module-sympde.topology.tests.test_field_2d">sympde.topology.tests.test_field_2d</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_gallery.html#module-sympde.topology.tests.test_gallery">sympde.topology.tests.test_gallery</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_logical_expr.html#module-sympde.topology.tests.test_logical_expr">sympde.topology.tests.test_logical_expr</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_mapping.html#module-sympde.topology.tests.test_mapping">sympde.topology.tests.test_mapping</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_space.html#module-sympde.topology.tests.test_space">sympde.topology.tests.test_space</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html#module-sympde.topology.tests.test_symbolic_expr">sympde.topology.tests.test_symbolic_expr</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_topology.html#module-sympde.topology.tests.test_topology">sympde.topology.tests.test_topology</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.utilities.html#module-sympde.utilities">sympde.utilities</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.utilities.tests.html#module-sympde.utilities.tests">sympde.utilities.tests</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.utilities.tests.test_utilities.html#module-sympde.utilities.tests.test_utilities">sympde.utilities.tests.test_utilities</a>
+</li>
+        <li><a href="documentation/_autosummary/sympde.utilities.utils.html#module-sympde.utilities.utils">sympde.utilities.utils</a>
+</li>
+      </ul></li>
+  </ul></td>
+</tr></table>
+
+<h2 id="P">P</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li>
+    psydac.api
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.html#module-psydac.api">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.html#module-psydac.api.ast">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.basic
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.basic.html#module-psydac.api.ast.basic">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.basic.html#module-psydac.api.ast.basic">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.evaluation
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.evaluation.html#module-psydac.api.ast.evaluation">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.evaluation.html#module-psydac.api.ast.evaluation">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.expr
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.expr.html#module-psydac.api.ast.expr">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.expr.html#module-psydac.api.ast.expr">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.fem
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.fem.html#module-psydac.api.ast.fem">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.fem.html#module-psydac.api.ast.fem">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.glt
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.glt.html#module-psydac.api.ast.glt">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.glt.html#module-psydac.api.ast.glt">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.linalg
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.linalg.html#module-psydac.api.ast.linalg">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.linalg.html#module-psydac.api.ast.linalg">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.nodes
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.nodes.html#module-psydac.api.ast.nodes">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.nodes.html#module-psydac.api.ast.nodes">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.parser
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.parser.html#module-psydac.api.ast.parser">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.parser.html#module-psydac.api.ast.parser">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.html#module-psydac.api.ast.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.tests.boundary
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.boundary.html#module-psydac.api.ast.tests.boundary">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.tests.system_1
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.system_1.html#module-psydac.api.ast.tests.system_1">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.tests.system_2
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.system_2.html#module-psydac.api.ast.tests.system_2">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.tests.system_3
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.system_3.html#module-psydac.api.ast.tests.system_3">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.tests.test_nodes
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.tests.test_nodes.html#module-psydac.api.ast.tests.test_nodes">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.ast.utilities
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.ast.utilities.html#module-psydac.api.ast.utilities">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.ast.utilities.html#module-psydac.api.ast.utilities">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.basic
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.basic.html#module-psydac.api.basic">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.basic.html#module-psydac.api.basic">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.discretization
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.discretization.html#module-psydac.api.discretization">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.discretization.html#module-psydac.api.discretization">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.equation
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.equation.html#module-psydac.api.equation">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.equation.html#module-psydac.api.equation">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.essential_bc
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.essential_bc.html#module-psydac.api.essential_bc">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.essential_bc.html#module-psydac.api.essential_bc">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.expr
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.expr.html#module-psydac.api.expr">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.expr.html#module-psydac.api.expr">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.feec
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.feec.html#module-psydac.api.feec">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.feec.html#module-psydac.api.feec">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.fem
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.fem.html#module-psydac.api.fem">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.fem.html#module-psydac.api.fem">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.glt
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.glt.html#module-psydac.api.glt">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.glt.html#module-psydac.api.glt">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.grid
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.grid.html#module-psydac.api.grid">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.grid.html#module-psydac.api.grid">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.postprocessing
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.postprocessing.html#module-psydac.api.postprocessing">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.postprocessing.html#module-psydac.api.postprocessing">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.printing
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.printing.html#module-psydac.api.printing">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.printing.pycode
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.printing.pycode.html#module-psydac.api.printing.pycode">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.api.printing.pycode.html#module-psydac.api.printing.pycode">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.settings
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.settings.html#module-psydac.api.settings">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.html#module-psydac.api.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.build_domain
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.build_domain.html#module-psydac.api.tests.build_domain">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_2d_biharmonic
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_biharmonic.html#module-psydac.api.tests.test_2d_biharmonic">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_2d_complex
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_complex.html#module-psydac.api.tests.test_2d_complex">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_2d_laplace
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_laplace.html#module-psydac.api.tests.test_2d_laplace">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_2d_mapping_biharmonic
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_mapping_biharmonic.html#module-psydac.api.tests.test_2d_mapping_biharmonic">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_2d_mapping_laplace
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_mapping_laplace.html#module-psydac.api.tests.test_2d_mapping_laplace">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_2d_mapping_poisson
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_mapping_poisson.html#module-psydac.api.tests.test_2d_mapping_poisson">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_2d_multipatch_mapping_maxwell
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_multipatch_mapping_maxwell.html#module-psydac.api.tests.test_2d_multipatch_mapping_maxwell">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_2d_multipatch_mapping_poisson
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_multipatch_mapping_poisson.html#module-psydac.api.tests.test_2d_multipatch_mapping_poisson">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_2d_multipatch_poisson
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_multipatch_poisson.html#module-psydac.api.tests.test_2d_multipatch_poisson">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_2d_navier_stokes
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_navier_stokes.html#module-psydac.api.tests.test_2d_navier_stokes">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_2d_poisson
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_2d_poisson.html#module-psydac.api.tests.test_2d_poisson">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_1d_compatible_spaces
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_1d_compatible_spaces.html#module-psydac.api.tests.test_api_1d_compatible_spaces">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_2d_compatible_spaces
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_compatible_spaces.html#module-psydac.api.tests.test_api_2d_compatible_spaces">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_2d_fields
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_fields.html#module-psydac.api.tests.test_api_2d_fields">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_2d_scalar_analytical_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_scalar_analytical_mapping.html#module-psydac.api.tests.test_api_2d_scalar_analytical_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_2d_system
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_system.html#module-psydac.api.tests.test_api_2d_system">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_2d_vector
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_vector.html#module-psydac.api.tests.test_api_2d_vector">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_2d_vector_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_vector_mapping.html#module-psydac.api.tests.test_api_2d_vector_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_2d_vector_multipatch_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_2d_vector_multipatch_mapping.html#module-psydac.api.tests.test_api_2d_vector_multipatch_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_3d_scalar
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_3d_scalar.html#module-psydac.api.tests.test_api_3d_scalar">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_3d_scalar_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_3d_scalar_mapping.html#module-psydac.api.tests.test_api_3d_scalar_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_3d_vector
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_3d_vector.html#module-psydac.api.tests.test_api_3d_vector">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_3d_vector_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_3d_vector_mapping.html#module-psydac.api.tests.test_api_3d_vector_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_expr_2d_scalar
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_expr_2d_scalar.html#module-psydac.api.tests.test_api_expr_2d_scalar">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_feec_1d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_feec_1d.html#module-psydac.api.tests.test_api_feec_1d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_feec_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_feec_2d.html#module-psydac.api.tests.test_api_feec_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_feec_3d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_feec_3d.html#module-psydac.api.tests.test_api_feec_3d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_glt_2d_scalar
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_glt_2d_scalar.html#module-psydac.api.tests.test_api_glt_2d_scalar">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_glt_2d_scalar_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_glt_2d_scalar_mapping.html#module-psydac.api.tests.test_api_glt_2d_scalar_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_api_glt_2d_vector
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_api_glt_2d_vector.html#module-psydac.api.tests.test_api_glt_2d_vector">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_assembly
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_assembly.html#module-psydac.api.tests.test_assembly">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_build_derham_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_build_derham_mapping.html#module-psydac.api.tests.test_build_derham_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_equation
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_equation.html#module-psydac.api.tests.test_equation">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_postprocessing
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_postprocessing.html#module-psydac.api.tests.test_postprocessing">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.test_quadorder
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.test_quadorder.html#module-psydac.api.tests.test_quadorder">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.tests.utils
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.tests.utils.html#module-psydac.api.tests.utils">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.api.utilities
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.api.utilities.html#module-psydac.api.utilities">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.cad
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.cad.html#module-psydac.cad">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.cad.cad
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.cad.cad.html#module-psydac.cad.cad">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cad.cad.html#module-psydac.cad.cad">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.cad.gallery
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.cad.gallery.html#module-psydac.cad.gallery">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cad.gallery.html#module-psydac.cad.gallery">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.cad.geometry
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.cad.geometry.html#module-psydac.cad.geometry">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cad.geometry.html#module-psydac.cad.geometry">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.cad.multipatch
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.cad.multipatch.html#module-psydac.cad.multipatch">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cad.multipatch.html#module-psydac.cad.multipatch">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.cad.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.cad.tests.html#module-psydac.cad.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.cad.tests.test_geometry
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.cad.tests.test_geometry.html#module-psydac.cad.tests.test_geometry">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.cad.utils
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.cad.utils.html#module-psydac.cad.utils">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cad.utils.html#module-psydac.cad.utils">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.cmd
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.cmd.html#module-psydac.cmd">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.cmd.mesh
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.cmd.mesh.html#module-psydac.cmd.mesh">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.cmd.mesh.html#module-psydac.cmd.mesh">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.core
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.core.html#module-psydac.core">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.core.bsplines
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.core.bsplines.html#module-psydac.core.bsplines">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.core.bsplines.html#module-psydac.core.bsplines">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.core.bsplines_kernels
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.core.bsplines_kernels.html#module-psydac.core.bsplines_kernels">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.core.bsplines_kernels.html#module-psydac.core.bsplines_kernels">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.core.field_evaluation_kernels
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.core.field_evaluation_kernels.html#module-psydac.core.field_evaluation_kernels">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.core.field_evaluation_kernels.html#module-psydac.core.field_evaluation_kernels">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.core.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.core.tests.html#module-psydac.core.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.core.tests.test_bsplines
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.core.tests.test_bsplines.html#module-psydac.core.tests.test_bsplines">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.core.tests.test_bsplines_pyccel
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.core.tests.test_bsplines_pyccel.html#module-psydac.core.tests.test_bsplines_pyccel">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.core.tests.test_field_evaluation_kernel
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.core.tests.test_field_evaluation_kernel.html#module-psydac.core.tests.test_field_evaluation_kernel">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.html#module-psydac.ddm">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.basic
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.basic.html#module-psydac.ddm.basic">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.basic.html#module-psydac.ddm.basic">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.blocking_data_exchanger
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.blocking_data_exchanger.html#module-psydac.ddm.blocking_data_exchanger">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.blocking_data_exchanger.html#module-psydac.ddm.blocking_data_exchanger">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.cart
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.cart.html#module-psydac.ddm.cart">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.cart.html#module-psydac.ddm.cart">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.interface_data_exchanger
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.interface_data_exchanger.html#module-psydac.ddm.interface_data_exchanger">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.interface_data_exchanger.html#module-psydac.ddm.interface_data_exchanger">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.nonblocking_data_exchanger
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.nonblocking_data_exchanger.html#module-psydac.ddm.nonblocking_data_exchanger">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.nonblocking_data_exchanger.html#module-psydac.ddm.nonblocking_data_exchanger">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.partition
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.partition.html#module-psydac.ddm.partition">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.partition.html#module-psydac.ddm.partition">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.petsc
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.petsc.html#module-psydac.ddm.petsc">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.petsc.html#module-psydac.ddm.petsc">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.html#module-psydac.ddm.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.tests.test_cart_1d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.test_cart_1d.html#module-psydac.ddm.tests.test_cart_1d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.tests.test_cart_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.test_cart_2d.html#module-psydac.ddm.tests.test_cart_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.tests.test_cart_3d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.test_cart_3d.html#module-psydac.ddm.tests.test_cart_3d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.tests.test_multicart_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.test_multicart_2d.html#module-psydac.ddm.tests.test_multicart_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.tests.test_partition
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.tests.test_partition.html#module-psydac.ddm.tests.test_partition">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.ddm.utilities
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.ddm.utilities.html#module-psydac.ddm.utilities">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.utilities.html#module-psydac.ddm.utilities">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.html#module-psydac.feec">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.derivatives
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.derivatives.html#module-psydac.feec.derivatives">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.derivatives.html#module-psydac.feec.derivatives">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.global_projectors
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.global_projectors.html#module-psydac.feec.global_projectors">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.global_projectors.html#module-psydac.feec.global_projectors">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.html#module-psydac.feec.multipatch">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.api
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.api.html#module-psydac.feec.multipatch.api">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.api.html#module-psydac.feec.multipatch.api">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.examples
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.html#module-psydac.feec.multipatch.examples">module</a>
+</li>
+      </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li>
+    psydac.feec.multipatch.examples.h1_source_pbms_conga_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.h1_source_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.h1_source_pbms_conga_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.examples.ppc_test_cases
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.examples.ppc_test_cases.html#module-psydac.feec.multipatch.examples.ppc_test_cases">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.fem_linear_operators
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.fem_linear_operators.html#module-psydac.feec.multipatch.fem_linear_operators">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.fem_linear_operators.html#module-psydac.feec.multipatch.fem_linear_operators">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.multipatch_domain_utilities
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.multipatch_domain_utilities.html#module-psydac.feec.multipatch.multipatch_domain_utilities">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.operators
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.operators.html#module-psydac.feec.multipatch.operators">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.operators.html#module-psydac.feec.multipatch.operators">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.plotting_utilities
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.plotting_utilities.html#module-psydac.feec.multipatch.plotting_utilities">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.utilities
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.utilities.html#module-psydac.feec.multipatch.utilities">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.utilities.html#module-psydac.feec.multipatch.utilities">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.multipatch.utils_conga_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.multipatch.utils_conga_2d.html#module-psydac.feec.multipatch.utils_conga_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.pull_push
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.pull_push.html#module-psydac.feec.pull_push">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.pull_push.html#module-psydac.feec.pull_push">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.pushforward
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.pushforward.html#module-psydac.feec.pushforward">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.pushforward.html#module-psydac.feec.pushforward">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.html#module-psydac.feec.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.tests.test_axis_projection
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_axis_projection.html#module-psydac.feec.tests.test_axis_projection">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.tests.test_commuting_projections
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_commuting_projections.html#module-psydac.feec.tests.test_commuting_projections">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.tests.test_commuting_projections_dual
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_commuting_projections_dual.html#module-psydac.feec.tests.test_commuting_projections_dual">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.tests.test_differentiation_matrices
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_differentiation_matrices.html#module-psydac.feec.tests.test_differentiation_matrices">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.tests.test_global_projectors
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_global_projectors.html#module-psydac.feec.tests.test_global_projectors">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.feec.tests.test_projections_parallel
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.feec.tests.test_projections_parallel.html#module-psydac.feec.tests.test_projections_parallel">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.html#module-psydac.fem">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.basic
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.basic.html#module-psydac.fem.basic">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.basic.html#module-psydac.fem.basic">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.context
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.context.html#module-psydac.fem.context">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.context.html#module-psydac.fem.context">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.grid
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.grid.html#module-psydac.fem.grid">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.grid.html#module-psydac.fem.grid">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.partitioning
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.partitioning.html#module-psydac.fem.partitioning">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.partitioning.html#module-psydac.fem.partitioning">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.projectors
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.projectors.html#module-psydac.fem.projectors">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.projectors.html#module-psydac.fem.projectors">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.splines
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.splines.html#module-psydac.fem.splines">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.splines.html#module-psydac.fem.splines">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tensor
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tensor.html#module-psydac.fem.tensor">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.tensor.html#module-psydac.fem.tensor">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.html#module-psydac.fem.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tests.analytical_profiles_1d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.analytical_profiles_1d.html#module-psydac.fem.tests.analytical_profiles_1d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tests.analytical_profiles_base
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.analytical_profiles_base.html#module-psydac.fem.tests.analytical_profiles_base">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tests.splines_error_bounds
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.splines_error_bounds.html#module-psydac.fem.tests.splines_error_bounds">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tests.test_eval_fields_parallel
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.test_eval_fields_parallel.html#module-psydac.fem.tests.test_eval_fields_parallel">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tests.test_product
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.test_product.html#module-psydac.fem.tests.test_product">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tests.test_spline_histopolation
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.test_spline_histopolation.html#module-psydac.fem.tests.test_spline_histopolation">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tests.test_spline_interpolation
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.test_spline_interpolation.html#module-psydac.fem.tests.test_spline_interpolation">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tests.test_splines
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.test_splines.html#module-psydac.fem.tests.test_splines">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tests.test_splines_par
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.test_splines_par.html#module-psydac.fem.tests.test_splines_par">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.tests.utilities
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.tests.utilities.html#module-psydac.fem.tests.utilities">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.fem.vector
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.fem.vector.html#module-psydac.fem.vector">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.vector.html#module-psydac.fem.vector">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.html#module-psydac.linalg">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.basic
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.basic.html#module-psydac.linalg.basic">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.basic.html#module-psydac.linalg.basic">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.block
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.block.html#module-psydac.linalg.block">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.block.html#module-psydac.linalg.block">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.direct_solvers
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.direct_solvers.html#module-psydac.linalg.direct_solvers">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.direct_solvers.html#module-psydac.linalg.direct_solvers">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.fft
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.fft.html#module-psydac.linalg.fft">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.fft.html#module-psydac.linalg.fft">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.kernels
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.kernels.html#module-psydac.linalg.kernels">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.html#module-psydac.linalg.kernels">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.kernels.axpy_kernels
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.kernels.axpy_kernels.html#module-psydac.linalg.kernels.axpy_kernels">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.axpy_kernels.html#module-psydac.linalg.kernels.axpy_kernels">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.kernels.inner_kernels
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.kernels.inner_kernels.html#module-psydac.linalg.kernels.inner_kernels">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.inner_kernels.html#module-psydac.linalg.kernels.inner_kernels">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.kernels.matvec_kernels
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.kernels.matvec_kernels.html#module-psydac.linalg.kernels.matvec_kernels">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.matvec_kernels.html#module-psydac.linalg.kernels.matvec_kernels">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.kernels.stencil2coo_kernels
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.kernels.stencil2coo_kernels.html#module-psydac.linalg.kernels.stencil2coo_kernels">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.stencil2coo_kernels.html#module-psydac.linalg.kernels.stencil2coo_kernels">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.kernels.transpose_kernels
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.kernels.transpose_kernels.html#module-psydac.linalg.kernels.transpose_kernels">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.transpose_kernels.html#module-psydac.linalg.kernels.transpose_kernels">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.kron
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.kron.html#module-psydac.linalg.kron">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.kron.html#module-psydac.linalg.kron">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.solvers
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.solvers.html#module-psydac.linalg.solvers">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.solvers.html#module-psydac.linalg.solvers">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.stencil
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.stencil.html#module-psydac.linalg.stencil">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.stencil.html#module-psydac.linalg.stencil">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.tests.html#module-psydac.linalg.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.tests.test_block
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.tests.test_block.html#module-psydac.linalg.tests.test_block">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.tests.test_fft
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.tests.test_fft.html#module-psydac.linalg.tests.test_fft">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.tests.test_kron_direct_solver
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.tests.test_kron_direct_solver.html#module-psydac.linalg.tests.test_kron_direct_solver">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.tests.test_kron_stencil_matrix
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.tests.test_kron_stencil_matrix.html#module-psydac.linalg.tests.test_kron_stencil_matrix">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.tests.test_linalg
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.tests.test_linalg.html#module-psydac.linalg.tests.test_linalg">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.tests.test_solvers
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.tests.test_solvers.html#module-psydac.linalg.tests.test_solvers">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.tests.test_stencil_interface_matrix
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.tests.test_stencil_interface_matrix.html#module-psydac.linalg.tests.test_stencil_interface_matrix">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.tests.test_stencil_matrix
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.tests.test_stencil_matrix.html#module-psydac.linalg.tests.test_stencil_matrix">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.tests.test_stencil_vector
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.tests.test_stencil_vector.html#module-psydac.linalg.tests.test_stencil_vector">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.tests.test_stencil_vector_space
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.tests.test_stencil_vector_space.html#module-psydac.linalg.tests.test_stencil_vector_space">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.topetsc
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.topetsc.html#module-psydac.linalg.topetsc">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.topetsc.html#module-psydac.linalg.topetsc">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.linalg.utilities
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.linalg.utilities.html#module-psydac.linalg.utilities">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.utilities.html#module-psydac.linalg.utilities">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.mapping.html#module-psydac.mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.mapping.discrete
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.mapping.discrete.html#module-psydac.mapping.discrete">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.mapping.discrete.html#module-psydac.mapping.discrete">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.mapping.discrete_gallery
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.mapping.discrete_gallery.html#module-psydac.mapping.discrete_gallery">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.mapping.discrete_gallery.html#module-psydac.mapping.discrete_gallery">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.mapping.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.mapping.tests.html#module-psydac.mapping.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.mapping.tests.test_discrete_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.mapping.tests.test_discrete_mapping.html#module-psydac.mapping.tests.test_discrete_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.mapping.tests.visual_test_discrete_mapping_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.mapping.tests.visual_test_discrete_mapping_2d.html#module-psydac.mapping.tests.visual_test_discrete_mapping_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.mapping.tests.visual_test_discrete_mapping_3d_surface
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.mapping.tests.visual_test_discrete_mapping_3d_surface.html#module-psydac.mapping.tests.visual_test_discrete_mapping_3d_surface">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.polar
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.polar.html#module-psydac.polar">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.polar.c1_cart
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.polar.c1_cart.html#module-psydac.polar.c1_cart">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.polar.c1_cart.html#module-psydac.polar.c1_cart">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.polar.c1_linops
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.polar.c1_linops.html#module-psydac.polar.c1_linops">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.polar.c1_linops.html#module-psydac.polar.c1_linops">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.polar.c1_projections
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.polar.c1_projections.html#module-psydac.polar.c1_projections">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.polar.c1_projections.html#module-psydac.polar.c1_projections">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.polar.c1_spaces
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.polar.c1_spaces.html#module-psydac.polar.c1_spaces">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.polar.c1_spaces.html#module-psydac.polar.c1_spaces">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.polar.dense
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.polar.dense.html#module-psydac.polar.dense">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.polar.dense.html#module-psydac.polar.dense">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.polar.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.polar.tests.html#module-psydac.polar.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.polar.tests.test_c1_linops
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.polar.tests.test_c1_linops.html#module-psydac.polar.tests.test_c1_linops">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.polar.tests.test_c1_projections
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.polar.tests.test_c1_projections.html#module-psydac.polar.tests.test_c1_projections">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.html#module-psydac.pyccel">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.html#module-psydac.pyccel.ast">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast.basic
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.basic.html#module-psydac.pyccel.ast.basic">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast.builtins
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.builtins.html#module-psydac.pyccel.ast.builtins">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast.core
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.core.html#module-psydac.pyccel.ast.core">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast.datatypes
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.datatypes.html#module-psydac.pyccel.ast.datatypes">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast.functionalexpr
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.functionalexpr.html#module-psydac.pyccel.ast.functionalexpr">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast.mathext
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.mathext.html#module-psydac.pyccel.ast.mathext">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast.numbers
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.numbers.html#module-psydac.pyccel.ast.numbers">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast.numpyext
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.numpyext.html#module-psydac.pyccel.ast.numpyext">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast.singleton
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.singleton.html#module-psydac.pyccel.ast.singleton">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast.type_inference
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.type_inference.html#module-psydac.pyccel.ast.type_inference">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.ast.utilities
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.ast.utilities.html#module-psydac.pyccel.ast.utilities">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.codegen
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.codegen.html#module-psydac.pyccel.codegen">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.codegen.printing
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.codegen.printing.html#module-psydac.pyccel.codegen.printing">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.codegen.printing.pycode
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.codegen.printing.pycode.html#module-psydac.pyccel.codegen.printing.pycode">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.errors
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.errors.html#module-psydac.pyccel.errors">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.errors.errors
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.errors.errors.html#module-psydac.pyccel.errors.errors">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.errors.messages
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.errors.messages.html#module-psydac.pyccel.errors.messages">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.symbolic
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.symbolic.html#module-psydac.pyccel.symbolic">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.pyccel.symbolic.lambdify
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.pyccel.symbolic.lambdify.html#module-psydac.pyccel.symbolic.lambdify">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.utilities
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.utilities.html#module-psydac.utilities">module</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.utilities.quadratures
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.utilities.quadratures.html#module-psydac.utilities.quadratures">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.utilities.quadratures.html#module-psydac.utilities.quadratures">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.utilities.utils
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.utilities.utils.html#module-psydac.utilities.utils">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.utilities.utils.html#module-psydac.utilities.utils">[1]</a>
+</li>
+      </ul></li>
+      <li>
+    psydac.utilities.vtk
+
+      <ul>
+        <li><a href="documentation/_autosummary/psydac.utilities.vtk.html#module-psydac.utilities.vtk">module</a>, <a href="psydac/docs/source/modules/STUBDIR/psydac.utilities.vtk.html#module-psydac.utilities.vtk">[1]</a>
+</li>
+      </ul></li>
+  </ul></td>
+</tr></table>
+
+<h2 id="S">S</h2>
+<table style="width: 100%" class="indextable genindextable"><tr>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li>
+    sympde.calculus
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.calculus.html#module-sympde.calculus">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.core
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.core.html#module-sympde.core">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.core.algebra
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.core.algebra.html#module-sympde.core.algebra">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.core.basic
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.core.basic.html#module-sympde.core.basic">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.core.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.core.tests.html#module-sympde.core.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.core.tests.test_utils
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.core.tests.test_utils.html#module-sympde.core.tests.test_utils">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.core.utils
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.core.utils.html#module-sympde.core.utils">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.html#module-sympde.expr">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.basic
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.basic.html#module-sympde.expr.basic">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.equation
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.equation.html#module-sympde.expr.equation">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.errors
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.errors.html#module-sympde.expr.errors">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.evaluation
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.evaluation.html#module-sympde.expr.evaluation">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.expr
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.expr.html#module-sympde.expr.expr">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.tests.html#module-sympde.expr.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.tests.test_equation_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.tests.test_equation_2d.html#module-sympde.expr.tests.test_equation_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.tests.test_essential_bc
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.tests.test_essential_bc.html#module-sympde.expr.tests.test_essential_bc">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.tests.test_expr
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.tests.test_expr.html#module-sympde.expr.tests.test_expr">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.tests.test_find_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.tests.test_find_2d.html#module-sympde.expr.tests.test_find_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.tests.test_newton_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.tests.test_newton_2d.html#module-sympde.expr.tests.test_newton_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.expr.tests.test_tensor_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html#module-sympde.expr.tests.test_tensor_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.exterior
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.exterior.html#module-sympde.exterior">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.exterior.calculus
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.exterior.calculus.html#module-sympde.exterior.calculus">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.exterior.compiler
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.exterior.compiler.html#module-sympde.exterior.compiler">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.exterior.datatype
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.exterior.datatype.html#module-sympde.exterior.datatype">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.exterior.form
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.exterior.form.html#module-sympde.exterior.form">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.exterior.inference
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.exterior.inference.html#module-sympde.exterior.inference">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.exterior.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.exterior.tests.html#module-sympde.exterior.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.exterior.tests.test_compiler
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.exterior.tests.test_compiler.html#module-sympde.exterior.tests.test_compiler">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.exterior.tests.test_datatype
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.exterior.tests.test_datatype.html#module-sympde.exterior.tests.test_datatype">module</a>
+</li>
+      </ul></li>
+  </ul></td>
+  <td style="width: 33%; vertical-align: top;"><ul>
+      <li>
+    sympde.exterior.tests.test_exterior
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.exterior.tests.test_exterior.html#module-sympde.exterior.tests.test_exterior">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.exterior.tests.test_inference
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.exterior.tests.test_inference.html#module-sympde.exterior.tests.test_inference">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.printing
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.printing.html#module-sympde.printing">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.printing.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.printing.tests.html#module-sympde.printing.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.printing.tests.test_latex
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.printing.tests.test_latex.html#module-sympde.printing.tests.test_latex">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.html#module-sympde.topology">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.analytical_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.analytical_mapping.html#module-sympde.topology.analytical_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.basic
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.basic.html#module-sympde.topology.basic">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.callable_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.callable_mapping.html#module-sympde.topology.callable_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.datatype
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.datatype.html#module-sympde.topology.datatype">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.derivatives
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.derivatives.html#module-sympde.topology.derivatives">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.domain
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.domain.html#module-sympde.topology.domain">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.mapping.html#module-sympde.topology.mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.measure
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.measure.html#module-sympde.topology.measure">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.space
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.space.html#module-sympde.topology.space">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.tests
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.html#module-sympde.topology.tests">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.tests.test_callable_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_callable_mapping.html#module-sympde.topology.tests.test_callable_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.tests.test_derivatives
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_derivatives.html#module-sympde.topology.tests.test_derivatives">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.tests.test_field_2d
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_field_2d.html#module-sympde.topology.tests.test_field_2d">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.tests.test_gallery
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_gallery.html#module-sympde.topology.tests.test_gallery">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.tests.test_logical_expr
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_logical_expr.html#module-sympde.topology.tests.test_logical_expr">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.tests.test_mapping
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_mapping.html#module-sympde.topology.tests.test_mapping">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.tests.test_space
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_space.html#module-sympde.topology.tests.test_space">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.tests.test_symbolic_expr
+
+      <ul>
+        <li><a href="documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html#module-sympde.topology.tests.test_symbolic_expr">module</a>
+</li>
+      </ul></li>
+      <li>
+    sympde.topology.tests.test_topology
+
+      <ul>
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+<li class="toctree-l2"><a class="reference internal" href="chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+    <h1>IGA-Python</h1>
+    <!-- Table of contents -->
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+            
+            <div>
+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#todo">TODO</a><ul class="nav section-nav flex-column">
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+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="iga-python">
+<h1>IGA-Python<a class="headerlink" href="#iga-python" title="Link to this heading">#</a></h1>
+<p>Welcome to IGA-Python, a tutorial for Isogeometric Analysis using Python.</p>
+<section id="todo">
+<h2>TODO<a class="headerlink" href="#todo" title="Link to this heading">#</a></h2>
+<ul class="simple">
+<li><p>Subdomains</p></li>
+<li><p>Geometry and Meshes</p></li>
+<li><p>More models</p></li>
+<li><p>Visualization</p></li>
+<li><p>Documentation using docstrings from sympde and psydac</p></li>
+<li><p>GLT</p></li>
+<li><p>DOcumentation using docstrings from gelato</p></li>
+</ul>
+<div class="toctree-wrapper compound">
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Introduction</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="chapter0/howto.html">What to expect from IGA-Python</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter0/cad.html">Computer Aided Design</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter0/iga.html">Isogeometric Analysis</a></li>
+</ul>
+</div>
+<div class="toctree-wrapper compound">
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="chapter1/install.html">Installation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter1/sympde.html">SymPDE concepts and their mathematical meaning</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+</div>
+<div class="toctree-wrapper compound">
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/advection-diffusion.html">Advection-diffusion equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/stokes.html">Mixed FEM for the Stokes problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+</div>
+<div class="toctree-wrapper compound">
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</div>
+<div class="toctree-wrapper compound">
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="chapter4/subdomains.html">Subdomains</a></li>
+</ul>
+</div>
+<div class="toctree-wrapper compound">
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/cfd.html">Computational Fluid Dynamics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
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+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/psydac.api.tests.test_equation.html">psydac.api.tests.test_equation</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.ast.mathext.html">psydac.pyccel.ast.mathext</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.ast.numbers.html">psydac.pyccel.ast.numbers</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.ast.numpyext.html">psydac.pyccel.ast.numpyext</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.ast.singleton.html">psydac.pyccel.ast.singleton</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.ast.type_inference.html">psydac.pyccel.ast.type_inference</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.ast.utilities.html">psydac.pyccel.ast.utilities</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.codegen.html">psydac.pyccel.codegen</a><ul>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.codegen.printing.html">psydac.pyccel.codegen.printing</a><ul>
+<li class="toctree-l5"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.codegen.printing.pycode.html">psydac.pyccel.codegen.printing.pycode</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.errors.html">psydac.pyccel.errors</a><ul>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.errors.errors.html">psydac.pyccel.errors.errors</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.errors.messages.html">psydac.pyccel.errors.messages</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.symbolic.html">psydac.pyccel.symbolic</a><ul>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/psydac.pyccel.symbolic.lambdify.html">psydac.pyccel.symbolic.lambdify</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2"><a class="reference internal" href="documentation/_autosummary/psydac.utilities.html">psydac.utilities</a><ul>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/psydac.utilities.quadratures.html">psydac.utilities.quadratures</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/psydac.utilities.utils.html">psydac.utilities.utils</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/psydac.utilities.vtk.html">psydac.utilities.vtk</a></li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+</div>
+</section>
+</section>
+
+    <script type="text/x-thebe-config">
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+        },
+        codeMirrorConfig: {
+            theme: "abcdef",
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+        kernelOptions: {
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new file mode 100644
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+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="chapter0/howto.html">What to expect from IGA-Python</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter0/cad.html">Computer Aided Design</a><input class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-1"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2 has-children"><a class="reference internal" href="chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="chapter0/bezier-curves.html">Bézier curves</a></li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="chapter0/bsplines.html">B-Splines</a><input class="toctree-checkbox" id="toctree-checkbox-3" name="toctree-checkbox-3" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-3"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="chapter0/bsplines-curves.html">B-Splines curves</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter0/bsplines-surfaces.html">B-Splines surfaces</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter0/iga.html">Isogeometric Analysis</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter0/fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter0/data-structure.html">Data Structure</a></li>
+
+<li class="toctree-l2"><a class="reference internal" href="chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="chapter1/install.html">Installation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter1/sympde.html">SymPDE concepts and their mathematical meaning</a><input class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-5"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter1/topology.html">Topology concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter1/space.html">Function Space concepts</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="documentation/sympde.html">SymPDE API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-13" name="toctree-checkbox-13" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-13"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.core.algebra.html">sympde.core.algebra</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.core.basic.html">sympde.core.basic</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="documentation/_autosummary/sympde.core.tests.html">sympde.core.tests</a><input class="toctree-checkbox" id="toctree-checkbox-15" name="toctree-checkbox-15" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-15"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.core.tests.test_utils.html">sympde.core.tests.test_utils</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.core.utils.html">sympde.core.utils</a></li>
+</ul>
+</li>
+<li class="toctree-l2"><a class="reference internal" href="documentation/_autosummary/sympde.calculus.html">sympde.calculus</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="documentation/_autosummary/sympde.topology.html">sympde.topology</a><input class="toctree-checkbox" id="toctree-checkbox-16" name="toctree-checkbox-16" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-16"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.topology.analytical_mapping.html">sympde.topology.analytical_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.topology.basic.html">sympde.topology.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.topology.callable_mapping.html">sympde.topology.callable_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.topology.datatype.html">sympde.topology.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.topology.derivatives.html">sympde.topology.derivatives</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.topology.domain.html">sympde.topology.domain</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.topology.tests.test_callable_mapping.html">sympde.topology.tests.test_callable_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.topology.tests.test_derivatives.html">sympde.topology.tests.test_derivatives</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.topology.tests.test_field_2d.html">sympde.topology.tests.test_field_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toctree-l2 has-children"><a class="reference internal" href="documentation/_autosummary/sympde.expr.html">sympde.expr</a><input class="toctree-checkbox" id="toctree-checkbox-18" name="toctree-checkbox-18" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-18"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.expr.basic.html">sympde.expr.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.expr.equation.html">sympde.expr.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.expr.tests.test_equation_2d.html">sympde.expr.tests.test_equation_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="documentation/_autosummary/sympde.printing.html">sympde.printing</a><input class="toctree-checkbox" id="toctree-checkbox-20" name="toctree-checkbox-20" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-20"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.printing.latex.html">sympde.printing.latex</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="documentation/_autosummary/sympde.printing.tests.html">sympde.printing.tests</a><input class="toctree-checkbox" id="toctree-checkbox-21" name="toctree-checkbox-21" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-21"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="documentation/_autosummary/sympde.printing.tests.test_latex.html">sympde.printing.tests.test_latex</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toctree-l2 has-children"><a class="reference internal" href="documentation/_autosummary/sympde.exterior.html">sympde.exterior</a><input class="toctree-checkbox" id="toctree-checkbox-22" name="toctree-checkbox-22" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-22"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.exterior.calculus.html">sympde.exterior.calculus</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1></h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
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+    </div>
+</div>
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+              
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+                <article class="bd-article">
+                  
+
+   <h1>Proof Index</h1>
+
+   <div class="modindex-jumpbox">
+   <a href="#cap-algo-decasteljau"><strong>algo-decasteljau</strong></a> | 
+   <a href="#cap-bezier-curve-ex-1"><strong>bezier-curve-ex-1</strong></a> | 
+   <a href="#cap-bezier-curve-ex-2"><strong>bezier-curve-ex-2</strong></a> | 
+   <a href="#cap-bezier-curve-ex-3"><strong>bezier-curve-ex-3</strong></a> | 
+   <a href="#cap-bezier-curve-ex-4"><strong>bezier-curve-ex-4</strong></a> | 
+   <a href="#cap-bezier-curve-ex-5"><strong>bezier-curve-ex-5</strong></a> | 
+   <a href="#cap-bezier-curve-ex-6"><strong>bezier-curve-ex-6</strong></a> | 
+   <a href="#cap-bezier-curve-ex-7"><strong>bezier-curve-ex-7</strong></a> | 
+   <a href="#cap-bsplines-curves-ex-1"><strong>bsplines-curves-ex-1</strong></a> | 
+   <a href="#cap-bsplines-curves-ex-2"><strong>bsplines-curves-ex-2</strong></a> | 
+   <a href="#cap-bsplines-curves-ex-3"><strong>bsplines-curves-ex-3</strong></a> | 
+   <a href="#cap-bsplines-curves-ex-4"><strong>bsplines-curves-ex-4</strong></a> | 
+   <a href="#cap-bsplines-curves-ex-5"><strong>bsplines-curves-ex-5</strong></a> | 
+   <a href="#cap-bsplines-curves-ex-6"><strong>bsplines-curves-ex-6</strong></a> | 
+   <a href="#cap-bsplines-curves-ex-7"><strong>bsplines-curves-ex-7</strong></a> | 
+   <a href="#cap-bsplines-curves-ex-8"><strong>bsplines-curves-ex-8</strong></a> | 
+   <a href="#cap-bsplines-ex-1"><strong>bsplines-ex-1</strong></a> | 
+   <a href="#cap-bsplines-ex-10"><strong>bsplines-ex-10</strong></a> | 
+   <a href="#cap-bsplines-ex-11"><strong>bsplines-ex-11</strong></a> | 
+   <a href="#cap-bsplines-ex-12"><strong>bsplines-ex-12</strong></a> | 
+   <a href="#cap-bsplines-ex-13"><strong>bsplines-ex-13</strong></a> | 
+   <a href="#cap-bsplines-ex-14"><strong>bsplines-ex-14</strong></a> | 
+   <a href="#cap-bsplines-ex-15"><strong>bsplines-ex-15</strong></a> | 
+   <a href="#cap-bsplines-ex-16"><strong>bsplines-ex-16</strong></a> | 
+   <a href="#cap-bsplines-ex-17"><strong>bsplines-ex-17</strong></a> | 
+   <a href="#cap-bsplines-ex-2"><strong>bsplines-ex-2</strong></a> | 
+   <a href="#cap-bsplines-ex-3"><strong>bsplines-ex-3</strong></a> | 
+   <a href="#cap-bsplines-ex-4"><strong>bsplines-ex-4</strong></a> | 
+   <a href="#cap-bsplines-ex-5"><strong>bsplines-ex-5</strong></a> | 
+   <a href="#cap-bsplines-ex-6"><strong>bsplines-ex-6</strong></a> | 
+   <a href="#cap-bsplines-ex-7"><strong>bsplines-ex-7</strong></a> | 
+   <a href="#cap-bsplines-ex-8"><strong>bsplines-ex-8</strong></a> | 
+   <a href="#cap-bsplines-ex-9"><strong>bsplines-ex-9</strong></a> | 
+   <a href="#cap-def-bspline"><strong>def-bspline</strong></a> | 
+   <a href="#cap-definition-0"><strong>definition-0</strong></a> | 
+   <a href="#cap-definition-15"><strong>definition-15</strong></a> | 
+   <a href="#cap-definition-5"><strong>definition-5</strong></a> | 
+   <a href="#cap-lemma-19"><strong>lemma-19</strong></a> | 
+   <a href="#cap-nurbs-curves-ex-1"><strong>nurbs-curves-ex-1</strong></a> | 
+   <a href="#cap-nurbs-curves-ex-2"><strong>nurbs-curves-ex-2</strong></a> | 
+   <a href="#cap-nurbs-curves-ex-3"><strong>nurbs-curves-ex-3</strong></a> | 
+   <a href="#cap-nurbs-curves-ex-4"><strong>nurbs-curves-ex-4</strong></a> | 
+   <a href="#cap-nurbs-curves-ex-5"><strong>nurbs-curves-ex-5</strong></a> | 
+   <a href="#cap-proposition-1"><strong>proposition-1</strong></a> | 
+   <a href="#cap-proposition-11"><strong>proposition-11</strong></a> | 
+   <a href="#cap-proposition-13"><strong>proposition-13</strong></a> | 
+   <a href="#cap-proposition-14"><strong>proposition-14</strong></a> | 
+   <a href="#cap-proposition-15"><strong>proposition-15</strong></a> | 
+   <a href="#cap-proposition-16"><strong>proposition-16</strong></a> | 
+   <a href="#cap-proposition-20"><strong>proposition-20</strong></a> | 
+   <a href="#cap-proposition-21"><strong>proposition-21</strong></a> | 
+   <a href="#cap-proposition-23"><strong>proposition-23</strong></a> | 
+   <a href="#cap-proposition-6"><strong>proposition-6</strong></a> | 
+   <a href="#cap-proposition-8"><strong>proposition-8</strong></a> | 
+   <a href="#cap-proposition-9"><strong>proposition-9</strong></a> | 
+   <a href="#cap-remark-1"><strong>remark-1</strong></a> | 
+   <a href="#cap-remark-10"><strong>remark-10</strong></a> | 
+   <a href="#cap-remark-12"><strong>remark-12</strong></a> | 
+   <a href="#cap-remark-17"><strong>remark-17</strong></a> | 
+   <a href="#cap-remark-18"><strong>remark-18</strong></a> | 
+   <a href="#cap-remark-22"><strong>remark-22</strong></a> | 
+   <a href="#cap-remark-7"><strong>remark-7</strong></a>
+   </div>
+
+   <table class="indextable modindextable">
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-algo-decasteljau"><td></td><td>
+       <strong>algo-decasteljau</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#algo-decasteljau"><code class="xref">algo-decasteljau</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>algorithm</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bezier-curve-ex-1"><td></td><td>
+       <strong>bezier-curve-ex-1</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#bezier-curve-ex-1"><code class="xref">bezier-curve-ex-1</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bezier-curve-ex-2"><td></td><td>
+       <strong>bezier-curve-ex-2</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#bezier-curve-ex-2"><code class="xref">bezier-curve-ex-2</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bezier-curve-ex-3"><td></td><td>
+       <strong>bezier-curve-ex-3</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#bezier-curve-ex-3"><code class="xref">bezier-curve-ex-3</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bezier-curve-ex-4"><td></td><td>
+       <strong>bezier-curve-ex-4</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#bezier-curve-ex-4"><code class="xref">bezier-curve-ex-4</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bezier-curve-ex-5"><td></td><td>
+       <strong>bezier-curve-ex-5</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#bezier-curve-ex-5"><code class="xref">bezier-curve-ex-5</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bezier-curve-ex-6"><td></td><td>
+       <strong>bezier-curve-ex-6</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#bezier-curve-ex-6"><code class="xref">bezier-curve-ex-6</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bezier-curve-ex-7"><td></td><td>
+       <strong>bezier-curve-ex-7</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#bezier-curve-ex-7"><code class="xref">bezier-curve-ex-7</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-curves-ex-1"><td></td><td>
+       <strong>bsplines-curves-ex-1</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#bsplines-curves-ex-1"><code class="xref">bsplines-curves-ex-1</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-curves-ex-2"><td></td><td>
+       <strong>bsplines-curves-ex-2</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#bsplines-curves-ex-2"><code class="xref">bsplines-curves-ex-2</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-curves-ex-3"><td></td><td>
+       <strong>bsplines-curves-ex-3</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#bsplines-curves-ex-3"><code class="xref">bsplines-curves-ex-3</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-curves-ex-4"><td></td><td>
+       <strong>bsplines-curves-ex-4</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#bsplines-curves-ex-4"><code class="xref">bsplines-curves-ex-4</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-curves-ex-5"><td></td><td>
+       <strong>bsplines-curves-ex-5</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#bsplines-curves-ex-5"><code class="xref">bsplines-curves-ex-5</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-curves-ex-6"><td></td><td>
+       <strong>bsplines-curves-ex-6</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#bsplines-curves-ex-6"><code class="xref">bsplines-curves-ex-6</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-curves-ex-7"><td></td><td>
+       <strong>bsplines-curves-ex-7</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#bsplines-curves-ex-7"><code class="xref">bsplines-curves-ex-7</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-curves-ex-8"><td></td><td>
+       <strong>bsplines-curves-ex-8</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#bsplines-curves-ex-8"><code class="xref">bsplines-curves-ex-8</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-1"><td></td><td>
+       <strong>bsplines-ex-1</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-1"><code class="xref">bsplines-ex-1</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-10"><td></td><td>
+       <strong>bsplines-ex-10</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-10"><code class="xref">bsplines-ex-10</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-11"><td></td><td>
+       <strong>bsplines-ex-11</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-11"><code class="xref">bsplines-ex-11</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-12"><td></td><td>
+       <strong>bsplines-ex-12</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-12"><code class="xref">bsplines-ex-12</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-13"><td></td><td>
+       <strong>bsplines-ex-13</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-13"><code class="xref">bsplines-ex-13</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-14"><td></td><td>
+       <strong>bsplines-ex-14</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-14"><code class="xref">bsplines-ex-14</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-15"><td></td><td>
+       <strong>bsplines-ex-15</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-15"><code class="xref">bsplines-ex-15</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-16"><td></td><td>
+       <strong>bsplines-ex-16</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-16"><code class="xref">bsplines-ex-16</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-17"><td></td><td>
+       <strong>bsplines-ex-17</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-17"><code class="xref">bsplines-ex-17</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-2"><td></td><td>
+       <strong>bsplines-ex-2</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-2"><code class="xref">bsplines-ex-2</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-3"><td></td><td>
+       <strong>bsplines-ex-3</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-3"><code class="xref">bsplines-ex-3</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-4"><td></td><td>
+       <strong>bsplines-ex-4</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-4"><code class="xref">bsplines-ex-4</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-5"><td></td><td>
+       <strong>bsplines-ex-5</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-5"><code class="xref">bsplines-ex-5</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-6"><td></td><td>
+       <strong>bsplines-ex-6</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-6"><code class="xref">bsplines-ex-6</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-7"><td></td><td>
+       <strong>bsplines-ex-7</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-7"><code class="xref">bsplines-ex-7</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-8"><td></td><td>
+       <strong>bsplines-ex-8</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-8"><code class="xref">bsplines-ex-8</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-bsplines-ex-9"><td></td><td>
+       <strong>bsplines-ex-9</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#bsplines-ex-9"><code class="xref">bsplines-ex-9</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-def-bspline"><td></td><td>
+       <strong>def-bspline</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#def-bspline"><code class="xref">def-bspline</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>definition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-definition-0"><td></td><td>
+       <strong>definition-0</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#definition-0"><code class="xref">definition-0</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>definition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-definition-15"><td></td><td>
+       <strong>definition-15</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#definition-15"><code class="xref">definition-15</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>definition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-definition-5"><td></td><td>
+       <strong>definition-5</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#definition-5"><code class="xref">definition-5</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>definition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-lemma-19"><td></td><td>
+       <strong>lemma-19</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#lemma-19"><code class="xref">lemma-19</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>lemma</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-nurbs-curves-ex-1"><td></td><td>
+       <strong>nurbs-curves-ex-1</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#nurbs-curves-ex-1"><code class="xref">nurbs-curves-ex-1</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-nurbs-curves-ex-2"><td></td><td>
+       <strong>nurbs-curves-ex-2</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#nurbs-curves-ex-2"><code class="xref">nurbs-curves-ex-2</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-nurbs-curves-ex-3"><td></td><td>
+       <strong>nurbs-curves-ex-3</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#nurbs-curves-ex-3"><code class="xref">nurbs-curves-ex-3</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-nurbs-curves-ex-4"><td></td><td>
+       <strong>nurbs-curves-ex-4</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#nurbs-curves-ex-4"><code class="xref">nurbs-curves-ex-4</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-nurbs-curves-ex-5"><td></td><td>
+       <strong>nurbs-curves-ex-5</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#nurbs-curves-ex-5"><code class="xref">nurbs-curves-ex-5</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>example</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-1"><td></td><td>
+       <strong>proposition-1</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier.html#proposition-1"><code class="xref">proposition-1</code></a> <em>(chapter0/bezier)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-11"><td></td><td>
+       <strong>proposition-11</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#proposition-11"><code class="xref">proposition-11</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-13"><td></td><td>
+       <strong>proposition-13</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#proposition-13"><code class="xref">proposition-13</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-14"><td></td><td>
+       <strong>proposition-14</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#proposition-14"><code class="xref">proposition-14</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-15"><td></td><td>
+       <strong>proposition-15</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#proposition-15"><code class="xref">proposition-15</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-16"><td></td><td>
+       <strong>proposition-16</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#proposition-16"><code class="xref">proposition-16</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-20"><td></td><td>
+       <strong>proposition-20</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#proposition-20"><code class="xref">proposition-20</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-21"><td></td><td>
+       <strong>proposition-21</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#proposition-21"><code class="xref">proposition-21</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-23"><td></td><td>
+       <strong>proposition-23</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#proposition-23"><code class="xref">proposition-23</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-6"><td></td><td>
+       <strong>proposition-6</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#proposition-6"><code class="xref">proposition-6</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-8"><td></td><td>
+       <strong>proposition-8</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#proposition-8"><code class="xref">proposition-8</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-proposition-9"><td></td><td>
+       <strong>proposition-9</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#proposition-9"><code class="xref">proposition-9</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>proposition</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-remark-1"><td></td><td>
+       <strong>remark-1</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#remark-1"><code class="xref">remark-1</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>remark</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-remark-10"><td></td><td>
+       <strong>remark-10</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines-curves.html#remark-10"><code class="xref">remark-10</code></a> <em>(chapter0/bsplines-curves)</em></td><td>
+       <em>remark</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-remark-12"><td></td><td>
+       <strong>remark-12</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#remark-12"><code class="xref">remark-12</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>remark</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-remark-17"><td></td><td>
+       <strong>remark-17</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#remark-17"><code class="xref">remark-17</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>remark</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-remark-18"><td></td><td>
+       <strong>remark-18</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#remark-18"><code class="xref">remark-18</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>remark</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-remark-22"><td></td><td>
+       <strong>remark-22</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bsplines.html#remark-22"><code class="xref">remark-22</code></a> <em>(chapter0/bsplines)</em></td><td>
+       <em>remark</em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-remark-7"><td></td><td>
+       <strong>remark-7</strong></td><td></td></tr>
+     <tr>
+       <td></td>
+       <td>
+       <a href="chapter0/bezier-curves.html#remark-7"><code class="xref">remark-7</code></a> <em>(chapter0/bezier-curves)</em></td><td>
+       <em>remark</em></td></tr>
+   </table>
+
+
+                </article>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.basic.html">sympde.topology.basic</a></li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Welcome to PSYDAC</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#table-of-contents">Table of contents</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#requirements">Requirements</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#linux-debian-ubuntu-mint">Linux Debian-Ubuntu-Mint</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#macos">macOS</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#other-operating-systems">Other operating systems</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#python-setup-and-project-download">Python setup and project download</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#installing-the-library">Installing the library</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#uninstall">Uninstall</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#running-tests">Running tests</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#speeding-up-psydac-s-core">Speeding up <strong>Psydac</strong>’s core</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#user-documentation">User documentation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#code-documentation">Code documentation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mesh-generation">Mesh Generation</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#example-of-usage">Example of usage</a></li>
+</ul>
+</li>
+</ul>
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+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="welcome-to-psydac">
+<h1>Welcome to PSYDAC<a class="headerlink" href="#welcome-to-psydac" title="Link to this heading">#</a></h1>
+<p><a class="reference external" href="https://github.com/pyccel/psydac/actions/workflows/continuous-integration.yml"><img alt="devel_tests" src="https://github.com/pyccel/psydac/actions/workflows/continuous-integration.yml/badge.svg" /></a> <a class="reference external" href="https://github.com/pyccel/psydac/actions/workflows/documentation.yml"><img alt="docs" src="https://github.com/pyccel/psydac/actions/workflows/documentation.yml/badge.svg" /></a></p>
+<p><strong>PSYDAC</strong> is a Python 3 Library for isogeometric analysis.</p>
+<section id="table-of-contents">
+<h2>Table of contents<a class="headerlink" href="#table-of-contents" title="Link to this heading">#</a></h2>
+<ul class="simple">
+<li><p><a class="reference internal" href="#requirements"><span class="xref myst">Requirements</span></a></p></li>
+<li><p><a class="reference internal" href="#python-setup-and-project-download"><span class="xref myst">Python setup and project download</span></a></p></li>
+<li><p><a class="reference internal" href="#installing-the-library"><span class="xref myst">Installing the library</span></a></p></li>
+<li><p><a class="reference internal" href="#uninstall"><span class="xref myst">Uninstall</span></a></p></li>
+<li><p><a class="reference internal" href="#running-tests"><span class="xref myst">Running tests</span></a></p></li>
+<li><p><a class="reference internal" href="#speeding-up-psydacs-core"><span class="xref myst">Speeding up Psydac’s core</span></a></p></li>
+<li><p><a class="reference internal" href="#user-documentation"><span class="xref myst">User Documentation</span></a></p></li>
+<li><p><a class="reference internal" href="#code-documentation"><span class="xref myst">Code Documentation</span></a></p></li>
+</ul>
+</section>
+<section id="requirements">
+<h2>Requirements<a class="headerlink" href="#requirements" title="Link to this heading">#</a></h2>
+<p>Psydac requires a certain number of components to be installed on the machine:</p>
+<ul class="simple">
+<li><p>Fortran and C compilers with OpenMP support</p></li>
+<li><p>OpenMP library</p></li>
+<li><p>BLAS and LAPACK libraries</p></li>
+<li><p>MPI library</p></li>
+<li><p>HDF5 library with MPI support</p></li>
+</ul>
+<p>The installations instructions depend on the operating system and on the packaging manager used.</p>
+<section id="linux-debian-ubuntu-mint">
+<h3>Linux Debian-Ubuntu-Mint<a class="headerlink" href="#linux-debian-ubuntu-mint" title="Link to this heading">#</a></h3>
+<p>To install all requirements on a Linux Ubuntu operating system, just use APT, the Advanced Packaging Tool:</p>
+<div class="highlight-sh notranslate"><div class="highlight"><pre><span></span>sudo<span class="w"> </span>apt<span class="w"> </span>update
+sudo<span class="w"> </span>apt<span class="w"> </span>install<span class="w"> </span>python3<span class="w"> </span>python3-dev<span class="w"> </span>python3-pip
+sudo<span class="w"> </span>apt<span class="w"> </span>install<span class="w"> </span>gcc<span class="w"> </span>gfortran
+sudo<span class="w"> </span>apt<span class="w"> </span>install<span class="w"> </span>libblas-dev<span class="w"> </span>liblapack-dev
+sudo<span class="w"> </span>apt<span class="w"> </span>install<span class="w"> </span>libopenmpi-dev<span class="w"> </span>openmpi-bin
+sudo<span class="w"> </span>apt<span class="w"> </span>install<span class="w"> </span>libomp-dev<span class="w"> </span>libomp5
+sudo<span class="w"> </span>apt<span class="w"> </span>install<span class="w"> </span>libhdf5-openmpi-dev
+</pre></div>
+</div>
+</section>
+<section id="macos">
+<h3>macOS<a class="headerlink" href="#macos" title="Link to this heading">#</a></h3>
+<p>To install all the requirements on a macOS operating system we recommend using <a class="reference external" href="https://brew.sh/">Homebrew</a>:</p>
+<div class="highlight-eh notranslate"><div class="highlight"><pre><span></span>brew update
+brew install gcc
+brew install openblas
+brew install lapack
+brew install open-mpi
+brew install libomp
+brew install hdf5-mpi
+</pre></div>
+</div>
+</section>
+<section id="other-operating-systems">
+<h3>Other operating systems<a class="headerlink" href="#other-operating-systems" title="Link to this heading">#</a></h3>
+<p>Please see the <a class="reference external" href="https://github.com/pyccel/pyccel#Requirements">instructions for the pyccel library</a> for further details.</p>
+</section>
+</section>
+<section id="python-setup-and-project-download">
+<h2>Python setup and project download<a class="headerlink" href="#python-setup-and-project-download" title="Link to this heading">#</a></h2>
+<p>We recommend creating a clean Python virtual environment using <a class="reference external" href="https://packaging.python.org/en/latest/guides/installing-using-pip-and-virtual-environments/#creating-a-virtual-environment">venv</a>:</p>
+<div class="highlight-sh notranslate"><div class="highlight"><pre><span></span>python3<span class="w"> </span>-m<span class="w"> </span>venv<span class="w"> </span>&lt;ENV-PATH&gt;
+</pre></div>
+</div>
+<p>where <code class="docutils literal notranslate"><span class="pre">&lt;ENV-PATH&gt;</span></code> is the location to create the virtual environment.
+(A new directory will be created at the required location.)</p>
+<p>In order to activate the environment from a new terminal session just run the command</p>
+<div class="highlight-sh notranslate"><div class="highlight"><pre><span></span><span class="nb">source</span><span class="w"> </span>&lt;ENV-PATH&gt;/bin/activate
+</pre></div>
+</div>
+<p>One can clone the Psydac repository at any location <code class="docutils literal notranslate"><span class="pre">&lt;ROOT-PATH&gt;</span></code> in the filesystem which does not require administrator privileges, using either</p>
+<div class="highlight-sh notranslate"><div class="highlight"><pre><span></span>git<span class="w"> </span>clone<span class="w"> </span>https://github.com/pyccel/psydac.git
+</pre></div>
+</div>
+<p>or</p>
+<div class="highlight-sh notranslate"><div class="highlight"><pre><span></span>git<span class="w"> </span>clone<span class="w"> </span>git@github.com:pyccel/psydac.git
+</pre></div>
+</div>
+<p>The latter command requires a GitHub account.</p>
+</section>
+<section id="installing-the-library">
+<h2>Installing the library<a class="headerlink" href="#installing-the-library" title="Link to this heading">#</a></h2>
+<p>Psydac depends on several Python packages, which should be installed in the newly created virtual environment.
+These dependencies can be installed from the cloned directory <code class="docutils literal notranslate"><span class="pre">&lt;ROOT-PATH&gt;/psydac</span></code> using</p>
+<div class="highlight-sh notranslate"><div class="highlight"><pre><span></span><span class="nb">export</span><span class="w"> </span><span class="nv">CC</span><span class="o">=</span><span class="s2">&quot;mpicc&quot;</span>
+<span class="nb">export</span><span class="w"> </span><span class="nv">HDF5_MPI</span><span class="o">=</span><span class="s2">&quot;ON&quot;</span>
+<span class="nb">export</span><span class="w"> </span><span class="nv">HDF5_DIR</span><span class="o">=</span>/path/to/parallel/hdf5
+
+python3<span class="w"> </span>-m<span class="w"> </span>pip<span class="w"> </span>install<span class="w"> </span>--upgrade<span class="w"> </span>pip
+python3<span class="w"> </span>-m<span class="w"> </span>pip<span class="w"> </span>install<span class="w"> </span>-r<span class="w"> </span>requirements.txt
+python3<span class="w"> </span>-m<span class="w"> </span>pip<span class="w"> </span>install<span class="w"> </span>-r<span class="w"> </span>requirements_extra.txt<span class="w"> </span>--no-build-isolation
+</pre></div>
+</div>
+<p>where the <code class="docutils literal notranslate"><span class="pre">HDF5_DIR</span></code> environment variable should store the absolute path to the <strong>installation point</strong> of your parallel HDF5 library, which can be found with</p>
+<div class="highlight-sh notranslate"><div class="highlight"><pre><span></span>h5pcc<span class="w"> </span>-showconfig
+</pre></div>
+</div>
+<p>or (on macOS)</p>
+<div class="highlight-sh notranslate"><div class="highlight"><pre><span></span>brew<span class="w"> </span>info<span class="w"> </span>hdf5-mpi
+</pre></div>
+</div>
+<p>At this point the Psydac library may be installed in <strong>standard mode</strong>, which copies the relevant files to the correct locations of the virtual environment, or in <strong>development mode</strong>, which only installs symbolic links to the Psydac directory. The latter mode allows one to effect the behavior of Psydac by modifying the source files.</p>
+<ul>
+<li><p><strong>Standard mode</strong>:</p>
+<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span>python3<span class="w"> </span>-m<span class="w"> </span>pip<span class="w"> </span>install<span class="w"> </span>.
+</pre></div>
+</div>
+</li>
+<li><p><strong>Development mode</strong>:</p>
+<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span>python3<span class="w"> </span>-m<span class="w"> </span>pip<span class="w"> </span>install<span class="w"> </span>--editable<span class="w"> </span>.
+</pre></div>
+</div>
+</li>
+</ul>
+</section>
+<section id="uninstall">
+<h2>Uninstall<a class="headerlink" href="#uninstall" title="Link to this heading">#</a></h2>
+<ul>
+<li><p><strong>Whichever the install mode</strong>:</p>
+<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span>python3<span class="w"> </span>-m<span class="w"> </span>pip<span class="w"> </span>uninstall<span class="w"> </span>psydac
+</pre></div>
+</div>
+</li>
+</ul>
+</section>
+<section id="running-tests">
+<h2>Running tests<a class="headerlink" href="#running-tests" title="Link to this heading">#</a></h2>
+<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span><span class="nb">export</span><span class="w"> </span><span class="nv">PSYDAC_MESH_DIR</span><span class="o">=</span>/path/to/psydac/mesh/
+python3<span class="w"> </span>-m<span class="w"> </span>pytest<span class="w"> </span>--pyargs<span class="w"> </span>psydac<span class="w"> </span>-m<span class="w"> </span><span class="s2">&quot;not parallel&quot;</span>
+python3<span class="w"> </span>/path/to/psydac/mpi_tester.py<span class="w"> </span>--pyargs<span class="w"> </span>psydac<span class="w"> </span>-m<span class="w"> </span><span class="s2">&quot;parallel&quot;</span>
+</pre></div>
+</div>
+</section>
+<section id="speeding-up-psydac-s-core">
+<h2>Speeding up <strong>Psydac</strong>’s core<a class="headerlink" href="#speeding-up-psydac-s-core" title="Link to this heading">#</a></h2>
+<p>Many of Psydac’s low-level Python functions can be translated to a compiled language using the <a class="reference external" href="https://github.com/pyccel/pyccel">Pyccel</a> transpiler. Currently, all of those functions are collected in modules which follow the name pattern <code class="docutils literal notranslate"><span class="pre">[module]_kernels.py</span></code>.</p>
+<p>The classical installation translates all kernel files to Fortran without user intervention. This does not happen in the case of an editable install, but the command <code class="docutils literal notranslate"><span class="pre">psydac-accelerate</span></code> is made available to the user instead. This command applies Pyccel to all the kernel files in the source directory, and the C language may be selected instead of Fortran (which is the default).</p>
+<ul>
+<li><p><strong>Only in development mode</strong>:</p>
+<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span>python3<span class="w"> </span>/path/to/psydac/psydac_accelerate.py<span class="w"> </span><span class="o">[</span>--language<span class="w"> </span>LANGUAGE<span class="o">]</span><span class="w"> </span><span class="o">[</span>--openmp<span class="o">]</span>
+</pre></div>
+</div>
+</li>
+</ul>
+</section>
+<section id="user-documentation">
+<h2>User documentation<a class="headerlink" href="#user-documentation" title="Link to this heading">#</a></h2>
+<ul class="simple">
+<li><p><a class="reference internal" href="output.html"><span class="std std-doc">Output formats</span></a></p></li>
+<li><p><a class="reference internal" href="#./examples/notebooks/"><span class="xref myst">Notebook examples</span></a></p></li>
+<li><p><a class="reference internal" href="#./examples/"><span class="xref myst">Other examples</span></a></p></li>
+</ul>
+</section>
+<section id="code-documentation">
+<h2>Code documentation<a class="headerlink" href="#code-documentation" title="Link to this heading">#</a></h2>
+<p>Find our latest code documentation <a class="reference external" href="https://pyccel.github.io/psydac/">here</a>.</p>
+</section>
+<section id="mesh-generation">
+<h2>Mesh Generation<a class="headerlink" href="#mesh-generation" title="Link to this heading">#</a></h2>
+<p>After installation, a command <code class="docutils literal notranslate"><span class="pre">psydac-mesh</span></code> will be available.</p>
+<section id="example-of-usage">
+<h3>Example of usage<a class="headerlink" href="#example-of-usage" title="Link to this heading">#</a></h3>
+<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span>psydac-mesh<span class="w"> </span>-n<span class="o">=</span><span class="s1">&#39;16,16&#39;</span><span class="w"> </span>-d<span class="o">=</span><span class="s1">&#39;3,3&#39;</span><span class="w"> </span>square<span class="w"> </span>mesh.h5
+</pre></div>
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+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.analytical_mapping.html">sympde.topology.analytical_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.basic.html">sympde.topology.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.callable_mapping.html">sympde.topology.callable_mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.datatype.html">sympde.topology.datatype</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.mapping.html">sympde.topology.mapping</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.measure.html">sympde.topology.measure</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.space.html">sympde.topology.space</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_field_2d.html">sympde.topology.tests.test_field_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_gallery.html">sympde.topology.tests.test_gallery</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_logical_expr.html">sympde.topology.tests.test_logical_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_mapping.html">sympde.topology.tests.test_mapping</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_space.html">sympde.topology.tests.test_space</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html">sympde.topology.tests.test_symbolic_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.topology.tests.test_topology.html">sympde.topology.tests.test_topology</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.html">sympde.expr</a><input class="toctree-checkbox" id="toctree-checkbox-18" name="toctree-checkbox-18" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-18"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.equation.html">sympde.expr.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.errors.html">sympde.expr.errors</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.evaluation.html">sympde.expr.evaluation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.expr.html">sympde.expr.expr</a></li>
+<li class="toctree-l3 has-children"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.html">sympde.expr.tests</a><input class="toctree-checkbox" id="toctree-checkbox-19" name="toctree-checkbox-19" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-19"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_equation_2d.html">sympde.expr.tests.test_equation_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_essential_bc.html">sympde.expr.tests.test_essential_bc</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_expr.html">sympde.expr.tests.test_expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_find_2d.html">sympde.expr.tests.test_find_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_newton_2d.html">sympde.expr.tests.test_newton_2d</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html">sympde.expr.tests.test_tensor_2d</a></li>
+</ul>
+</li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.printing.latex.html">sympde.printing.latex</a></li>
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+</li>
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+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.compiler.html">sympde.exterior.compiler</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.datatype.html">sympde.exterior.datatype</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.form.html">sympde.exterior.form</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.inference.html">sympde.exterior.inference</a></li>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_compiler.html">sympde.exterior.tests.test_compiler</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_datatype.html">sympde.exterior.tests.test_datatype</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_exterior.html">sympde.exterior.tests.test_exterior</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/sympde.exterior.tests.test_inference.html">sympde.exterior.tests.test_inference</a></li>
+</ul>
+</li>
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+</ul>
+</li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../documentation/psydac.html">Psydac API Reference</a><input class="toctree-checkbox" id="toctree-checkbox-26" name="toctree-checkbox-26" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-26"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.expr.html">psydac.api.ast.expr</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.fem.html">psydac.api.ast.fem</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.glt.html">psydac.api.ast.glt</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.linalg.html">psydac.api.ast.linalg</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.nodes.html">psydac.api.ast.nodes</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.parser.html">psydac.api.ast.parser</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.tests.html">psydac.api.ast.tests</a></li>
+<li class="toctree-l4"><a class="reference internal" href="../documentation/_autosummary/psydac.api.ast.utilities.html">psydac.api.ast.utilities</a></li>
+</ul>
+</li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.basic.html">psydac.api.basic</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.discretization.html">psydac.api.discretization</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.equation.html">psydac.api.equation</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../documentation/_autosummary/psydac.api.essential_bc.html">psydac.api.essential_bc</a></li>
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+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="todo">
+<h1>TODO<a class="headerlink" href="#todo" title="Link to this heading">#</a></h1>
+<ul class="simple">
+<li><p>mv psydac/feec/tests/todo_test_derivatives.py to psydac/feec/tests/test_derivatives.py once it is fixed</p></li>
+<li><p>Add GMRES solver to ‘psydac.linalg.solvers’</p></li>
+<li><p>Add unit tests for Cart subcommunicators in ‘psydac.ddm.cart’</p></li>
+<li><p>Create parallel ‘KroneckerLinearOperator’ in ‘psydac.linalg.kronecker’ using Cart subcommunicators</p></li>
+<li><p>Create object of type ‘psydac.fem.psydacines.SplineSpace’ with Cart</p></li>
+<li><p>Extend functionality of ‘SplineSpace’ class:
+. add (private?) methods ‘init_fem’ and ‘init_collocation’
+. call methods above when required
+. add method ‘compute_interpolant( self, values, field )’
+. implement ‘eval_field_gradient’ method</p></li>
+<li><p>Create ‘SplineMapping’ class in module ‘psydac.mapping.discrete’</p></li>
+<li><p>add a section in documentation about hdf5 installation (serial/parallel) + h5-tools (install+usage)</p></li>
+</ul>
+<section id="core">
+<h2>Core<a class="headerlink" href="#core" title="Link to this heading">#</a></h2>
+<ul class="simple">
+<li><p>interface to <em>psydac_eval_psydacines_ders</em></p></li>
+</ul>
+</section>
+<section id="api">
+<h2>API<a class="headerlink" href="#api" title="Link to this heading">#</a></h2>
+<ul class="simple">
+<li><p>reorganize code in api (no change to codgen)</p></li>
+<li><p>additional tests and unit tests in api/tests</p></li>
+<li><p>for the moment the codegen is not using the support from space:</p>
+<ul>
+<li><p>allow <em>fem_context</em> to be called with grid and degrees, and return a discrete space and a None for mapping.</p></li>
+<li><p>use support in codegen/interface</p></li>
+</ul>
+</li>
+<li><p>normal vector in 3d</p></li>
+<li><p>periodic bc in codegen</p></li>
+<li><p>init_fem is called everytime wa call discretize: maybe we should first do some checks (nderiv is ok norder etc)</p></li>
+<li><p>hessian not yet implemented in sympde and api</p></li>
+<li><p>add other solvers to the solver_driver (only cg is available now)</p></li>
+<li><p>however, we need to clear the cache of sympy after 1d, 2d and 3d tests, otherwise pytest will crash.</p></li>
+<li><p>add sympde and pyccel install procedure; maybe wget to download the requierements files as requirements_pyccel.txt etc then call pip3</p></li>
+<li><p>api tests should be moved to examples</p></li>
+<li><p>shall we add the Boundayr object as an attribut of SplineMapping? what to do if we don’t use a mapping? we can also add the notion of a patch …</p></li>
+<li><p>DiscreteModel is not up to date; changes need to be done because of the SumForm concept that was introduced lately. in fact, things will become even easier. a Model will be like a namespace or a Module.</p></li>
+<li><p>add BoundaryDomain to define the whole boundary of a domain</p></li>
+</ul>
+</section>
+</section>
+
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+  <p><strong>PSYDAC</strong> is a Python/Fortran 2003 library for B-Splines/NURBS and Computer Aided Design Algorithms.</p>
+<p><strong>PSYDAC</strong> can be used in three different ways:</p>
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new file mode 100644
index 0000000..70c6cfc
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@@ -0,0 +1,962 @@
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+<li class="toctree-l3"><a class="reference internal" href="../../../chapter0/bsplines-surfaces.html">B-Splines surfaces</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter0/iga.html">Isogeometric Analysis</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter0/fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter0/data-structure.html">Data Structure</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter1/install.html">Installation</a></li>
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+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
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+</li>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<p><strong>PSYDAC</strong> is a Python/Fortran 2003 library for B-Splines/NURBS and Computer Aided Design Algorithms.</p>
+<p><strong>PSYDAC</strong> can be used in three different ways:</p>
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+<li><p><em>Fortran 90/95</em> subroutines through the file <strong>fortran/src/bsplines/bspline.F90</strong></p></li>
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+<li class="toctree-l2 has-children"><a class="reference internal" href="../../../chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter0/bezier-curves.html">Bézier curves</a></li>
+</ul>
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+<li class="toctree-l2 has-children"><a class="reference internal" href="../../../chapter0/bsplines.html">B-Splines</a><input class="toctree-checkbox" id="toctree-checkbox-3" name="toctree-checkbox-3" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-3"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter0/iga.html">Isogeometric Analysis</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter0/fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter0/data-structure.html">Data Structure</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter1/install.html">Installation</a></li>
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+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
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+</li>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../../chapter5/mhd.html">MHD</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+    <h1>DeRham sequences</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#pullbacks">Pullbacks</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discrete-spaces">Discrete Spaces</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#projectors">Projectors</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discrete-derham-sequence-for-b-splines">Discrete DeRham sequence for B-Splines</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#d-case">1d case</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">2d case</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">3d case</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#algebraic-identities">Algebraic identities</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="derham-sequences">
+<h1>DeRham sequences<a class="headerlink" href="#derham-sequences" title="Link to this heading">#</a></h1>
+<p>here without boundary conditions</p>
+<div class="math notranslate nohighlight">
+\[\mathbb{R} \hookrightarrow \Hgrad  \xrightarrow{\quad \Grad \quad}  \Hcurl  \xrightarrow{\quad \Curl \quad}   \Hdiv  \xrightarrow{\quad \Div \quad}  \Ltwo  \xrightarrow{\quad} 0\]</div>
+<section id="pullbacks">
+<h2>Pullbacks<a class="headerlink" href="#pullbacks" title="Link to this heading">#</a></h2>
+<p>In the case where the physical domain <span class="math notranslate nohighlight">\(\Omega := \mathcal{F}(\hat{\Omega})\)</span> is the <em>image</em> of a <em>logical</em> domain <span class="math notranslate nohighlight">\(\hat{\Omega}\)</span> by a smooth mapping <span class="math notranslate nohighlight">\(\mathcal{F}\)</span> (at least <span class="math notranslate nohighlight">\(\mathcal{C}^1\)</span>), we have the following <em>parallel</em> diagrams</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{array}{ccccccc}
+\Hgrad &amp; \xrightarrow{\quad \Grad \quad} &amp; \Hcurl &amp; \xrightarrow{\quad \Curl \quad} &amp;  \Hdiv &amp; \xrightarrow{\quad \Div \quad} &amp; \Ltwo \\
+\igrad \Bigg\uparrow   &amp;     &amp; \icurl \Bigg\uparrow  &amp;   &amp; \idiv \Bigg\uparrow &amp;  &amp; \iltwo \Bigg\uparrow       \\
+\HgradLogical &amp; \xrightarrow{\quad \Grad \quad} &amp; \HcurlLogical &amp; \xrightarrow{\quad \Curl \quad} &amp;  \HdivLogical &amp; \xrightarrow{\quad \Div \quad} &amp; \LtwoLogical \\
+%
+\end{array}\end{split}\]</div>
+<p>Where the <em>mappings</em> <span class="math notranslate nohighlight">\(\igrad, \icurl, \idiv\)</span> and <span class="math notranslate nohighlight">\(\iltwo\)</span> are called <strong>pullbacks</strong> and are given by</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\phi (x) :=&amp; \igrad \hat{\phi} (\hat{x}) = \hat{\phi}(\mathcal{F}^{-1}(x))
+\\
+\Psi (x) :=&amp; \icurl \hat{\Psi} (\hat{x}) = \left( D \mathcal{F} \right)^{-T} \hat{\Psi}(\mathcal{F}^{-1}(x))
+\\
+\Phi (x) :=&amp; \idiv \hat{\Phi} (\hat{x})  = \frac{1}{J} D \mathcal{F} \hat{\Phi}(\mathcal{F}^{-1}(x))
+\\
+\rho (x) :=&amp; \iltwo \hat{\rho} (\hat{x}) = \hat{\rho}(\mathcal{F}^{-1}(x))\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(D \mathcal{F}\)</span> is the <strong>jacobian matrix</strong> of the mapping <span class="math notranslate nohighlight">\(\mathcal{F}\)</span>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The <em>pullbacks</em> <span class="math notranslate nohighlight">\(\igrad, \icurl, \idiv\)</span> and <span class="math notranslate nohighlight">\(\iltwo\)</span> are <strong>isomorphisms</strong> between the corresponding spaces.</p>
+</div>
+</section>
+<section id="discrete-spaces">
+<h2>Discrete Spaces<a class="headerlink" href="#discrete-spaces" title="Link to this heading">#</a></h2>
+<p>Let us suppose that we have a sequence of finite subspaces for each of the spaces involved in the DeRham sequence. The discrete DeRham sequence stands for the following commutative diagram between continuous and discrete spaces</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{array}{ccccccc}
+\Hgrad &amp; \xrightarrow{\quad \Grad \quad} &amp; \Hcurl &amp; \xrightarrow{\quad \Curl \quad} &amp;  \Hdiv &amp; \xrightarrow{\quad \Div \quad} &amp; \Ltwo \\
+\Pigrad \Bigg\downarrow   &amp;     &amp; \Picurl \Bigg\downarrow  &amp;   &amp; \Pidiv \Bigg\downarrow &amp;  &amp; \Piltwo \Bigg\downarrow       \\
+\Vgrad &amp; \xrightarrow{\quad \Grad \quad} &amp; \Vcurl &amp; \xrightarrow{\quad \Curl \quad}  &amp; \Vdiv &amp; \xrightarrow{\quad \Div \quad} &amp; \Vltwo    \\
+%
+\end{array}\end{split}\]</div>
+<p>When using a Finite Elements methods, we often deal with a reference element, and thus we need also to apply the <em>pullbacks</em> on the discrete spaces. In fact, we have again the following parallel diagram</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{array}{ccccccc}
+\Vgrad &amp; \xrightarrow{\quad \Grad \quad} &amp; \Vcurl &amp; \xrightarrow{\quad \Curl \quad} &amp;  \Vdiv &amp; \xrightarrow{\quad \Div \quad} &amp; \Vltwo \\
+\igrad \Bigg\uparrow   &amp;     &amp; \icurl \Bigg\uparrow  &amp;   &amp; \idiv \Bigg\uparrow &amp;  &amp; \iltwo \Bigg\uparrow       \\
+\VgradLogical &amp; \xrightarrow{\quad \Grad \quad} &amp; \VcurlLogical &amp; \xrightarrow{\quad \Curl \quad} &amp;  \VdivLogical &amp; \xrightarrow{\quad \Div \quad} &amp; \VltwoLogical \\
+%
+\end{array}\end{split}\]</div>
+<p>Since, the <em>pullbacks</em> are <strong>isomorphisms</strong> in the previous diagram, we can define a <strong>one-to-one</strong> correspondance</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\phi :=&amp; \igrad \hat{\phi}, \quad \phi \in \Vgrad, \hat{\phi} \in \VgradLogical
+\\
+\Psi :=&amp; \icurl \hat{\Psi}, \quad \Psi \in \Vcurl, \hat{\Psi} \in \VcurlLogical
+\\
+\Phi :=&amp; \idiv \hat{\Phi}, \quad \Phi \in \Vdiv, \hat{\Phi} \in \VdivLogical
+\\
+\rho :=&amp; \iltwo \hat{\rho}, \quad \rho \in \Vltwo, \hat{\rho} \in \VltwoLogical\end{split}\]</div>
+<p>We have then, the following results</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\Grad \phi =&amp; \icurl \Grad \hat{\phi} , \quad \phi \in \Vgrad
+\\
+\Curl \Psi =&amp; \idiv \Curl \hat{\Psi} , \quad \Psi \in \Vcurl
+\\
+\Div \Phi =&amp; \iltwo \Div \hat{\Phi} , \quad \Phi \in \Vdiv\end{split}\]</div>
+</section>
+<section id="projectors">
+<h2>Projectors<a class="headerlink" href="#projectors" title="Link to this heading">#</a></h2>
+<p>In some cases, one may need to define projectors on <em>smooth</em> functions</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{array}{ccccccc}
+\Cinfinity &amp; \xrightarrow{\quad \Grad \quad} &amp; \Cinfinity &amp; \xrightarrow{\quad \Curl \quad}  &amp; \Cinfinity &amp; \xrightarrow{\quad \Div \quad} &amp; \Cinfinity    \\
+\Pigrad \Bigg\downarrow   &amp;     &amp; \Picurl \Bigg\downarrow  &amp;   &amp; \Pidiv \Bigg\downarrow &amp;  &amp; \Piltwo \Bigg\downarrow       \\
+\Vgrad &amp; \xrightarrow{\quad \Grad \quad} &amp; \Vcurl &amp; \xrightarrow{\quad \Curl \quad}  &amp; \Vdiv &amp; \xrightarrow{\quad \Div \quad} &amp; \Vltwo    \\
+\end{array}\end{split}\]</div>
+</section>
+<section id="discrete-derham-sequence-for-b-splines">
+<h2>Discrete DeRham sequence for B-Splines<a class="headerlink" href="#discrete-derham-sequence-for-b-splines" title="Link to this heading">#</a></h2>
+<p>Buffa et al <span id="id1">[]</span> show the construction of a discrete DeRham sequence using B-Splines, (here without boundary conditions)</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{array}{ccccccc}
+\Hgrad &amp; \xrightarrow{\quad \Grad \quad} &amp; \Hcurl &amp; \xrightarrow{\quad \Curl \quad}  &amp; \Hdiv &amp; \xrightarrow{\quad \Div \quad} &amp; \Ltwo    \\
+\Pigrad \Bigg\downarrow   &amp;     &amp; \Picurl \Bigg\downarrow  &amp;   &amp; \Pidiv \Bigg\downarrow &amp;  &amp; \Piltwo \Bigg\downarrow       \\
+\Vgradspline &amp;  \xrightarrow{\quad \Grad \quad}&amp;   \Vcurlspline &amp;  \xrightarrow{\quad \Curl \quad}  &amp;  \Vdivspline&amp;   \xrightarrow{\quad \Div \quad} &amp;  \Vltwospline \\
+\end{array}\end{split}\]</div>
+<section id="d-case">
+<h3>1d case<a class="headerlink" href="#d-case" title="Link to this heading">#</a></h3>
+<ol class="arabic simple">
+<li><p>DeRham sequence is reduced to</p></li>
+</ol>
+<div class="math notranslate nohighlight">
+\[\mathbb{R} \hookrightarrow
+\underbrace{\mathcal{S}^{p}}_{\VgradLogical}  \xrightarrow{\quad \Grad \quad}
+\underbrace{\mathcal{S}^{p-1}}_{\VltwoLogical}  \xrightarrow{\quad} 0\]</div>
+<ol class="arabic simple" start="2">
+<li><p>The recursion formula for derivative writes</p></li>
+</ol>
+<div class="math notranslate nohighlight">
+\[{N_i^p}'(t)=D_i^{p}(t)-D_{i+1}^{p}(t)
+\quad \mbox{where} \quad
+D_{i}^{p}(t) = \frac{p}{t_{i+p+1}-t_i}N_i^{p-1}(t)\]</div>
+<ol class="arabic simple" start="3">
+<li><p>we have <span class="math notranslate nohighlight">\(\mathcal{S}^{p-1} = \mathbf{span}\{ N_i^{p-1}, 1 \leq i \leq n-1 \} = \mathbf{span}\{ D_i^p, 1 \leq i \leq n-1 \}\)</span> which is a change of basis as a diagonal matrix</p></li>
+<li><p>Now if <span class="math notranslate nohighlight">\(u \in S^p\)</span>, with and expansion <span class="math notranslate nohighlight">\(u = \sum_i u_i N_i^p\)</span>, we have</p></li>
+</ol>
+<div class="math notranslate nohighlight">
+\[  \begin{align*}
+  u^{\prime} = \sum_i u_i \left( N_i^p \right)^{\prime} = \sum_i (-u_{i-1} + u_i) D_i^p
+%  \label{}
+  \end{align*}\]</div>
+<ol class="arabic simple" start="5">
+<li><p>If we introduce the B-Splines coefficients vector <span class="math notranslate nohighlight">\(\mathbf{u} := \left( u_i \right)_{1 \leq i \leq n}\)</span> (and <span class="math notranslate nohighlight">\(\mathbf{u}^{\star}\)</span> for the derivative), we have</p></li>
+</ol>
+<div class="math notranslate nohighlight">
+\[\mathbf{u}^{\star} = D \mathbf{u}\]</div>
+<p>where <span class="math notranslate nohighlight">\(D\)</span> is the incidence matrix (of entries <span class="math notranslate nohighlight">\(-1\)</span> and <span class="math notranslate nohighlight">\(+1\)</span>)</p>
+<aside class="topic">
+<p class="topic-title">Discrete derivatives:</p>
+<div class="math notranslate nohighlight">
+\[\mathcal{G} = D\]</div>
+</aside>
+</section>
+<section id="id2">
+<h3>2d case<a class="headerlink" href="#id2" title="Link to this heading">#</a></h3>
+<p>In <em>2d</em>, the are two De-Rham complexes:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{array}{ccccc}
+\Hgrad &amp; \xrightarrow{\quad \Grad \quad} &amp; \Hcurl &amp; \xrightarrow{\quad \Rots \quad} &amp; \Ltwo \\
+\Pigrad \Bigg\downarrow   &amp;     &amp; \Picurl \Bigg\downarrow  &amp;   &amp; \Piltwo \Bigg\downarrow   \\
+\Vgrad &amp; \xrightarrow{\quad \Grad \quad} &amp; \Vcurl &amp; \xrightarrow{\quad \Rots \quad} &amp; \Vltwo \\
+\end{array}\end{split}\]</div>
+<p>and</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{array}{ccccc}
+\Hgrad &amp; \xrightarrow{\quad \Curl \quad} &amp; \Hdiv &amp; \xrightarrow{\quad \Div \quad} &amp; \Ltwo \\
+\Pigrad \Bigg\downarrow   &amp;     &amp; \Pidiv \Bigg\downarrow  &amp;   &amp; \Piltwo \Bigg\downarrow   \\
+\Vgrad &amp; \xrightarrow{\quad \Grad \quad} &amp; \Vdiv &amp; \xrightarrow{\quad \Div \quad} &amp; \Vltwo \\
+\end{array}\end{split}\]</div>
+<p>Let <span class="math notranslate nohighlight">\(I\)</span> be the identity matrix, we have</p>
+<aside class="topic">
+<p class="topic-title">Discrete derivatives:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathcal{G} =
+\begin{pmatrix}
+  D \otimes I
+  \\
+  I \otimes D
+\end{pmatrix}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathcal{C} =
+\begin{pmatrix}
+  I \otimes D
+  \\
+- D \otimes I
+\end{pmatrix}
+\quad \mbox{[scalar curl],} \quad
+\mathcal{C} =
+\begin{pmatrix}
+- I \otimes D
+ &amp;
+  D \otimes I
+\end{pmatrix}
+\quad \mbox{[vectorial curl]}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\mathcal{D} =
+\begin{pmatrix}
+  D \otimes I
+ &amp;
+  I \otimes D
+\end{pmatrix}\]</div>
+</aside>
+</section>
+<section id="id3">
+<h3>3d case<a class="headerlink" href="#id3" title="Link to this heading">#</a></h3>
+<aside class="topic">
+<p class="topic-title">Discrete derivatives:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathcal{G} =
+\begin{pmatrix}
+  D \otimes I \otimes I
+  \\
+  I \otimes D \otimes I
+  \\
+  I \otimes I \otimes D
+\end{pmatrix}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathcal{C} =
+\begin{pmatrix}
+  0    &amp;    - I \otimes I \otimes D     &amp;     I \otimes D \otimes I
+  \\
+  I \otimes I \otimes D   &amp;    0   &amp;   - D \otimes I \otimes I
+  \\
+  - I \otimes D \otimes I  &amp; D \otimes I \otimes I &amp; 0
+\end{pmatrix}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\mathcal{D} =
+\begin{pmatrix}
+  D \otimes I \otimes I
+  &amp;
+  I \otimes D \otimes I
+  &amp;
+  I \otimes I \otimes D
+\end{pmatrix}\]</div>
+</aside>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>From now on, we will denote the discrete derivative by <span class="math notranslate nohighlight">\(\mathbb{D}_k\)</span> for the one going from <span class="math notranslate nohighlight">\(V_k\)</span> to <span class="math notranslate nohighlight">\(V_{k+1}\)</span>.</p>
+</div>
+</section>
+</section>
+<section id="algebraic-identities">
+<h2>Algebraic identities<a class="headerlink" href="#algebraic-identities" title="Link to this heading">#</a></h2>
+<p>Let us consider the discretization of the exterior derivative</p>
+<div class="math notranslate nohighlight">
+\[\omega^{k+1} = \diff  \omega^k\]</div>
+<p>multiplying by a test function <span class="math notranslate nohighlight">\(\eta^{k+1}\)</span> and integrating over the whole computation domain, we get</p>
+<div class="math notranslate nohighlight">
+\[\left( \eta^{k+1}, \omega^{k+1} \right)_{k+1} = \left( \eta^{k+1}, \diff \omega^{k} \right)_{k+1}\]</div>
+<p>let <span class="math notranslate nohighlight">\(E^{k+1}\)</span>, <span class="math notranslate nohighlight">\(W^{k}\)</span> and <span class="math notranslate nohighlight">\(W^{k+1}\)</span> be the vector representation of <span class="math notranslate nohighlight">\(\eta^{k+1}\)</span>, <span class="math notranslate nohighlight">\(\omega^{k}\)</span> and  <span class="math notranslate nohighlight">\(\omega^{k+1}\)</span>. We get</p>
+<div class="math notranslate nohighlight">
+\[{E^{k+1}}^T M_{k+1} W^{k+1} = {E^{k+1}}^T D_{k+1,k} W^{k}\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[D_{k+1,k} = \left( \left( \eta^{k+1}_i, \diff \omega^{k}_j \right)_{k+1} \right)_{i,j}\]</div>
+<p>On the other hand, using the coderivative, we get</p>
+<div class="math notranslate nohighlight">
+\[\left( \eta^{k+1}, \omega^{k+1} \right)_{k+1} = \left( \delta \eta^{k+1}, \omega^{k} \right)_{k} + BC\]</div>
+<p>Let us now introduce the following matrix</p>
+<div class="math notranslate nohighlight">
+\[D_{k,k+1} = \left( \left( \delta \eta^{k+1}_i, \omega^{k}_j \right)_{k} \right)_{i,j}\]</div>
+<p>hence,</p>
+<div class="math notranslate nohighlight">
+\[{E^{k+1}}^T D_{k,k+1} W^{k} = \left( \mathbb{D}^{\star}_{k+1} E^{k+1} \right)^T M_{k} W^{k}\]</div>
+<p>Therefor, we have the following important result</p>
+<aside class="topic">
+<p class="topic-title">Proposition:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(D_{k+1,k} = D_{k,k+1} + BC\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(D_{k+1,k} = M_{k+1} \mathbb{D}^T_k\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(D_{k,k+1} = {\mathbb{D}^{\star}_{k+1}}^T M_{k}\)</span></p></li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Exterior Algebra</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#basis">Basis</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#orientation-and-volume-form">Orientation and Volume form</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exterior-calculus-on-manifolds-and-differential-forms">Exterior Calculus on manifolds and Differential forms</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sobolev-spaces-of-differential-forms">Sobolev spaces of differential forms</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cohomology-and-de-rham-complex">Cohomology and De Rham Complex</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cohomology-with-boundary-conditions">Cohomology with boundary conditions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#homological-algebra-and-hilbert-complexes">Homological Algebra and Hilbert complexes</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#homological-algebra">Homological Algebra</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#cycles-and-boundaries-of-the-de-rham-complex">Cycles and boundaries of the De Rham complex</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-hodge-decomposition">The Hodge decomposition</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="exterior-algebra">
+<h1>Exterior Algebra<a class="headerlink" href="#exterior-algebra" title="Link to this heading">#</a></h1>
+<p>Let <span class="math notranslate nohighlight">\(V\)</span> be a real vector space of dimension <span class="math notranslate nohighlight">\(n\)</span>.</p>
+<aside class="topic">
+<p class="topic-title">Definition, Alternating algebraic forms:</p>
+<p>For each <span class="math notranslate nohighlight">\(k\)</span>, we define <span class="math notranslate nohighlight">\(\Alt^k V\)</span> as the space of alternating <span class="math notranslate nohighlight">\(k\)</span>-linear maps <span class="math notranslate nohighlight">\(V \times \cdots \times V \rightarrow \mathbb{R}\)</span>.</p>
+</aside>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\Alt^0 = \mathbb{R}\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\Alt^1 = V^{*}\)</span> is the dual space of <span class="math notranslate nohighlight">\(V\)</span> (the space of covectors)</p></li>
+</ul>
+</div>
+<aside class="topic">
+<p class="topic-title">Definition, Exterior product:</p>
+<p>For <span class="math notranslate nohighlight">\(\omega \in \Alt^j\)</span> and <span class="math notranslate nohighlight">\(\eta \in \Alt^k\)</span>, their exterior (wedge) product is given by:</p>
+<div class="math notranslate nohighlight">
+\[(\omega \wedge \eta ) (v_1, \cdots, v_{j+k}) = \sum_{\sigma} (\mathrm{sign}~ \sigma)
+\omega (v_{\sigma(1)}, \cdots, v_{\sigma(j)})
+\eta (v_{\sigma(j+1)}, \cdots, v_{\sigma(j+k)})\]</div>
+<p>for all <span class="math notranslate nohighlight">\(v_i \in V\)</span>. Where the sum is over all permutations <span class="math notranslate nohighlight">\(\sigma\)</span> of <span class="math notranslate nohighlight">\(\{ 1,\cdots,j+k \}\)</span>,
+for which <span class="math notranslate nohighlight">\(\sigma(1)&lt; \cdots &lt;\sigma(j)\)</span> and <span class="math notranslate nohighlight">\(\sigma(j+1)&lt; \cdots &lt;\sigma(j+k)\)</span>.</p>
+</aside>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<ul class="simple">
+<li><p>The exterior product is <strong>bilinear</strong>, <strong>associative</strong>,</p></li>
+<li><p><strong>anti-commutative</strong>: <span class="math notranslate nohighlight">\(\eta \wedge \omega = (-1)^{jk} \omega \wedge \eta\)</span> for all <span class="math notranslate nohighlight">\(\omega \in \Alt^j\)</span> and <span class="math notranslate nohighlight">\(\eta \in \Alt^k\)</span>.</p></li>
+</ul>
+</div>
+<aside class="topic">
+<p class="topic-title">Definition, Grassmann Algebra:</p>
+<p>Grassmann Algebra is defined by:</p>
+<div class="math notranslate nohighlight">
+\[\Alt V := \bigoplus_k \Alt^k V\]</div>
+<p>This is a <strong>anti-commutative graded algebra</strong>. Also called <strong>Exterior Algebra</strong> of <span class="math notranslate nohighlight">\(V^{*}\)</span></p>
+</aside>
+<p>In the case of <span class="math notranslate nohighlight">\(V=\mathbb{R}^n\)</span>, we have:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\Alt V^0 \sim \mathbb{R}\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\Alt V^1 \sim \mathbb{R}^n\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\Alt V^{n-1} \sim \mathbb{R}^n\)</span>, using Riesz representation theorem,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\Alt V^n \sim \mathbb{R}\)</span>, using the map <span class="math notranslate nohighlight">\(v \longmapsto \det(v,v_1,\cdots,v_{n-1})\)</span>.</p></li>
+</ul>
+<section id="basis">
+<h2>Basis<a class="headerlink" href="#basis" title="Link to this heading">#</a></h2>
+<p>Let <span class="math notranslate nohighlight">\(v_1,\cdots,v_n\)</span> be a basis of <span class="math notranslate nohighlight">\(V\)</span> and <span class="math notranslate nohighlight">\(\mu_1,\cdots,\mu_n\)</span> the associated dual basis for <span class="math notranslate nohighlight">\(V^*\)</span> (<span class="math notranslate nohighlight">\(\mu_i(v_j) = \delta_{ij}\)</span>).</p>
+<p>For any increasing permutations <span class="math notranslate nohighlight">\(\sigma, \rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[\mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)} (v_{\rho(1)}, \cdots, v_{\rho(k)}) = \chi_{\sigma,\rho}\]</div>
+<p>thus the <span class="math notranslate nohighlight">\(\binom {n}{k}\)</span> algebraic <span class="math notranslate nohighlight">\(k\)</span>-forms <span class="math notranslate nohighlight">\(\mu_{\sigma(1)} \wedge \cdots \wedge \mu_{\sigma(k)}\)</span>,
+form a basis for <span class="math notranslate nohighlight">\(\Alt^k V\)</span> and <span class="math notranslate nohighlight">\(\dim \Alt^k V = \binom {n}{k}\)</span>.</p>
+<aside class="topic">
+<p class="topic-title">Definition, Interior product:</p>
+<p>Let <span class="math notranslate nohighlight">\(\omega\)</span> be a <span class="math notranslate nohighlight">\(k\)</span>-form, and <span class="math notranslate nohighlight">\(v \in V\)</span>. The <strong>interior product</strong> of <span class="math notranslate nohighlight">\(\omega\)</span> and <span class="math notranslate nohighlight">\(v\)</span> is the <span class="math notranslate nohighlight">\((k-1)\)</span>-form  <span class="math notranslate nohighlight">\(\omega \lrcorner v\)</span> defined by:</p>
+<div class="math notranslate nohighlight">
+\[\omega \lrcorner v (v_1,\cdots,v_{k-1}) = \omega (v,v_1,\cdots,v_{k-1})\]</div>
+</aside>
+<ul class="simple">
+<li><p>We have for <span class="math notranslate nohighlight">\(\omega \in \Alt^k V\)</span>, <span class="math notranslate nohighlight">\(\eta \in \Alt^l V\)</span> and <span class="math notranslate nohighlight">\(v \in V\)</span>:</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[(\omega \wedge \eta) \lrcorner v = (\omega \lrcorner v)\wedge \eta + (-1)^k \omega \wedge (\eta \lrcorner v)\]</div>
+<aside class="topic">
+<p class="topic-title">Definition, Inner product:</p>
+<p>If <span class="math notranslate nohighlight">\(V\)</span> is has an inner product, then <span class="math notranslate nohighlight">\(\Alt^k V\)</span> is endowed with an inner product given by:</p>
+<div class="math notranslate nohighlight">
+\[(\omega , \eta) = \sum_{\rho} \omega (v_{\rho(1)}, \cdots, v_{\rho(k)}) \eta (v_{\rho(1)}, \cdots, v_{\rho(k)}), ~~~\forall \omega, \eta \in \Alt^k V.\]</div>
+<p>where the sum is over increasing sequences <span class="math notranslate nohighlight">\(\rho : \{ 1,\cdots,k \} \longrightarrow \{ 1,\cdots,n \}\)</span>, and <span class="math notranslate nohighlight">\(v_1, \cdots,v_n\)</span> is any orthonormal basis.</p>
+</aside>
+</section>
+<section id="orientation-and-volume-form">
+<h2>Orientation and Volume form<a class="headerlink" href="#orientation-and-volume-form" title="Link to this heading">#</a></h2>
+<aside class="topic">
+<p class="topic-title">Definition, Pullback:</p>
+<p>A linear transformation of vector spaces <span class="math notranslate nohighlight">\(L: V \rightarrow W\)</span> induces a transformation
+<span class="math notranslate nohighlight">\(L^{*}: \Alt W \rightarrow \Alt V\)</span>, called the <strong>pullback</strong>, and given by:</p>
+<div class="math notranslate nohighlight">
+\[L^{*} \omega (v_{1}, \cdots, v_{k}) = \omega (L v_{1}, \cdots, L v_{k}),~~~~~ \forall \omega \in \Alt^k W,~~~ v_{1}, \cdots, v_{k} \in V\]</div>
+</aside>
+<ul>
+<li><p>The pullback acts <strong>contravariantly</strong>: if <span class="math notranslate nohighlight">\(U \xrightarrow{~K~} V \xrightarrow{~L~} W\)</span> then,</p>
+<div class="math notranslate nohighlight">
+\[\Alt W \xrightarrow{~K^{*}~} \Alt V \xrightarrow{~L^{*}~} \Alt U\]</div>
+</li>
+<li><p><span class="math notranslate nohighlight">\(L^{*} (\omega \wedge \eta) = L^{*} \omega \wedge L^{*} \eta\)</span></p></li>
+</ul>
+<p>Let V be a subspace of W. For the inclusion <span class="math notranslate nohighlight">\(\imath_V : V \longrightarrow W\)</span>, we can define its pullback <span class="math notranslate nohighlight">\(\imath_V^{*}\)</span>:
+this is a <strong>surjection</strong> of <span class="math notranslate nohighlight">\(\Alt W\)</span> onto <span class="math notranslate nohighlight">\(\Alt V\)</span>.</p>
+<p>If W has an inner product and <span class="math notranslate nohighlight">\(\pi_V : W \longrightarrow V\)</span> is the orthogonal projection. We can define its pullback <span class="math notranslate nohighlight">\(\pi_V^{*}\)</span> :
+this an <strong>injection</strong> of <span class="math notranslate nohighlight">\(\Alt V\)</span> onto <span class="math notranslate nohighlight">\(\Alt W\)</span>.</p>
+<p>Let us consider the composition :  <span class="math notranslate nohighlight">\(W\)</span> shortstack{<span class="math notranslate nohighlight">\(\pi_V\)</span> \ <span class="math notranslate nohighlight">\(\longrightarrow\)</span>} <span class="math notranslate nohighlight">\(V\)</span> shortstack{<span class="math notranslate nohighlight">\(\imath_V\)</span> \ <span class="math notranslate nohighlight">\(\longrightarrow\)</span>} <span class="math notranslate nohighlight">\(W\)</span>, and its pullback <span class="math notranslate nohighlight">\(\pi_V^* \imath_V^*\)</span>.</p>
+<aside class="topic">
+<p class="topic-title">Definition, The tangential and normal parts:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\pi_V^* \imath_V^*\)</span> associates for each <span class="math notranslate nohighlight">\(\omega \in \Alt^k\)</span> its <strong>tangential</strong> part <span class="math notranslate nohighlight">\(\omega_{\parallel}\)</span> with respect to <span class="math notranslate nohighlight">\(V\)</span> :</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[(\pi_V^* \imath_V^* \omega) (v_1,\cdots,v_k) = \omega (\pi_V v_1, \cdots, \pi_V v_k), ~~~~~\forall v_1,\cdots,v_k \in W.\]</div>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\omega - \pi_V^* \imath_V^* \omega\)</span> associates for each <span class="math notranslate nohighlight">\(\omega \in \Alt^k\)</span> its <strong>normal</strong> part <span class="math notranslate nohighlight">\(\omega_{\perp}\)</span> with respect to <span class="math notranslate nohighlight">\(V\)</span>.</p></li>
+</ul>
+</aside>
+<p>The <strong>tangential part</strong> of <span class="math notranslate nohighlight">\(\omega\)</span> vanishes if and only if the image of <span class="math notranslate nohighlight">\(\omega\)</span> in <span class="math notranslate nohighlight">\(\Alt^k V\)</span> vanishes.</p>
+<p>Let <span class="math notranslate nohighlight">\(V\)</span> be an oriented inner product space, with volume form <span class="math notranslate nohighlight">\(\mbox{vol}\)</span>. Let <span class="math notranslate nohighlight">\(\omega \in \Alt^k V\)</span>.
+We can define a new linear map <span class="math notranslate nohighlight">\(L_{\omega}\)</span> as the composition of <span class="math notranslate nohighlight">\(\Alt^{n-k} V \longrightarrow \Alt^n V\)</span> such as:</p>
+<div class="math notranslate nohighlight">
+\[\mu \longmapsto \omega \wedge \mu\]</div>
+<p>and the canonical isomorphism of <span class="math notranslate nohighlight">\(\Alt^n V\)</span> onto <span class="math notranslate nohighlight">\(\mathbb{R}\)</span>, and using the Riesz representation theorem,
+there exists an element <span class="math notranslate nohighlight">\(\star \omega \in \Alt^{n-k} V\)</span> such that : <span class="math notranslate nohighlight">\(L_{\omega} (\mu) = (\star \omega , \mu)\)</span>, <em>i.e.</em>:</p>
+<div class="math notranslate nohighlight">
+\[\omega \wedge \mu = (\star \omega , \mu) \mbox{vol}, ~~~\omega \in \Alt^{k}, ~\mu \in \Alt^{n-k}\]</div>
+<aside class="topic">
+<p class="topic-title">Definition, The Hodge star operation:</p>
+<p>The linear map which maps <span class="math notranslate nohighlight">\(\Alt^k V\)</span> onto <span class="math notranslate nohighlight">\(\Alt^{n-k} V\)</span> <span class="math notranslate nohighlight">\(\omega \longmapsto \star \omega\)</span> is called the <strong>Hodge star</strong> operator.</p>
+</aside>
+<ul class="simple">
+<li><p>If <span class="math notranslate nohighlight">\(e_1,\cdots,e_n\)</span> is any positively oriented orthonormal basis, and <span class="math notranslate nohighlight">\(\sigma\)</span> a permutation, we have</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[\omega(e_{\sigma(1)}, \cdots, e_{\sigma(k)}) = (\mathrm{sign} \sigma) \star \omega(e_{\sigma(k+1)}, \cdots, e_{\sigma(n)})\]</div>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\star \star \omega = (-1)^{k(n-k)} \omega, ~~~\forall \omega \in \Alt^k V\)</span>, thus the Hodge star is an <strong>isometry</strong>.</p></li>
+<li><p><span class="math notranslate nohighlight">\((\star \omega)_{\parallel} = \star (\omega_{\perp})\)</span> and <span class="math notranslate nohighlight">\((\star \omega)_{\perp} = \star (\omega_{\parallel})\)</span></p></li>
+<li><p>the image of <span class="math notranslate nohighlight">\(\star \omega\)</span> in <span class="math notranslate nohighlight">\(\Alt^k V\)</span> vanishes if and only if <span class="math notranslate nohighlight">\(\omega_{\perp}\)</span> vanishes.</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{tabular}{|c|l|}
+  \hline
+ $\Alt^0 \mathbb{R}^3 \cong \mathbb{R}$ &amp;  $c \leftrightarrow c$ \\
+%   \hline
+ $\Alt^1 \mathbb{R}^3 \cong \mathbb{R}^3$ &amp; $u_1 \diff x_1 + u_2 \diff x_2 + u_3 \diff x_3 \leftrightarrow u$ \\
+%    \hline
+ $\Alt^2 \mathbb{R}^3 \cong \mathbb{R}^3$ &amp; $u_3 \diff x_1 \wedge \diff x_2 - u_2 \diff x_1 \wedge \diff x_3 + u_1 \diff x_2 \wedge \diff x_3 +  \leftrightarrow u$ \\
+%   \hline
+ $\Alt^3 \mathbb{R}^3 \cong \mathbb{R}$ &amp; $c \diff x_1 \wedge \diff x_2 \wedge \diff x_3 \leftrightarrow c$  \\
+  \hline
+\end{tabular}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{tabular}{|c|l|}
+  \hline
+ $ \wedge : \Alt^1 \mathbb{R}^3 \times \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3$
+&amp;  $\times : \mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}^3$
+\\
+ $ \wedge : \Alt^1 \mathbb{R}^3 \times \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3$
+&amp;  $\cdot : \mathbb{R}^3 \times \mathbb{R}^3 \longrightarrow \mathbb{R}$
+\\
+  \hline
+\end{tabular}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{tabular}{|c|l|}
+  \hline
+ $ L^* : \Alt^0 \mathbb{R}^3 \longrightarrow \Alt^0 \mathbb{R}^3 $ &amp; $\id : \mathbb{R} \longrightarrow \mathbb{R}$
+\\
+ $ L^* : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^1 \mathbb{R}^3 $ &amp; $L^T : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$
+\\
+ $ L^* : \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ &amp; $(\det L )L^{-1} : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$
+\\
+ $ L^* : \Alt^3 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3 $ &amp; $(\det L) : \mathbb{R} \longrightarrow \mathbb{R}$ ~~~($c \longmapsto c \det L$)
+\\
+  \hline
+\end{tabular}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{tabular}{|c|l|}
+  \hline
+ $ \lrcorner v : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^0 \mathbb{R}^3 $ &amp; $v \cdot : \mathbb{R}^3 \longrightarrow \mathbb{R}$
+\\
+ $ \lrcorner v : \Alt^2 \mathbb{R}^3 \longrightarrow \Alt^1 \mathbb{R}^3 $ &amp; $v \times : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$
+\\
+ $ \lrcorner v : \Alt^3 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ &amp; $v : \mathbb{R} \longrightarrow \mathbb{R}^3$ ~~~($c \longmapsto c v$)
+\\
+  \hline
+\end{tabular}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{tabular}{|c|l|}
+  \hline
+ inner product on $\Alt^k \mathbb{R}^3$ induced  &amp; dot product on $\mathbb{R}$ and $\mathbb{R}^3$
+\\
+ by dot product on $\mathbb{R}^3$ &amp;
+\\
+ $\volume = \diff x_1 \wedge \diff x_2 \wedge \diff x_3$ &amp; $(v_1,v_2,v_3) \longmapsto \det(v_1|v_2|v_3)$
+\\
+  \hline
+\end{tabular}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{tabular}{|c|l|}
+  \hline
+ $ \star : \Alt^0 \mathbb{R}^3 \longrightarrow \Alt^3 \mathbb{R}^3 $ &amp; $\id : \mathbb{R} \longrightarrow \mathbb{R}$
+\\
+ $ \star : \Alt^1 \mathbb{R}^3 \longrightarrow \Alt^2 \mathbb{R}^3 $ &amp; $\id : \mathbb{R}^3 \longrightarrow \mathbb{R}^3$
+\\
+  \hline
+\end{tabular}\end{split}\]</div>
+</section>
+<section id="exterior-calculus-on-manifolds-and-differential-forms">
+<h2>Exterior Calculus on manifolds and Differential forms<a class="headerlink" href="#exterior-calculus-on-manifolds-and-differential-forms" title="Link to this heading">#</a></h2>
+<p>Let <span class="math notranslate nohighlight">\(\Omega\)</span> be a smooth manifold, of dimension <span class="math notranslate nohighlight">\(n\)</span>.</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\forall x \in \Omega\)</span> we denote by <span class="math notranslate nohighlight">\(T_x \Omega\)</span> the tangent space. This is a vector space of dimension <span class="math notranslate nohighlight">\(n\)</span>,</p></li>
+<li><p>tangent bundle <span class="math notranslate nohighlight">\(\{ (x,v), ~~ x \in \Omega, v \in T_x \Omega \}\)</span>,</p></li>
+<li><p>Applying the exterior algebra to the tangent spaces, we obtain the exterior forms bundle, whose elements are pairs <span class="math notranslate nohighlight">\((x,\mu)\)</span> with <span class="math notranslate nohighlight">\(x \in \Omega\)</span> and <span class="math notranslate nohighlight">\(\mu \in \Alt^k T_x \Omega\)</span>.</p></li>
+<li><p>a <strong>differential</strong> <span class="math notranslate nohighlight">\(k\)</span>-form <span class="math notranslate nohighlight">\(\omega\)</span> is a section of this bundle. This is a map which associates to each <span class="math notranslate nohighlight">\(x \in \Omega\)</span> an element <span class="math notranslate nohighlight">\(\omega_x \in \Alt^k T_x \Omega\)</span>,</p></li>
+<li><p>if the map <span class="math notranslate nohighlight">\(\mathcal{L}_{\omega}^k : x \longmapsto \omega_x (v_1(x), \cdots, v_k(x))\)</span> is smooth (whenever <span class="math notranslate nohighlight">\(v_i\)</span> are smooth), we say that <span class="math notranslate nohighlight">\(\omega\)</span> is a smooth differential <span class="math notranslate nohighlight">\(k\)</span>-form,</p></li>
+<li><p>we define <span class="math notranslate nohighlight">\(\Lambda^k(\Omega)\)</span> the space of all smooth <span class="math notranslate nohighlight">\(k\)</span>-forms on <span class="math notranslate nohighlight">\(\Omega\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\Lambda^0(\Omega) = \mathcal{C}^{\infty}(\Omega)\)</span>,</p></li>
+<li><p>if the map <span class="math notranslate nohighlight">\(\mathcal{L}_{\omega}^k\)</span> is <span class="math notranslate nohighlight">\(\mathcal{C}^{m}(\Omega)\)</span>, we define differential <span class="math notranslate nohighlight">\(k\)</span>-forms with less smoothness <span class="math notranslate nohighlight">\(\mathcal{C}^{m} \Lambda^k (\Omega)\)</span>.</p></li>
+</ul>
+<p>Let <span class="math notranslate nohighlight">\(\Omega\)</span> be a smooth manifold, of dimension <span class="math notranslate nohighlight">\(n\)</span>.</p>
+<aside class="topic">
+<p class="topic-title">Exterior product:</p>
+<p>if <span class="math notranslate nohighlight">\(\omega \in \Lambda^k(\Omega)\)</span> and <span class="math notranslate nohighlight">\(\eta \in \Lambda^j(\Omega)\)</span>, we may define <span class="math notranslate nohighlight">\(\omega \wedge \eta\)</span> as  <span class="math notranslate nohighlight">\((\omega \wedge \eta)_x = \omega_x \wedge \eta_x\)</span> and the Grassmann algebra <span class="math notranslate nohighlight">\(\Lambda(\Omega) := \bigoplus_k \Lambda^k(\Omega)\)</span></p>
+</aside>
+<p>Differential forms can be differentiated and integrated, without recourse to any additional structure, such as a metric or a measure.</p>
+<aside class="topic">
+<p class="topic-title">Exterior differentiation:</p>
+<blockquote>
+<div><p>For each <span class="math notranslate nohighlight">\(\omega \in \Lambda^k(\Omega)\)</span>, can define the <span class="math notranslate nohighlight">\((k+1)\)</span>-form <span class="math notranslate nohighlight">\(\diff \omega \in \Lambda^{k+1}(\Omega)\)</span>, such as:</p>
+</div></blockquote>
+<div class="math notranslate nohighlight">
+\[\diff\omega_x(v_1,\cdots,v_{k+1}) = \sum_{j=1}^{k+1} (-1)^j \partial_{v_j} \omega_x(v_1,\cdots,\hat{v_j},\cdots,v_{k+1})\]</div>
+<p>where the hat is used to indicated a suppressed argument.</p>
+<p>This defines a graded linear operator of degree <span class="math notranslate nohighlight">\(+1\)</span>, of <span class="math notranslate nohighlight">\(\Lambda(\Omega)\)</span> onto <span class="math notranslate nohighlight">\(\Lambda(\Omega)\)</span>.</p>
+</aside>
+<p>We have the following properties:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\diff \circ \diff = 0\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(\diff (\omega \wedge \eta) = \diff \omega \wedge \eta + (-1)^k \omega \wedge \diff \eta, ~~\forall \omega \in \Lambda^k(\Omega), \eta \in \Lambda^j(\Omega)\)</span>,</p></li>
+<li><p>(Pullback) let <span class="math notranslate nohighlight">\(\phi\)</span> be a smooth map of <span class="math notranslate nohighlight">\(\Omega\)</span> onto <span class="math notranslate nohighlight">\(\Omega^{\prime}\)</span>. Then <span class="math notranslate nohighlight">\(\phi^*(\omega \wedge \eta) = \phi^*(\omega) \wedge \phi^*(\eta)\)</span> and <span class="math notranslate nohighlight">\(\phi^* (\diff \omega) = \diff (\phi^* \omega)\)</span>,</p></li>
+<li><p>(Interior product) the interior product of a differential <span class="math notranslate nohighlight">\(k\)</span>-form <span class="math notranslate nohighlight">\(\omega\)</span> with a vector field <span class="math notranslate nohighlight">\(v\)</span>,</p></li>
+<li><p>we obtain a <span class="math notranslate nohighlight">\((k-1)\)</span>-form by : <span class="math notranslate nohighlight">\((\omega \lrcorner v)_x := \omega_x \lrcorner v_x\)</span>,</p></li>
+<li><p>(Trace operator) the pullback <span class="math notranslate nohighlight">\(i_{\partial \Omega}^*\)</span> of <span class="math notranslate nohighlight">\(i_{\partial \Omega}\)</span> is the trace operator <span class="math notranslate nohighlight">\(\trace\)</span></p></li>
+</ul>
+<aside class="topic">
+<p class="topic-title">Integration:</p>
+<ul class="simple">
+<li><p>If <span class="math notranslate nohighlight">\(f\)</span> is an oriented, piecewise smooth <span class="math notranslate nohighlight">\(k\)</span>-dimensional submanifold of <span class="math notranslate nohighlight">\(\Omega\)</span>, and <span class="math notranslate nohighlight">\(\omega\)</span> is a continuous <span class="math notranslate nohighlight">\(k\)</span>-form, then th integral <span class="math notranslate nohighlight">\(\int_f \omega\)</span> is well defined :</p>
+<ul>
+<li><p>[0-forms] can be evaluated at points,</p></li>
+<li><p>[1-forms] can be integrated over directed curves,</p></li>
+<li><p>[2-forms] can be integrated over directed surfaces,</p></li>
+</ul>
+</li>
+<li><p>(Inner product) The <span class="math notranslate nohighlight">\(L^2\)</span>-inner product of two differential <span class="math notranslate nohighlight">\(k\)</span>-forms on an oriented Riemannian manifold <span class="math notranslate nohighlight">\(\Omega\)</span> is defined as :</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[(\omega,\eta)_{L^2 \Lambda^k} = \int_{\Omega} (\omega_x,\eta_x) \volume = \int \omega \wedge \star \eta\]</div>
+<p>The completion of <span class="math notranslate nohighlight">\(\Lambda^k(\Omega)\)</span> in the corresponding norm defines the Hilbert space <span class="math notranslate nohighlight">\(L^2 \Lambda^k(\Omega)\)</span>.</p>
+</aside>
+<p>We have the following results:</p>
+<ul class="simple">
+<li><p>(Integration) if <span class="math notranslate nohighlight">\(\phi\)</span> is an orientation-preserving diffeomorphism, then</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[\int_{\Omega} \phi^* \omega = \int_{\Omega^{\prime}} \omega, ~~~ \forall \omega \in \Lambda^n(\Omega^{\prime})\]</div>
+<aside class="topic">
+<p class="topic-title">Theorem, Stokes theorem:</p>
+<p>If <span class="math notranslate nohighlight">\(\Omega\)</span> is an oriented <span class="math notranslate nohighlight">\(n\)</span>-manifold with boundary <span class="math notranslate nohighlight">\(\partial \Omega\)</span>, then</p>
+<div class="math notranslate nohighlight">
+\[\int_{\Omega} \diff \omega = \int_{\partial \Omega} \trace \omega, ~~~ \forall \omega \in \Lambda^{n-1}(\Omega)\]</div>
+</aside>
+<aside class="topic">
+<p class="topic-title">Theorem, Integration by parts:</p>
+<p>If <span class="math notranslate nohighlight">\(\Omega\)</span> is an oriented <span class="math notranslate nohighlight">\(n\)</span>-manifold with boundary <span class="math notranslate nohighlight">\(\partial \Omega\)</span>, then</p>
+<div class="math notranslate nohighlight">
+\[\int_{\Omega} \diff \omega \wedge \eta = (-1)^{k-1} \int_{\Omega} \omega \wedge \diff \eta + \int_{\partial \Omega} \trace \omega \wedge \trace \eta, ~~~ \forall \omega \in \Lambda^{k}(\Omega), \eta \in \Lambda^{n-k-1}(\Omega)\]</div>
+</aside>
+</section>
+<section id="sobolev-spaces-of-differential-forms">
+<h2>Sobolev spaces of differential forms<a class="headerlink" href="#sobolev-spaces-of-differential-forms" title="Link to this heading">#</a></h2>
+<p>As for the classical case, we can define the Sobolev spaces as:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(H^s \Lambda^k(\Omega)\)</span> is the space of differential <span class="math notranslate nohighlight">\(k\)</span>-forms such that <span class="math notranslate nohighlight">\(\mathcal{L}_{\omega}^k \in H^s(\Omega)\)</span>.</p></li>
+<li><p><span class="math notranslate nohighlight">\(H \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \diff \omega \in L^2 \Lambda^{k+1}(\Omega) \}\)</span>. The associated norm is :</p></li>
+</ul>
+<blockquote>
+<div><div class="math notranslate nohighlight">
+\[\| \omega \|_{H \Lambda^k}^2 = \| \omega \|_{H \Lambda}^2 := \| \omega \|_{L^2 \Lambda^k}^2 + \| \diff \omega \|_{L^2 \Lambda^{k+1}}^2\]</div>
+</div></blockquote>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(H \Lambda^{0}(\Omega)\)</span> coincides with <span class="math notranslate nohighlight">\(H^1 \Lambda^{0}(\Omega)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(H \Lambda^{n}(\Omega)\)</span> coincides with <span class="math notranslate nohighlight">\(L^2 \Lambda^{n}(\Omega)\)</span>,</p></li>
+<li><p>for <span class="math notranslate nohighlight">\(0 &lt; k &lt; n\)</span>, we have <span class="math notranslate nohighlight">\(H^1 \Lambda^k(\Omega) \subset H \Lambda^k(\Omega) \subset L^2 \Lambda^k(\Omega)\)</span>, strictly.</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{tabular}{|c|c c c c c|}
+  \hline
+ $k$ &amp; $\Lambda^k$ &amp; $H \Lambda^k$ &amp; $\diff \omega$ &amp; $\int_f \omega$ &amp; $\kappa \omega$
+\\
+ \hline
+&amp; &amp; &amp; &amp; &amp; \\
+ 0 &amp; $\mathcal{C}^{\infty}$ &amp; $H^1$ &amp; $\nabla \omega$ &amp; $\omega(f)$ &amp; $0$
+\\
+ 1 &amp; $\mathcal{C}^{\infty}(\mathbb{R}^3)$ &amp; $H(\rots,\mathbb{R}^3)$ &amp; $\rots \omega$ &amp; $\int_f \omega \cdot t \diff \mathcal{H}_1$ &amp; $x \longmapsto x \cdot \omega(x)$
+\\
+ 2 &amp; $\mathcal{C}^{\infty}(\mathbb{R}^3)$ &amp; $H(\divs, \mathbb{R}^3)$ &amp; $\divs \omega$ &amp; $\int_f \omega \cdot n \diff \mathcal{H}_2$ &amp; $x \longmapsto x \times \omega(x)$
+\\
+ 3 &amp; $\mathcal{C}^{\infty}$ &amp; $L^2$ &amp; $0$ &amp; $\int_f \omega \diff \mathcal{H}_3$ &amp; $x \longmapsto x \omega(x)$
+\\
+&amp; &amp; &amp; &amp; &amp; \\
+  \hline
+\end{tabular}\end{split}\]</div>
+</section>
+<section id="cohomology-and-de-rham-complex">
+<h2>Cohomology and De Rham Complex<a class="headerlink" href="#cohomology-and-de-rham-complex" title="Link to this heading">#</a></h2>
+<p>The De Rham complex is the sequence of spaces and mappings</p>
+<div class="math notranslate nohighlight">
+\[0 \xrightarrow{\quad} \Lambda^0(\Omega)  \xrightarrow{~\diff~}  \Lambda^1(\Omega)  \xrightarrow{~\diff~}   \cdots  \xrightarrow{~\diff~}  \Lambda^n(\Omega)  \xrightarrow{\quad} 0\]</div>
+<p>Since, <span class="math notranslate nohighlight">\(\diff \circ \diff = 0\)</span>, we have</p>
+<div class="math notranslate nohighlight">
+\[\mathcal{R}(\diff : \Lambda^{k-1}(\Omega) \longrightarrow \Lambda^k(\Omega)) \subset \mathcal{N}(\diff : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k+1}(\Omega))\]</div>
+<p>If <span class="math notranslate nohighlight">\(\Omega\)</span> is an oriented Riemannian manifold, we have the following cohomology:</p>
+<div class="math notranslate nohighlight">
+\[0 \xrightarrow{\quad} H \Lambda^0(\Omega)  \xrightarrow{~\diff~}  H \Lambda^1(\Omega)  \xrightarrow{~\diff~}   \cdots  \xrightarrow{~\diff~}  H \Lambda^n(\Omega)  \xrightarrow{\quad} 0\]</div>
+<p>The <em>coderivative operator</em> <span class="math notranslate nohighlight">\(\delta : \Lambda^{k}(\Omega) \longrightarrow \Lambda^{k-1}(\Omega)\)</span> is defined as:</p>
+<div class="math notranslate nohighlight">
+\[\star \delta \omega = (-1)^k \diff \star \omega,~~~ \omega \in \Lambda^k(\Omega)\]</div>
+<ul class="simple">
+<li><p>we have</p></li>
+</ul>
+<blockquote>
+<div><div class="math notranslate nohighlight">
+\[(\diff \omega , \eta ) = (\omega , \delta \eta )  + \int_{\partial \Omega} \trace \omega \wedge \trace \eta,  ~~~ \forall \omega \in \Lambda^{k}(\Omega), \eta \in \Lambda^{k+1}(\Omega),\]</div>
+</div></blockquote>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\delta\)</span> is a graded linear operator of degree <span class="math notranslate nohighlight">\(-1\)</span>.</p></li>
+<li><p><span class="math notranslate nohighlight">\(\delta\)</span> is the formal adjoint of <span class="math notranslate nohighlight">\(\diff\)</span> whenever <span class="math notranslate nohighlight">\(\omega\)</span> or <span class="math notranslate nohighlight">\(\eta\)</span> vanishes near the boundary.</p></li>
+<li><p>we define the spaces</p></li>
+</ul>
+<blockquote>
+<div><div class="math notranslate nohighlight">
+\[H^* \Lambda^k(\Omega) = \{ \omega \in L^2 \Lambda^k(\Omega),~~ \delta \omega \in L^2 \Lambda^{k-1}(\Omega) \}.\]</div>
+<p>we have <span class="math notranslate nohighlight">\(H^* \Lambda^k(\Omega) = \star H \Lambda^{n-k}(\Omega)\)</span>.</p>
+</div></blockquote>
+<ul class="simple">
+<li><p>we obtain the dual complex</p></li>
+</ul>
+<blockquote>
+<div><div class="math notranslate nohighlight">
+\[0 \xleftarrow{\quad} H^* \Lambda^0(\Omega)  \xleftarrow{~\delta~}  H^* \Lambda^1(\Omega)  \xleftarrow{~\delta~}   \cdots  \xleftarrow{~\delta~}  H^* \Lambda^n(\Omega)  \xleftarrow{\quad} 0\]</div>
+</div></blockquote>
+</section>
+<section id="cohomology-with-boundary-conditions">
+<h2>Cohomology with boundary conditions<a class="headerlink" href="#cohomology-with-boundary-conditions" title="Link to this heading">#</a></h2>
+<p>Let <span class="math notranslate nohighlight">\(\Lambda_0^k(\Omega)\)</span> be the subspace of <span class="math notranslate nohighlight">\(\Lambda^k(\Omega)\)</span> of smooth <span class="math notranslate nohighlight">\(k\)</span>-forms with compact support. We have <span class="math notranslate nohighlight">\(\diff \Lambda_0^k \subset \Lambda_0^{k+1}\)</span>.</p>
+<p>The De Rham complex with the compact support is</p>
+<div class="math notranslate nohighlight">
+\[0 \xrightarrow{\quad} \Lambda^0_0(\Omega)  \xrightarrow{~\diff~}  \Lambda^1_0(\Omega)  \xrightarrow{~\diff~}   \cdots  \xrightarrow{~\diff~}  \Lambda^n_0(\Omega)  \xrightarrow{\quad} 0\]</div>
+<p>Recall that the closure of <span class="math notranslate nohighlight">\(\Lambda_0^k(\Omega)\)</span> in <span class="math notranslate nohighlight">\(H \Lambda^k(\Omega)\)</span> is</p>
+<div class="math notranslate nohighlight">
+\[H_0 \Lambda^k(\Omega) = \{ \omega \in H \Lambda^k(\Omega),~~ \trace \omega =0\}.\]</div>
+<p>The <span class="math notranslate nohighlight">\(L^2\)</span> version of the last complex is</p>
+<div class="math notranslate nohighlight">
+\[0 \xrightarrow{\quad} H_0 \Lambda^0(\Omega)  \xrightarrow{~\diff~}  H_0 \Lambda^1(\Omega)  \xrightarrow{~\diff~}   \cdots  \xrightarrow{~\diff~}  H_0 \Lambda^n(\Omega)  \xrightarrow{\quad} 0\]</div>
+<aside class="topic">
+<p class="topic-title">Definition, Harmonic forms:</p>
+<p>The harmonic <span class="math notranslate nohighlight">\(k\)</span>-forms are the differential <span class="math notranslate nohighlight">\(k\)</span>-forms that verify the differential equations</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\left\{
+    \begin{aligned}
+      \diff \omega &amp;=&amp; 0,\\
+      \delta \omega &amp;=&amp; 0,\\
+      \trace \star \omega &amp;=&amp; 0.\\
+    \end{aligned}
+  \right.\end{split}\]</div>
+<p>this defines the following space,</p>
+<div class="math notranslate nohighlight">
+\[\mathfrak{H}^k (\Omega) = \{ \omega \in H \Lambda^k(\Omega) \cap H_0^* \Lambda^k(\Omega),~~\diff \omega = 0, \delta \omega = 0 \}\]</div>
+</aside>
+<p>We can also define the following space,</p>
+<div class="math notranslate nohighlight">
+\[\mathfrak{H}_0^k (\Omega) = \{ \omega \in H_0 \Lambda^k(\Omega) \cap H^* \Lambda^k(\Omega),~~\diff \omega = 0, \delta \omega = 0 \}\]</div>
+<p>As we can see, <span class="math notranslate nohighlight">\(\star \mathfrak{H}^k (\Omega) = \mathfrak{H}_0^{n-k} (\Omega)\)</span>.</p>
+<aside class="topic">
+<p class="topic-title">Proposition, Poincaré duality:</p>
+<p>There is an isomorphism between the <span class="math notranslate nohighlight">\(k\)</span> th De Rham cohomology space and the <span class="math notranslate nohighlight">\((n-k)\)</span> th cohomology space with boundary conditions.</p>
+</aside>
+</section>
+<section id="homological-algebra-and-hilbert-complexes">
+<h2>Homological Algebra and Hilbert complexes<a class="headerlink" href="#homological-algebra-and-hilbert-complexes" title="Link to this heading">#</a></h2>
+<section id="homological-algebra">
+<h3>Homological Algebra<a class="headerlink" href="#homological-algebra" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>A cochain complex is a sequence of vector spaces and linear maps</p></li>
+</ul>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(k\)</span>-cocycles <span class="math notranslate nohighlight">\(\mathfrak{Z}^k := \mathcal{N}(d_k)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(k\)</span>-coboundaries <span class="math notranslate nohighlight">\(\mathfrak{B}^k := \mathcal{R}(d_{k-1})\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(k\)</span>-cohomology <span class="math notranslate nohighlight">\(\mathcal{H}^k(V) := \mathfrak{Z}^k / \mathfrak{B}^k\)</span>,</p></li>
+<li><p>we say that the sequence is <strong>exact</strong>, if the <strong>cohomology vanishes</strong> (<em>i.e.</em> <span class="math notranslate nohighlight">\(\forall~k,~~ \mathcal{H}^k(V) = \{0\}\)</span>),</p></li>
+<li><p>Given two cochain complexes <span class="math notranslate nohighlight">\(V,V^{\prime}\)</span>, a <strong>cochain map</strong> <span class="math notranslate nohighlight">\(f =(f_k)\)</span> (such as <span class="math notranslate nohighlight">\(\diff^{\prime}_k f_k = f_{k+1} \diff_k\)</span>)</p></li>
+</ul>
+<blockquote>
+<div><div class="math notranslate nohighlight">
+\[\begin{split}\begin{array}{ccccccccc}
+\cdots &amp; \longrightarrow &amp; V_{k-1} &amp; \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} &amp; V_{k} &amp; \mbox{\shortstack{$\diff_k$ \\ $\longrightarrow$}} &amp; V_{k+1} &amp; \longrightarrow~\cdots \\
+ &amp; &amp; \downarrow f_{k-1} &amp; &amp; \downarrow f_{k} &amp;  &amp; \downarrow f_{k+1} &amp; &amp; \\
+\cdots &amp; \longrightarrow &amp; V_{k-1}^{\prime} &amp; \mbox{\shortstack{$\diff_{k-1}^{\prime}$ \\ $\longrightarrow$}} &amp; V_{k}^{\prime} &amp; \mbox{\shortstack{$\diff_k^{\prime}$ \\ $\longrightarrow$}} &amp; V_{k+1}^{\prime} &amp; \longrightarrow~\cdots
+\end{array}\end{split}\]</div>
+</div></blockquote>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(f_k\)</span> maps <span class="math notranslate nohighlight">\(k\)</span>-cochains to <span class="math notranslate nohighlight">\(k\)</span>-cochains and <span class="math notranslate nohighlight">\(k\)</span>-coboundaries to <span class="math notranslate nohighlight">\(k\)</span>-coboundaries, thus induces a map <span class="math notranslate nohighlight">\(\mathcal{H}^k(f) : \mathcal{H}^k(V) \longrightarrow \mathcal{H}^k(V^{\prime})\)</span>.</p></li>
+</ul>
+<p>Let <span class="math notranslate nohighlight">\(V^{\prime} \subset V\)</span> be two cochain complexes,</p>
+<ul>
+<li><p>The inclusion <span class="math notranslate nohighlight">\(\imath_V\)</span> is a cochain map and thus induces a map of cohomology <span class="math notranslate nohighlight">\(\mathcal{H}^k(V^{\prime}) \longrightarrow \mathcal{H}^k(V)\)</span>,</p></li>
+<li><p>If there exists a cochain projection of <span class="math notranslate nohighlight">\(V\)</span> onto <span class="math notranslate nohighlight">\(V^{\prime}\)</span>, (this leads to <span class="math notranslate nohighlight">\(\pi \circ \imath = \id_{V^{\prime}}\)</span>) so <span class="math notranslate nohighlight">\(\mathcal{H}^k(\pi) \circ \mathcal{H}^k(\imath) = \id_{\mathcal{H}^k(V^{\prime})}\)</span>.</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{array}{ccccccc}
+\cdots &amp; \longrightarrow &amp; V_{k-1} &amp; \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} &amp; V_{k} &amp; \longrightarrow~\cdots \\
+ &amp; &amp; \pi_{k-1} \downarrow \uparrow \imath &amp; &amp; \pi_{k} \downarrow \uparrow \imath &amp; &amp; \\
+\cdots &amp; \longrightarrow &amp; V_{k-1}^{\prime} &amp; \mbox{\shortstack{$\diff_{k-1}$ \\ $\longrightarrow$}} &amp; V_{k}^{\prime} &amp; \longrightarrow~\cdots
+\end{array}\end{split}\]</div>
+</li>
+</ul>
+<p>Thus, <span class="math notranslate nohighlight">\(\mathcal{H}^k(\imath)\)</span> is <strong>injective</strong> and <span class="math notranslate nohighlight">\(\mathcal{H}^k(\pi)\)</span> is <strong>surjective</strong>. Hence, if one of the cohomology spaces <span class="math notranslate nohighlight">\(\mathcal{H}^k(V)\)</span> vanishes,
+then so does <span class="math notranslate nohighlight">\(\mathcal{H}^k(V^{\prime})\)</span></p>
+</section>
+<section id="cycles-and-boundaries-of-the-de-rham-complex">
+<h3>Cycles and boundaries of the De Rham complex<a class="headerlink" href="#cycles-and-boundaries-of-the-de-rham-complex" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(k\)</span>-cocycles</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[\mathfrak{Z}^k = \{ \omega \in H\Lambda^k(\Omega),~~ \diff \omega = 0 \}, ~~~ \mathfrak{Z}^{*k} = \{ \omega \in H^*\Lambda^k(\Omega),~~ \delta \omega = 0 \},\]</div>
+<div class="math notranslate nohighlight">
+\[\mathfrak{Z}_0^k = \{ \omega \in H_0\Lambda^k(\Omega),~~ \diff \omega = 0 \}, ~~~ \mathfrak{Z}_0^{*k} = \{ \omega \in H_0^*\Lambda^k(\Omega),~~ \delta \omega = 0 \},\]</div>
+<div class="math notranslate nohighlight">
+\[\]</div>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(k\)</span>-coboundaries</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[\mathfrak{B}^k = \diff H\Lambda^{k-1}(\Omega), ~~~ \mathfrak{B}^{* k} = \delta \Lambda^{k+1}(\Omega),\]</div>
+<div class="math notranslate nohighlight">
+\[\mathfrak{B}_0^k = \diff H_0\Lambda^{k-1}(\Omega), ~~~ \mathfrak{B}_0^{* k} = \delta \Lambda_0^{k+1}(\Omega),\]</div>
+<ul class="simple">
+<li><p>each of the spaces of cycles is closed in <span class="math notranslate nohighlight">\(\mathcal{H} \Lambda^k(\Omega)\)</span> (<span class="math notranslate nohighlight">\(\mathcal{H}^* \Lambda^k(\Omega)\)</span>), as well in <span class="math notranslate nohighlight">\(L^2 \Lambda^k(\Omega)\)</span>.</p></li>
+<li><p>each of the spaces of boundaries is closed in <span class="math notranslate nohighlight">\(L^2 \Lambda^k(\Omega)\)</span>.</p></li>
+<li><p>let <span class="math notranslate nohighlight">\(\perp\)</span> denotes the orthogonal complement in <span class="math notranslate nohighlight">\(L^2 \Lambda^k(\Omega)\)</span>,</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[\mathfrak{Z}^{k \perp} \subset \mathfrak{B}^{k \perp} = \mathfrak{Z}_0^{* k} , ~~~ \mathfrak{Z}^{* k \perp} \subset \mathfrak{B}^{* k \perp} = \mathfrak{Z}_0^{k}\]</div>
+<div class="math notranslate nohighlight">
+\[\mathfrak{Z}_0^{k \perp} \subset \mathfrak{B}_0^{k \perp} = \mathfrak{Z}^{* k} , ~~~ \mathfrak{Z}_0^{* k \perp} \subset \mathfrak{B}_0^{* k \perp} = \mathfrak{Z}^{k}\]</div>
+</section>
+<section id="the-hodge-decomposition">
+<h3>The Hodge decomposition<a class="headerlink" href="#the-hodge-decomposition" title="Link to this heading">#</a></h3>
+<p>There are two Hodge decompositions, with different boundary conditions,</p>
+<ol class="arabic">
+<li><div class="math notranslate nohighlight">
+\[L^2 \Lambda^k(\Omega)
+=
+\underbrace{\mathfrak{B}^{k}}_{\mathfrak{Z}_0^{* k\perp}}
+\oplus
+\underbrace{\mathfrak{H}^{k}
+\oplus
+\mathfrak{B}_0^{* k}}_{\mathfrak{Z}_0^{* k}=\mathfrak{B}^{k\perp}}
+=
+\overbrace{\mathfrak{B}^{k}
+\oplus
+\mathfrak{H}^{k}}^{\mathfrak{Z}^{k}=\mathfrak{B}_0^{* k\perp}}
+\oplus
+\overbrace{\mathfrak{B}_0^{* k}}^{\mathfrak{Z}^{k\perp}}\]</div>
+</li>
+<li><div class="math notranslate nohighlight">
+\[L^2 \Lambda^k(\Omega)
+=
+\underbrace{\mathfrak{B}_0^{k}}_{\mathfrak{Z}^{* k\perp}}
+\oplus
+\underbrace{\mathfrak{H}_0^{k}
+\oplus
+\mathfrak{B}^{* k}}_{\mathfrak{Z}^{* k}=\mathfrak{B}_0^{k\perp}}
+=
+\overbrace{\mathfrak{B}_0^{k}
+\oplus
+\mathfrak{H}_0^{k}}^{\mathfrak{Z}_0^{k}=\mathfrak{B}^{* k\perp}}
+\oplus
+\overbrace{\mathfrak{B}^{* k}}^{\mathfrak{Z}_0^{k\perp}}\]</div>
+</li>
+</ol>
+</section>
+</section>
+<section id="summary">
+<h2>Summary<a class="headerlink" href="#summary" title="Link to this heading">#</a></h2>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{tabular}{|c||c|c|c|c|}
+ \hline
+ $\omega^k \in \Lambda^k(\Omega)$             &amp; $k=0$
+                                              &amp; $k=1$
+                                              &amp; $k=2$
+                                              &amp; $k=3$
+ \\
+ \hline
+ $\diff \omega^k$                             &amp; $\Grad u$
+                                              &amp; $\Curl \uu$
+                                              &amp; $\Div \uu$
+                                              &amp; $-$
+ \\
+ $\delta \omega^k$                            &amp; $-$
+                                              &amp; $-\Div \uu$
+                                              &amp; $\Curl \uu$
+                                              &amp; $-\Grad u$
+ \\
+ $\mathfrak{i}_{\boldsymbol{\beta}} \omega^k$ &amp; $-$
+                                              &amp; $\boldsymbol{\beta} \cdot \uu$
+                                              &amp; $\uu \times \boldsymbol{\beta}$
+                                              &amp; $u \boldsymbol{\beta}$
+ \\
+ $\mathfrak{j}_{\boldsymbol{\beta}} \omega^k$ &amp; $u \boldsymbol{\beta}$
+                                              &amp; $-\uu \times \boldsymbol{\beta}$
+                                              &amp; $\boldsymbol{\beta} \cdot \uu$
+                                              &amp; $-$
+ \\
+ $L_{\boldsymbol{\beta}} \omega^k$            &amp; $\boldsymbol{\beta} \cdot \Grad u$
+                                              &amp; $\Grad \left(\boldsymbol{\beta} \cdot \uu \right)  + \left(\Curl \uu \right) \times \boldsymbol{\beta}$
+                                              &amp; $\Curl \left(\uu \times \boldsymbol{\beta} \right) + \boldsymbol{\beta} \Div \uu$
+                                              &amp; $\Div \left( u \boldsymbol{\beta} \right)$
+ \\
+ $\mathcal{L}_{\boldsymbol{\beta}} \omega^k$  &amp; $-\Div \left( u \boldsymbol{\beta} \right)$
+                                              &amp; $-\Curl \left(\uu \times \boldsymbol{\beta} \right) - \boldsymbol{\beta} \Div \uu$
+                                              &amp; $-\Grad \left(\boldsymbol{\beta} \cdot \uu \right)  - \left(\Curl \uu \right) \times \boldsymbol{\beta}$
+                                              &amp; $-\boldsymbol{\beta} \cdot \Grad u$
+ \\
+ \hline
+ $\tr \omega^k$                               &amp; $u(\xx)$
+                                              &amp; $\uu(\xx) \times \nn(\xx)$
+                                              &amp; $\uu(\xx) \cdot  \nn(\xx)$
+                                              &amp; $-$
+ \\
+ \hline
+ \hline
+ $H \Lambda^k(\Omega)$                        &amp; $\Hgrad$
+                                              &amp; $\Hcurl$
+                                              &amp; $\Hdiv$
+                                              &amp; $\Ltwo$
+ \\
+ $V_k$                                        &amp; $\Vgrad$
+                                              &amp; $\Vcurl$
+                                              &amp; $\Vdiv$
+                                              &amp; $\Vltwo$
+ \\
+ \hline
+\end{tabular}\end{split}\]</div>
+<p class="rubric">References</p>
+<span class="target" id="id1"></span></section>
+</section>
+
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+    <i class="fa-solid fa-list"></i> Contents
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#basis">Basis</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#orientation-and-volume-form">Orientation and Volume form</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exterior-calculus-on-manifolds-and-differential-forms">Exterior Calculus on manifolds and Differential forms</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sobolev-spaces-of-differential-forms">Sobolev spaces of differential forms</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#cohomology-and-de-rham-complex">Cohomology and De Rham Complex</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-hodge-decomposition">The Hodge decomposition</a></li>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#where-do-the-glts-come-from">Where do the GLTs come from?</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#spectral-preliminaries">Spectral preliminaries</a></li>
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+  <section id="glt">
+<h1>GLT<a class="headerlink" href="#glt" title="Link to this heading">#</a></h1>
+<section id="where-do-the-glts-come-from">
+<h2>Where do the GLTs come from?<a class="headerlink" href="#where-do-the-glts-come-from" title="Link to this heading">#</a></h2>
+<p>The main aim of this paragraph is to present a crucial example that highlights the importance of the GLT algebra when dealing with linear systems coming from the discretization of PDEs. Let us start with some preliminaries. In detail, we will recall the notion of symbol of a matrix-sequence and the basic idea behind the GLT theory.</p>
+<section id="spectral-preliminaries">
+<h3>Spectral preliminaries<a class="headerlink" href="#spectral-preliminaries" title="Link to this heading">#</a></h3>
+<p>The following one is a rather informal definition of symbol of a matrix-sequence.</p>
+<p><strong>example</strong>:</p>
+<p>When <span class="math notranslate nohighlight">\(d_n=n\)</span>, <span class="math notranslate nohighlight">\(d=1\)</span>, <span class="math notranslate nohighlight">\(D=[0,\pi]\)</span>, <span class="math notranslate nohighlight">\(\{A_n\}_n\sim_\lambda f\)</span> means</p>
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new file mode 100644
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+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>B-Splines and NURBS</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#b-splines-series">B-Splines series</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#knots-vector-families">Knots vector families</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#b-spline-curve">B-Spline curve</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#multivariate-tensor-product-splines">Multivariate tensor product splines</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#deriving-a-b-spline-curve">Deriving a B-spline curve</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#proposition">proposition</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#example">Example</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#knot-insertion">Knot insertion</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#order-elevation">Order elevation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#translation">Translation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rotation">Rotation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#scaling">Scaling</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="b-splines-and-nurbs">
+<h1>B-Splines and NURBS<a class="headerlink" href="#b-splines-and-nurbs" title="Link to this heading">#</a></h1>
+<p>We start this section by recalling some basic properies about B-splines curves and surfaces. We also recall some fundamental algorithms (knot insertion and degree elevation).</p>
+<p>For a basic introduction to the subject, we refer to the books <span id="id1">[]</span> and <span id="id2">[]</span>.</p>
+<p>A B-Splines family, <span class="math notranslate nohighlight">\((N_i)_{ 1 \leqslant i \leqslant n}\)</span> of order <span class="math notranslate nohighlight">\(k\)</span>, can be generated using a non-decreasing sequence <strong>of knots</strong> <span class="math notranslate nohighlight">\(T=(t_i)_{1\leqslant i \leqslant n + k}\)</span>.</p>
+<section id="b-splines-series">
+<h2>B-Splines series<a class="headerlink" href="#b-splines-series" title="Link to this heading">#</a></h2>
+<p>The j-th B-Spline of order <span class="math notranslate nohighlight">\(k\)</span> is defined by the recurrence relation:</p>
+<div class="math notranslate nohighlight">
+\[N_j^k = w_j^k N_j^{k-1} + ( 1 - w_{j+1}^k ) N_{j+1}^{k-1}\]</div>
+<p>where,</p>
+<div class="math notranslate nohighlight">
+\[w_j^k (x) = \frac{x-t_j}{t_{j+k-1}-t_{j}} \hspace{2cm} N_j^1(x) = \chi_{ \left[ t_j, t_{j+1} \right[ }(x)\]</div>
+<p>for <span class="math notranslate nohighlight">\(k \geq 1\)</span> and <span class="math notranslate nohighlight">\(1 \leq j \leq n\)</span>.</p>
+<p>We note some important properties of a B-splines basis:</p>
+<ul class="simple">
+<li><p>B-splines are piecewise polynomial of degree <span class="math notranslate nohighlight">\(p=k-1\)</span>,</p></li>
+<li><p>Compact support; the support of <span class="math notranslate nohighlight">\(N_j^k\)</span> is contained in <span class="math notranslate nohighlight">\(\left[ t_j, t_{j+k} \right]\)</span> ,</p></li>
+<li><p>If <span class="math notranslate nohighlight">\(x \in~ ] t_j,t_{j+1} [\)</span>, then only the <em>B-splines</em> <span class="math notranslate nohighlight">\(\{ N_{j-k+1}^k,\cdots,N_{j}^k \}\)</span> are non vanishing at <span class="math notranslate nohighlight">\(x\)</span>,</p></li>
+<li><p>Positivity: <span class="math notranslate nohighlight">\(\forall j \in \{1,\cdots,n \}~~N_j(x) &gt;0, ~~\forall x \in ] t_j, t_{j+k} [\)</span>,</p></li>
+<li><p>Partition of unity  <span class="math notranslate nohighlight">\(\sum_{i=1}^n N_i^{k}(x) = 1, \forall x \in \mathbb{R}\)</span>,</p></li>
+<li><p>Local linear independence,</p></li>
+<li><p>If a knot <span class="math notranslate nohighlight">\(t_i\)</span> has a multiplicity <span class="math notranslate nohighlight">\(m_i\)</span> then the B-spline is <span class="math notranslate nohighlight">\(\mathcal{C}^{(p-m_i)}\)</span> at <span class="math notranslate nohighlight">\(t_i\)</span>.</p></li>
+</ul>
+</section>
+<section id="knots-vector-families">
+<h2>Knots vector families<a class="headerlink" href="#knots-vector-families" title="Link to this heading">#</a></h2>
+<p>There are two kind of <strong>knots vectors</strong>, called <strong>clamped</strong> and <strong>unclamped</strong>. Both families contains <strong>uniform</strong> and <strong>non-uniform</strong> sequences.</p>
+<p>The following are examples of such knots vectors</p>
+<ol class="arabic simple">
+<li><p><strong>Clamped knots</strong> (open knots vector)</p></li>
+</ol>
+<ul class="simple">
+<li><p>uniform</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[\begin{split}T_1 &amp;= \{0, 0, 0, 1, 2, 3, 4, 5, 5, 5 \}
+\\
+T_2 &amp;= \{-0.2, -0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 0.8 \}\end{split}\]</div>
+<a class="reference internal image-reference" href="../../../_images/bsplines_t1_p2.png"><img alt="../../../_images/bsplines_t1_p2.png" src="../../../_images/bsplines_t1_p2.png" style="width: 8cm; height: 8cm;" /></a>
+<a class="reference internal image-reference" href="../../../_images/bsplines_t2_p2.png"><img alt="../../../_images/bsplines_t2_p2.png" src="../../../_images/bsplines_t2_p2.png" style="width: 8cm; height: 8cm;" /></a>
+<ul class="simple">
+<li><p>non-uniform</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[\begin{split}T_3 &amp;= \{0, 0, 0, 1, 3, 4, 5, 5, 5 \}
+\\
+T_4 &amp;= \{-0.2, -0.2, 0.4, 0.6, 0.8, 0.8 \}\end{split}\]</div>
+<a class="reference internal image-reference" href="../../../_images/bsplines_t3_p2.png"><img alt="../../../_images/bsplines_t3_p2.png" src="../../../_images/bsplines_t3_p2.png" style="width: 8cm; height: 8cm;" /></a>
+<a class="reference internal image-reference" href="../../../_images/bsplines_t4_p2.png"><img alt="../../../_images/bsplines_t4_p2.png" src="../../../_images/bsplines_t4_p2.png" style="width: 8cm; height: 8cm;" /></a>
+<ol class="arabic simple" start="2">
+<li><p><strong>Unclamped knots</strong></p></li>
+</ol>
+<ul class="simple">
+<li><p>uniform</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[\begin{split}T_5 &amp;= \{0, 1, 2, 3, 4, 5, 6, 7 \}
+\\
+T_6 &amp;= \{-0.2, 0.0, 0.2, 0.4, 0.6, 0.8, 1.0 \}\end{split}\]</div>
+<a class="reference internal image-reference" href="../../../_images/bsplines_t5_p2.png"><img alt="../../../_images/bsplines_t5_p2.png" src="../../../_images/bsplines_t5_p2.png" style="width: 8cm; height: 8cm;" /></a>
+<a class="reference internal image-reference" href="../../../_images/bsplines_t6_p2.png"><img alt="../../../_images/bsplines_t6_p2.png" src="../../../_images/bsplines_t6_p2.png" style="width: 8cm; height: 8cm;" /></a>
+<ul class="simple">
+<li><p>non-uniform</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[\begin{split}T_7 &amp;= \{0, 0, 3, 4, 7, 8, 9 \}
+\\
+T_8 &amp;= \{-0.2, 0.2, 0.4, 0.6, 1.0, 2.0, 2.5 \}\end{split}\]</div>
+<a class="reference internal image-reference" href="../../../_images/bsplines_t7_p2.png"><img alt="../../../_images/bsplines_t7_p2.png" src="../../../_images/bsplines_t7_p2.png" style="width: 8cm; height: 8cm;" /></a>
+<a class="reference internal image-reference" href="../../../_images/bsplines_t8_p2.png"><img alt="../../../_images/bsplines_t8_p2.png" src="../../../_images/bsplines_t8_p2.png" style="width: 8cm; height: 8cm;" /></a>
+</section>
+<section id="b-spline-curve">
+<h2>B-Spline curve<a class="headerlink" href="#b-spline-curve" title="Link to this heading">#</a></h2>
+<p>The B-spline curve in <span class="math notranslate nohighlight">\(\mathbb{R}^d\)</span> associated to knots vector <span class="math notranslate nohighlight">\(T=(t_i)_{1\leqslant i \leqslant n + k}\)</span> and the control polygon <span class="math notranslate nohighlight">\((\mathbf{P}_i)_{ 1 \leqslant i \leqslant n}\)</span> is defined by :</p>
+<div class="math notranslate nohighlight">
+\[\mathcal{C}(t) = \sum_{i=1}^n N_i^k(t) \textbf{P}_i\]</div>
+<p>In (Fig. ref{figBSplineCurve}), we give an example of a quadratic B-Spline curve, and its corresponding knot vector and control points.</p>
+<a class="reference internal image-reference" href="../../../_images/courbe_bsplines.png"><img alt="../../../_images/courbe_bsplines.png" src="../../../_images/courbe_bsplines.png" style="width: 8cm; height: 8cm;" /></a>
+<a class="reference internal image-reference" href="../../../_images/basis_fct_p2_N5.png"><img alt="../../../_images/basis_fct_p2_N5.png" src="../../../_images/basis_fct_p2_N5.png" style="width: 8cm; height: 8cm;" /></a>
+<p>We have the following properties for a <em>B-spline</em> curve:</p>
+<ul class="simple">
+<li><p>If <span class="math notranslate nohighlight">\(n=k\)</span>, then <span class="math notranslate nohighlight">\(\mathcal{C}\)</span> is just a B'ezier-curve,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathcal{C}\)</span> is a piecewise polynomial curve,</p></li>
+<li><p>The curve interpolates its extremas if the associated multiplicity of the first and the last knot are maximum (<em>i.e.</em> equal to <span class="math notranslate nohighlight">\(k\)</span>), <em>i.e.</em> open knot vector,</p></li>
+<li><p>Invariance with respect to affine transformations,</p></li>
+<li><p>Strong convex-hull property:</p></li>
+</ul>
+<p>if <span class="math notranslate nohighlight">\(t_i \leq t \leq t_{i+1}\)</span>, then <span class="math notranslate nohighlight">\(\mathcal{C}(t)\)</span> is inside the convex-hull associated to the control points <span class="math notranslate nohighlight">\(\mathbf{P}_{i-p},\cdots,\mathbf{P}_{i}\)</span>,</p>
+<ul class="simple">
+<li><p>Local modification : moving the <span class="math notranslate nohighlight">\(i^{th}\)</span> control point <span class="math notranslate nohighlight">\(\mathbf{P}_{i}\)</span> affects <span class="math notranslate nohighlight">\(\mathcal{C}(t)\)</span>, only in the interval <span class="math notranslate nohighlight">\([t_i,t_{i+k}]\)</span>,</p></li>
+<li><p>The control polygon approaches the behavior of the curve.</p></li>
+</ul>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>In order to model a singular curve, we can use multiple control points : <span class="math notranslate nohighlight">\(\mathbf{P}_{i}=\mathbf{P}_{i+1}\)</span>.</p>
+</div>
+</section>
+<section id="multivariate-tensor-product-splines">
+<h2>Multivariate tensor product splines<a class="headerlink" href="#multivariate-tensor-product-splines" title="Link to this heading">#</a></h2>
+<p>Let us consider <span class="math notranslate nohighlight">\(d\)</span> knot vectors <span class="math notranslate nohighlight">\(\mathcal{T} = \{T^1,T^2,\cdots,T^d\}\)</span>. For simplicity, we consider that these knot vectors are open, which means that <span class="math notranslate nohighlight">\(k\)</span> knots on each side are duplicated so that the spline is interpolating on the boundary, and of bounds <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(1\)</span>. In the sequel we will use the notation <span class="math notranslate nohighlight">\(I=[0,1]\)</span>.
+Each knot vector <span class="math notranslate nohighlight">\(T^i\)</span>, will generate a basis for a Schoenberg space, <span class="math notranslate nohighlight">\(\mathcal{S}_{k_{i}}(T^i,I)\)</span>. The tensor product of all these spaces is also a Schoenberg space, namely <span class="math notranslate nohighlight">\(\mathcal{S}_{\mathbf{k}}(\mathcal{T})\)</span>, where <span class="math notranslate nohighlight">\(\mathbf{k}=\{k_1,\cdots,k_d\}\)</span>. The cube <span class="math notranslate nohighlight">\(\mathcal{P}=I^d=[0,1]^d\)</span>, will be referred to as a patch.</p>
+<p>The basis for <span class="math notranslate nohighlight">\(\mathcal{S}_{\mathbf{k}}(\mathcal{T})\)</span> is defined by a tensor product :</p>
+<div class="math notranslate nohighlight">
+\[N_{\mathbf{i}}^{\mathbf{k}} := N_{i_1}^{k_1} \otimes N_{i_2}^{k_2} \otimes \cdots \otimes N_{i_d}^{k_d}\]</div>
+<p>where, <span class="math notranslate nohighlight">\(\mathbf{i}=\{i_1,\cdots , i_d \}\)</span>.</p>
+<p>A typical cell from <span class="math notranslate nohighlight">\(\mathcal{P}\)</span> is a cube of the form : <span class="math notranslate nohighlight">\(Q_{\mathbf{i}}=[\xi_{i_1}, \xi_{i_1+1}] \otimes \cdots \otimes [\xi_{i_d}, \xi_{i_d+1}]\)</span>.</p>
+</section>
+<section id="deriving-a-b-spline-curve">
+<h2>Deriving a B-spline curve<a class="headerlink" href="#deriving-a-b-spline-curve" title="Link to this heading">#</a></h2>
+<p>The derivative of a B-spline curve is obtained as:</p>
+<div class="math notranslate nohighlight">
+\[ \mathcal{C}^{\prime}(t) = \sum_{i=1}^{n} {N_{i}^{k}}^{\prime}(t) \mathbf{P}_i = \sum_{i=1}^{n} \left(\frac{p}{t_{i+p}-t_{i}}N_{i}^{k-1}(t) \mathbf{P}_i - \frac{p}{t_{i+1+p}-t_{i+1}}N_{i+1}^{k-1}(t) \mathbf{P}_i \right)
+= \sum_{i=1}^{n-1} {N_{i}^{k-1}}^{\ast}(t) \mathbf{Q}_i\]</div>
+<p>where <span class="math notranslate nohighlight">\(\mathbf{Q}_i = p \frac{\mathbf{P}_{i+1} - \mathbf{P}_i}{t_{i+1+p}-t_{i+1}}\)</span>, and <span class="math notranslate nohighlight">\(\{{N_{i}^{k-1}}^{\ast},~~1 \leq i \leq n-1\}\)</span> are generated using the knot vector <span class="math notranslate nohighlight">\(T^{\ast}\)</span>, which is obtained from <span class="math notranslate nohighlight">\(T\)</span> by reducing by one the multiplicity of the first and the last knot (in the case of open knot vector), <em>i.e.</em> by removing the first and the last knot.</p>
+<p>More generally, by introducing the B-splines family <span class="math notranslate nohighlight">\(\{ {N_{i}^{k-j}}^{\ast}, 1 \leq i \leq n-j \}\)</span> generated by the knots vector <span class="math notranslate nohighlight">\(T^{j^{\ast}}\)</span> obtained from <span class="math notranslate nohighlight">\(T\)</span> by removing the first and the last knot <span class="math notranslate nohighlight">\(j\)</span> times, we have the following result:</p>
+<section id="proposition">
+<h3>proposition<a class="headerlink" href="#proposition" title="Link to this heading">#</a></h3>
+<p>The <span class="math notranslate nohighlight">\(j^{th}\)</span> derivative of the curve <span class="math notranslate nohighlight">\(\mathcal{C}\)</span> is given by</p>
+<div class="math notranslate nohighlight">
+\[\mathcal{C}^{(j)}(t) = \sum_{i=1}^{n-j} {N_{i}^{k-j}}^{\ast}(t) \mathbf{P}_i^{(j)}`\]</div>
+<p>where, for <span class="math notranslate nohighlight">\(j&gt;0\)</span></p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{P}_i^{(j)} = \frac{p-j+1}{t_{i+p+1}-t_{i+j}} \left( \mathbf{P}_{i+1}^{(j-1)} - \mathbf{P}_i^{(j-1)} \right)
+\\
+\mbox{and} ~ ~ ~ \mathbf{P}_i^{(0)} = \mathbf{P}_i.\end{split}\]</div>
+<p>By denoting <span class="math notranslate nohighlight">\(\mathcal{C}^{\prime}\)</span> and <span class="math notranslate nohighlight">\(\mathcal{C}^{\prime\prime}\)</span> the first and second derivative of the B-spline curve <span class="math notranslate nohighlight">\(\mathcal{C}\)</span>, it is easy to show that:</p>
+<p>We have,</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\mathcal{C}^{\prime}(0) = \frac{p}{t_{p+2}} \left(\mathbf{P}_{2} - \mathbf{P}_1\right)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathcal{C}^{\prime}(1) = \frac{p}{1-t_{n}} \left(\mathbf{P}_{n} - \mathbf{P}_{n-1}\right)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathcal{C}^{\prime\prime}(0) = \frac{p(p-1)}{t_{p+2}} \left( \frac{1}{t_{p+2}}\mathbf{P}_{1} - \{ \frac{1}{t_{p+2}} + \frac{1}{t_{p+3}} \} \mathbf{P}_2 + \frac{1}{t_{p+3}}\mathbf{P}_{3} \right)\)</span>,</p></li>
+<li><p><span class="math notranslate nohighlight">\(\mathcal{C}^{\prime\prime}(1) = \frac{p(p-1)}{1-t_{n}} \left( \frac{1}{1-t_{n}}\mathbf{P}_{n} - \{ \frac{1}{1-t_{n}} + \frac{1}{1-t_{n-1}} \} \mathbf{P}_{n-1} + \frac{1}{1-t_{n-1}}\mathbf{P}_{n-2} \right)\)</span>.</p></li>
+</ul>
+</section>
+<section id="example">
+<h3>Example<a class="headerlink" href="#example" title="Link to this heading">#</a></h3>
+<p>Let us consider the quadratic B-spline curve associated to the knots vector <span class="math notranslate nohighlight">\(T=\{000~\frac{2}{5}~\frac{3}{5}~111 \}\)</span> and the control points <span class="math notranslate nohighlight">\(\{ P_i, 1 \leq i \leq 5 \}\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[\mathcal{C}(t) = \sum_{i=1}^{5} {N_{i}^{3}}^{\prime}(t) \mathbf{P}_i\]</div>
+<p>we have,</p>
+<div class="math notranslate nohighlight">
+\[\mathcal{C}^{\prime}(t) = \sum_{i=1}^{4} {N_{i}^{2}}^{\ast}(t) \mathbf{Q}_i\]</div>
+<p>where</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{Q}_1 = 5 \{\mathbf{P}_{2} - \mathbf{P}_1\}, ~~~~\mathbf{Q}_2 = \frac{10}{3} \{ \mathbf{P}_{3} - \mathbf{P}_2\},
+\\
+\mathbf{Q}_3 = \frac{10}{3} \{ \mathbf{P}_{4} - \mathbf{P}_3\},~~~~\mathbf{Q}_4 = 5 \{\mathbf{P}_{5} - \mathbf{P}_4\}.\end{split}\]</div>
+<p>The <em>B-splines</em> <span class="math notranslate nohighlight">\(\{ {N_{i}^{2}}^{\ast},~~1 \leq i \leq 4\}\)</span> are associated to the knot vector <span class="math notranslate nohighlight">\(T^{\ast}=\{00~\frac{2}{5}~\frac{3}{5}~11 \}\)</span>.</p>
+<p class="rubric">Fundamental geometric operations</p>
+<p>By inserting new knots into the knot vector, we add new control points without changing the shape of the B-Spline curve. This can be done using the DeBoor algorithm <span id="id3">[]</span>. We can also elevate the degree of the B-Spline family and keep unchanged the curve <span id="id4">[]</span>. In (Fig. ref{refinement_curve_B_Spline}), we apply these algorithms on a quadratic B-Spline curve and we show the position of the new control points.</p>
+</section>
+</section>
+<section id="knot-insertion">
+<h2>Knot insertion<a class="headerlink" href="#knot-insertion" title="Link to this heading">#</a></h2>
+<p>After modification, we denote by <span class="math notranslate nohighlight">\(\widetilde{n}, \widetilde{k}, \widetilde{T}\)</span> the new parameters. <span class="math notranslate nohighlight">\((\textbf{Q}_i)\)</span> are the new control points.</p>
+<p>One can insert a new knot <span class="math notranslate nohighlight">\(t\)</span>, where <span class="math notranslate nohighlight">\(t_j \leqslant t &lt; t_{j+1}\)</span>. For this purpose we use the DeBoor algorithm <span id="id5">[]</span>:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\widetilde{n} = n+1
+\\
+\widetilde{k} = k
+\\
+\widetilde{T} = \{ t_1,.., t_j, t, t_{j+1},.., t_{n+k}\}
+\\
+\alpha_i = \left\{\begin{array}{cc}1 &amp; 1 \leqslant i \leqslant j-k+1 \\\frac{t-t_i}{t_{i+k-1}-t_i} &amp; j-k+2 \leqslant i \leqslant j \\0 &amp; j+1 \leqslant i \end{array}\right.
+\\
+\textbf{Q}_i = \alpha_i \textbf{P}_i + (1-\alpha_i) \textbf{P}_{i-1}\end{split}\]</div>
+<p>Many other algorithms exist, like blossoming for fast insertion algorithm. For more details about this topic, we refer to <span id="id6">[]</span>.</p>
+</section>
+<section id="order-elevation">
+<h2>Order elevation<a class="headerlink" href="#order-elevation" title="Link to this heading">#</a></h2>
+<p>We can elevate the order of the basis, without changing the curve. Several algorithms exist for this purpose. We used the one by Huang et al. <span id="id7">[]</span>, <span id="id8">[]</span>.</p>
+<p>A quadratic B-spline curve and its control points. The knot vector is <span class="math notranslate nohighlight">\(T = \{ 000, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 1 1 \}\)</span>.</p>
+<a class="reference internal image-reference" href="../../../_images/curve.png"><img alt="../../../_images/curve.png" src="../../../_images/curve.png" style="width: 8cm; height: 8cm;" /></a>
+<p>The curve after a h-refinement by inserting the knots <span class="math notranslate nohighlight">\(\{ 0.15, 0.35\}\)</span> while the degree is kept equal to <span class="math notranslate nohighlight">\(2\)</span>.</p>
+<a class="reference internal image-reference" href="../../../_images/curve_p0_n9.png"><img alt="../../../_images/curve_p0_n9.png" src="../../../_images/curve_p0_n9.png" style="width: 8cm; height: 8cm;" /></a>
+<p>The curve after a p-refinement, the degree was raised by <span class="math notranslate nohighlight">\(1\)</span> (using cubic B-splines).</p>
+<a class="reference internal image-reference" href="../../../_images/curve_p2_n0.png"><img alt="../../../_images/curve_p2_n0.png" src="../../../_images/curve_p2_n0.png" style="width: 8cm; height: 8cm;" /></a>
+<p>The curve after duplicating the multiplicity of the internal knots <span class="math notranslate nohighlight">\(\{ \frac{1}{4}, \frac{1}{2}, \frac{3}{4} \}\)</span>,
+this leads to a B'ezier description. We can then, split the curve into <span class="math notranslate nohighlight">\(4\)</span> pieces (sub-domains), each one will corresponds to a quadratic B'ezier curve.</p>
+<a class="reference internal image-reference" href="../../../_images/curve_p0_n3_bezier.png"><img alt="../../../_images/curve_p0_n3_bezier.png" src="../../../_images/curve_p0_n3_bezier.png" style="width: 8cm; height: 8cm;" /></a>
+</section>
+<section id="translation">
+<h2>Translation<a class="headerlink" href="#translation" title="Link to this heading">#</a></h2>
+</section>
+<section id="rotation">
+<h2>Rotation<a class="headerlink" href="#rotation" title="Link to this heading">#</a></h2>
+</section>
+<section id="scaling">
+<h2>Scaling<a class="headerlink" href="#scaling" title="Link to this heading">#</a></h2>
+<p class="rubric">References</p>
+<span class="target" id="id9"></span></section>
+</section>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#knots-vector-families">Knots vector families</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#b-spline-curve">B-Spline curve</a></li>
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+</ul>
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#knot-insertion">Knot insertion</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#order-elevation">Order elevation</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#translation">Translation</a></li>
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+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bezier-curves.html">Bézier curves</a></li>
+</ul>
+</li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter0/bsplines.html">B-Splines</a><input class="toctree-checkbox" id="toctree-checkbox-3" name="toctree-checkbox-3" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-3"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-curves.html">B-Splines curves</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-surfaces.html">B-Splines surfaces</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter0/iga.html">Isogeometric Analysis</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/fem.html">Introduction to B-Splines FEM</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-1d.html">B-splines FEM solver for Poisson equation (1D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/data-structure.html">Data Structure</a></li>
+
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/install.html">Installation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter1/sympde.html">SymPDE concepts and their mathematical meaning</a><input class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-5"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../chapter1/rules.html">Algebraic and differential operators evalution rules</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter1/poisson.html">Your first code using SymPDE &amp; PsyDAC</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson.html">The Poisson equation</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/poisson-nitsche.html">The Poisson equation with weak imposition of Dirichlet conditions</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-general-form.html">Elliptic equation in the general form</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/biharmonic.html">The Biharmonic problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/advection-diffusion-stabilized.html">Stabilized advection-diffusion equation</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="../../chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/power_law_fluid_flow_in_a_channel.html">Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/electromagnetics.html">Electromagnetics</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/mhd.html">MHD</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/material-science.html">Material Science</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../chapter5/multiphysics.html">Multiphysics</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>First Steps with PSYDAC</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#install-psydac">Install PSYDAC</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#examples">Examples</a></li>
+</ul>
+            </nav>
+        </div>
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+
+              
+                
+<div id="searchbox"></div>
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+  <section id="first-steps-with-psydac">
+<h1>First Steps with PSYDAC<a class="headerlink" href="#first-steps-with-psydac" title="Link to this heading">#</a></h1>
+<p>This document is meant to give a tutorial-like overview of PSYDAC.</p>
+<p>The green arrows designate “more info” links leading to advanced sections about
+the described task.</p>
+<p>By reading this tutorial, you’ll be able to:</p>
+<ul class="simple">
+<li><p>compile a simple <em>PSYDAC</em> file</p></li>
+<li><p>get familiar with parallel programing paradigms</p></li>
+<li><p>create, modify and build a <em>PSYDAC</em> project.</p></li>
+</ul>
+<section id="install-psydac">
+<h2>Install PSYDAC<a class="headerlink" href="#install-psydac" title="Link to this heading">#</a></h2>
+</section>
+<section id="examples">
+<h2>Examples<a class="headerlink" href="#examples" title="Link to this heading">#</a></h2>
+<p>In this section, we describe some features of <em>PSYDAC</em> on simple examples.</p>
+<p>See <code class="xref download docutils literal notranslate"><span class="pre">script</span></code>.</p>
+</section>
+</section>
+
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+<li class="toctree-l2"><a class="reference internal" href="../../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+  <p>Welcome to PSYDAC’s documentation!</p>
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+<p>This documentation is still under construction.
+For the time being, its purpose is to assist the developers.
+Find our GitHub repository <a class="reference external" href="https://github.com/pyccel/psydac/tree/devel/">here</a>.</p>
+</div>
+<p>Psydac is a Python 3 library for isogeometric analysis (IGA). It uses high-order tensor-product splines, multi-patch mapped
+domains, and hybrid MPI-OpenMP parallelization.</p>
+<p>In order to use Psydac, the user provides a geometry analytically or through an input file, and then defines the model equations
+in symbolic form (weak formulation) using <a class="reference external" href="https://github.com/pyccel/sympde">SymPDE</a>, which provides the mathematical
+expressions and checks the semantic validity of the model.</p>
+<p>Once a finite element discretization has been chosen, Psydac maps the abstract concepts to concrete objects, the basic building
+blocks being MPI-distributed vectors and matrices.
+For all the computationally intensive operations (assembly of matrices, vectors, and norms, etc.), Psydac generates ad-hoc
+Python code which is then accelerated to Fortran speed using <a class="reference external" href="https://github.com/pyccel/pyccel">Pyccel</a>.</p>
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diff --git a/psydac/docs/source/modules/STUBDIR/psydac.api.ast.basic.html b/psydac/docs/source/modules/STUBDIR/psydac.api.ast.basic.html
new file mode 100644
index 0000000..d8d1524
--- /dev/null
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+  <section id="module-psydac.api.ast.expr">
+<span id="psydac-api-ast-expr"></span><h1>psydac.api.ast.expr<a class="headerlink" href="#module-psydac.api.ast.expr" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_atoms_expr</span></code>(atom, basis, indices, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_scalar_field</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_vector_field</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ExprInterface</span></code>(kernel[, name, mapping, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ExprKernel</span></code>(expr, space[, name, mapping, ...])</p></td>
+<td><p></p></td>
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+  <section id="module-psydac.api.ast.fem">
+<span id="psydac-api-ast-fem"></span><h1>psydac.api.ast.fem<a class="headerlink" href="#module-psydac.api.ast.fem" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_diag_len</span></code>(p, md, mc)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">convert</span></code>(dtype)</p></td>
+<td><p>This function returns the index of a Function Space in a 3D DeRham sequence</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">expand</span></code>(args)</p></td>
+<td><p>This function expands vector functions into indexed functions</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">expand_hdiv_hcurl</span></code>(args)</p></td>
+<td><p>This function expands vector functions of type hdiv and hculr into indexed functions</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_degrees</span></code>(funcs, space)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_multiplicity</span></code>(funcs, space)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_nquads</span></code>(Vh)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">regroup</span></code>(tests)</p></td>
+<td><p>This function regourps the test/trial functions by their Function Space</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">toInteger</span></code>(a)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AST</span></code>(expr, terminal_expr, spaces[, ...])</p></td>
+<td><p>The Ast class transforms a terminal expression returned from sympde into a DefNode</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DefNode</span></code>(name, arguments, local_variables, ...)</p></td>
+<td><p>DefNode represents a function definition where it contains the arguments and the body</p></td>
+</tr>
+</tbody>
+</table>
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+  <section id="module-psydac.api.ast.linalg">
+<span id="psydac-api-ast-linalg"></span><h1>psydac.api.ast.linalg<a class="headerlink" href="#module-psydac.api.ast.linalg" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Mod</span></code>(a, b)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_diag_len</span></code>(p, md, mc[, return_padding])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">toInteger</span></code>(a)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">variable_to_sympy</span></code>(x)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearOperatorDot</span></code>(ndim, block_shape[, comm])</p></td>
+<td><p>Generate the Matrix Vector Product function for a BlockLinearOperator,StencilMatrix or StencilInterfaceMatrix.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorDot</span></code>(ndim[, backend])</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
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+  <section id="module-psydac.api.ast.nodes">
+<span id="psydac-api-ast-nodes"></span><h1>psydac.api.ast.nodes<a class="headerlink" href="#module-psydac.api.ast.nodes" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_itergener</span></code>(a, index)</p></td>
+<td><p>Create the generator and the iterator based on a and the index</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_logical_expressions</span></code>(u, nderiv[, lhs])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_number_derivatives</span></code>(expr)</p></td>
+<td><p>returns the number of partial derivatives in expr.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_string</span></code>(n)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">toInteger</span></code>(a)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AddNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Allocate</span></code>(arr, shape)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AndNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ArityType</span></code>()</p></td>
+<td><p>Base class representing a form type: bilinear/linear/functional</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Array</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ArrayNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">AtomicNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BaseNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasisAtom</span></code>(expr)</p></td>
+<td><p>Used to describe a temporary for the basis coefficient or in the kernel.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BilinearArity</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Block</span></code>(body)</p></td>
+<td><p>This class represents a Block of statements</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockLinearOperatorNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockScalarLocalBasis</span></code>([trials, tests, expr, ...])</p></td>
+<td><p>This is used to describe a block of scalar dofs over an element</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockStencilMatrixGlobalBasis</span></code>(trials, tests, ...)</p></td>
+<td><p>used to describe local dof over an element as a block stencil matrix</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockStencilMatrixLocalBasis</span></code>(trials, tests, ...)</p></td>
+<td><p>used to describe local dof over an element as a block stencil matrix</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockStencilVectorGlobalBasis</span></code>(tests, pads, ...)</p></td>
+<td><p>used to describe local dof over an element as a block stencil matrix</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockStencilVectorLocalBasis</span></code>(tests, pads, expr)</p></td>
+<td><p>used to describe local dof over an element as a block stencil matrix</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CoefficientBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputeKernelExpr</span></code>(expr[, weights])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputeLogical</span></code>(expr[, weights, lhs])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputeLogicalBasis</span></code>(expr[, lhs])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputeNode</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputePhysical</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComputePhysicalBasis</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Element</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ElementOf</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">EqNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">EvalField</span></code>(domain, atoms, q_index, l_index, ...)</p></td>
+<td><p>This function computes atomic expressions needed to evaluate  EvaluteField/VectorField final expression</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">EvalMapping</span></code>(domain, quads, indices_basis, ...)</p></td>
+<td><p>This function computes atomic expressions needed to evaluate  EvalMapping final expression.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Evaluation</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ExprNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Expression</span></code>(*args)</p></td>
+<td><p>The Expression class gives us the possibility to create specific instructions for some dimension, where the generated code is not in a vectorized form.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FieldEvaluation</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FloorDiv</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FunctionalArity</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GeneratorBase</span></code>(target, dummies)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GeometryAtom</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GeometryExpr</span></code>(expr[, dtype])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GeometryExpressions</span></code>(M, nderiv[, dtype])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalElementBasis</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalSpanArray</span></code>(target[, index])</p></td>
+<td><p>This represents the global span array</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalTensorQuadratureBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalTensorQuadratureGrid</span></code>(*args)</p></td>
+<td><p>This class represents the quadrature points and weights in a domain.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalTensorQuadratureTestBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalTensorQuadratureTrialBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalThreadEnds</span></code>([index])</p></td>
+<td><p>This represents the threads ends over the decomposed domain</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalThreadSizes</span></code>([index])</p></td>
+<td><p>This represents the number of elements owned by a thread</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalThreadSpanArray</span></code>(target[, index])</p></td>
+<td><p>This represents the global span array of each thread</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalThreadStarts</span></code>([index])</p></td>
+<td><p>This represents the threads starts over the decomposed domain</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Grid</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IfNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexDerivative</span></code>([length])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexDof</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexDofTest</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexDofTrial</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexElement</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexInnerDofTest</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexNode</span></code>([start, stop, length, index])</p></td>
+<td><p>Base class representing one index of an iterator</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexOuterDofTest</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IndexQuadrature</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IntDivNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IteratorBase</span></code>(target[, dummies])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthDof</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthDofTest</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthDofTrial</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthElement</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthInnerDofTest</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthNode</span></code>([target, index])</p></td>
+<td><p>Base class representing one length of an iterator</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthOuterDofTest</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LengthQuadrature</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearArity</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalElementBasis</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalIndexElement</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalSpanArray</span></code>(target[, index])</p></td>
+<td><p>This represents the local span array</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalTensorQuadratureBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalTensorQuadratureGrid</span></code>(*args)</p></td>
+<td><p>This class represents the element wise quadrature points and weights in a domain.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalTensorQuadratureTestBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalTensorQuadratureTrialBasis</span></code>(target[, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalThreadEnds</span></code>([index])</p></td>
+<td><p>This represents the local threads ends over the decomposed domain</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LocalThreadStarts</span></code>([index])</p></td>
+<td><p>This represents the local threads starts over the decomposed domain</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalBasisValue</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalGeometryValue</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogicalValueNode</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Loop</span></code>(iterable, index, *[, stmts, mask, ...])</p></td>
+<td><p>class to describe a dimensionless loop of an iterator over a generator.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MappingEvaluation</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Mask</span></code>(axis, ext)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixCoordsFromRank</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixGlobalBasis</span></code>(target, test[, dtype])</p></td>
+<td><p>used to describe global dof</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixLocalBasis</span></code>(target[, dtype])</p></td>
+<td><p>used to describe local dof over an element</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixQuadrature</span></code>(target[, dtype])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MatrixRankFromCoords</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Max</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Min</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MulNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NeighbourThreadCoordinates</span></code>([start, stop, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NotNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NumThreads</span></code>([target, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Pads</span></code>(tests[, trials, tests_degree, ...])</p></td>
+<td><p>This represents the global pads</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ParallelBlock</span></code>([default, private, shared, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Pattern</span></code>(*args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PhysicalBasisValue</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PhysicalGeometryValue</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PhysicalValueNode</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PlusGlobalTensorQuadratureGrid</span></code>(*args)</p></td>
+<td><p>This class represents the quadrature points and weights in the plus side of an interface.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PlusLocalTensorQuadratureGrid</span></code>(*args)</p></td>
+<td><p>This class represents the element wise quadrature points and weights in the plus side of an interface.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PlusTensorQuadrature</span></code>(*args)</p></td>
+<td><p>This class represents the quadrature point and weight in the plus side of an interface.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ProductGenerator</span></code>(target, dummies)</p></td>
+<td><p>This class represent an element of an array with arbitrary number of dimensions.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ProductIteration</span></code>(iterator, generator)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ProductIterator</span></code>(target[, dummies])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RAT</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RankDimension</span></code>()</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">RankNode</span></code>()</p></td>
+<td><p>Base class representing a rank of an iterator</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Reduce</span></code>(op, rhs, lhs, loop)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Reduction</span></code>(op, expr[, lhs])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Reset</span></code>(var[, expr])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ScalarLocalBasis</span></code>([u, v, tag, dtype])</p></td>
+<td><p>This is used to describe scalar dof over an element</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ScalarNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Span</span></code>(target[, index])</p></td>
+<td><p>This represents the span of a basis in an element</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SpanArray</span></code>(target[, index])</p></td>
+<td><p>This represents the global span array</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SplitArray</span></code>(target, positions, lengths)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilMatrixGlobalBasis</span></code>(u, v, pads[, tag, ...])</p></td>
+<td><p>used to describe local dof over an element as a stencil matrix</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilMatrixLocalBasis</span></code>(u, v, pads[, tag, dtype])</p></td>
+<td><p>used to describe local dof over an element as a stencil matrix</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilVectorGlobalBasis</span></code>(v, pads[, tag, dtype])</p></td>
+<td><p>used to describe local dof over an element as a stencil vector</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilVectorLocalBasis</span></code>(v, pads[, tag, dtype])</p></td>
+<td><p>used to describe local dof over an element as a stencil vector</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StrictLessThanNode</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorAdd</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorAssignExpr</span></code>(lhs, rhs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorExpression</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorGenerator</span></code>(target, dummies)</p></td>
+<td><p>This class represent an array list of array elements with arbitrary number of dimensions.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorIntDiv</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorInteger</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorIteration</span></code>(iterator, generator)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorIterator</span></code>(target[, dummies])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorMax</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorMul</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorQuadrature</span></code>(*args)</p></td>
+<td><p>This class represents the quadrature point and weight in a domain.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorQuadratureBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorQuadratureTestBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorQuadratureTrialBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorTestBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorTrialBasis</span></code>(target)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ThreadCoordinates</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ThreadId</span></code>([start, stop, length, index])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ValueNode</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorAssign</span></code>(lhs, rhs[, op])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">WeightedVolumeQuadrature</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">WhileLoop</span></code>(condition, body)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Zeros</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ZerosLike</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
+    <script type="text/x-thebe-config">
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new file mode 100644
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+                <article class="bd-article">
+                  
+  <section id="module-psydac.api.ast.parser">
+<span id="psydac-api-ast-parser"></span><h1>psydac.api.ast.parser<a class="headerlink" href="#module-psydac.api.ast.parser" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_scalar_array</span></code>(var)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">parse</span></code>(expr, settings[, backend])</p></td>
+<td><p>This function takes a Psydac Ast and returns a Pyccel Ast</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_string</span></code>(n)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Parser</span></code>(settings[, backend])</p></td>
+<td><p>This class takes a Psyadac Ast and transforms it to a Pyccel Ast by calling the Parser.doit method</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Shape</span></code>(*args)</p></td>
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+  <section id="module-psydac.api.ast.utilities">
+<span id="psydac-api-ast-utilities"></span><h1>psydac.api.ast.utilities<a class="headerlink" href="#module-psydac.api.ast.utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">build_pyccel_type_annotations</span></code>(args[, order])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">build_pythran_types_header</span></code>(name, args[, order])</p></td>
+<td><p>builds a types decorator from a list of arguments (of FunctionDef)</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_atoms_expr</span></code>(atomic_exprs, ...)</p></td>
+<td><p>This function computes atomic expressions needed to evaluate  the Kernel final expression</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_atoms_expr_field</span></code>(atomic_exprs, ...)</p></td>
+<td><p>This function computes atomic expressions needed to evaluate  EvaluteField/VectorField final expression</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_atoms_expr_mapping</span></code>(atomic_exprs, ...)</p></td>
+<td><p>This function computes atomic expressions needed to evaluate  EvalMapping final expression</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_boundary_jacobian</span></code>(parent_namespace, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_normal_vector</span></code>(parent_namespace, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_tangent_vector</span></code>(parent_namespace, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">filter_loops</span></code>(indices, ranges, body, boundary)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">filter_product</span></code>(indices, args, boundary)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">fusion_loops</span></code>(loops)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_name</span></code>(lhs)</p></td>
+<td><p>Given a list of variable return the meaningful part of the name of the first variable that has a _name attribute.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_mapping</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">logical2physical</span></code>(expr)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">math_atoms_as_str</span></code>(expr[, lib])</p></td>
+<td><p>Given a Sympy expression, find all known mathematical atoms (functions and constants) that need to be imported from a math library (e.g. Numpy) when generating Python code.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_string</span></code>(n)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">rationalize_eval_mapping</span></code>(mapping, nderiv, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">select_loops</span></code>(indices, ranges, body, boundary)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">variables</span></code>(names, dtype, **args)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../../../chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="../../../../../chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../../../../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../../../../../chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../../chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../../chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../../../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.api.basic">
+<span id="psydac-api-basic"></span><h1>psydac.api.basic<a class="headerlink" href="#module-psydac.api.basic" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicCodeGen</span></code>(expr, *[, folder, comm, root, ...])</p></td>
+<td><p>Basic class for any discrete concept that needs code generation.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasicDiscrete</span></code>(expr, kernel_expr, *[, ...])</p></td>
+<td><p>mapping is the symbolic mapping here.</p></td>
+</tr>
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+</table>
+</section>
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+  <section id="module-psydac.api.discretization">
+<span id="psydac-api-discretization"></span><h1>psydac.api.discretization<a class="headerlink" href="#module-psydac.api.discretization" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">change_dtype</span></code>(V, dtype)</p></td>
+<td><p>Given a FemSpace V, change its underlying vector_space (i.e. the space of its coefficients) so that it matches the required data type.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize</span></code>(a, *args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize_derham</span></code>(derham, domain_h[, ...])</p></td>
+<td><p>Create a discrete De Rham sequence from a symbolic one.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize_domain</span></code>(domain, *[, filename, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize_space</span></code>(V, domain_h, *[, degree, ...])</p></td>
+<td><p>This function creates the discretized space starting from the symbolic space.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">reduce_space_degrees</span></code>(V, Vh, *[, basis, sequence])</p></td>
+<td><p>This function takes a tensor FEM space Vh and reduces some degrees in order to obtain a tensor FEM space Wh that matches the symbolic space V in a certain sequence of spaces.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.api.equation">
+<span id="psydac-api-equation"></span><h1>psydac.api.equation<a class="headerlink" href="#module-psydac.api.equation" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">l2_boundary_projection</span></code>(equation)</p></td>
+<td><p>Create an auxiliary equation (weak formulation) that solves for the boundary trace of the solution by performing an L2 projection of inhomogeneous essential boundary conditions.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteEquation</span></code>(expr, *args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearSystem</span></code>(lhs, rhs)</p></td>
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+  <section id="module-psydac.api.essential_bc">
+<span id="psydac-api-essential-bc"></span><h1>psydac.api.essential_bc<a class="headerlink" href="#module-psydac.api.essential_bc" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">apply_essential_bc</span></code>(a, *bcs, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">apply_essential_bc_BlockLinearOperator</span></code>(a, bc, *)</p></td>
+<td><p>Apply homogeneous dirichlet boundary conditions in nD.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">apply_essential_bc_BlockVector</span></code>(a, bc, *[, ...])</p></td>
+<td><p>Apply homogeneous dirichlet boundary conditions in nD.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">apply_essential_bc_kronecker_dense_matrix</span></code>(a, ...)</p></td>
+<td><p>This function applies the homogeneous boundary condition to the Kronecker product matrix objects,</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">apply_essential_bc_stencil</span></code>(a, *, axis, ext, ...)</p></td>
+<td><p>This function applies the homogeneous boundary condition to the Stencil objects,</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">check_boundary_type</span></code>(bc)</p></td>
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+                  
+  <section id="module-psydac.api.fem">
+<span id="psydac-api-fem"></span><h1>psydac.api.fem<a class="headerlink" href="#module-psydac.api.fem" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">collect_spaces</span></code>(space, *args)</p></td>
+<td><p>This function collect the arguments used in the assembly function</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_diag_len</span></code>(p, md, mc)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_quad_grids_arguments</span></code>(grid[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_test_space_arguments</span></code>(basis_values)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_trial_space_arguments</span></code>(basis_values)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">do_nothing</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">extract_stencil_mats</span></code>(mats)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_nquads</span></code>(Vh)</p></td>
+<td><p>This function is to be deleted in the future, as we intend to make the number of quadrature points a property of DiscreteLinearForms, DiscreteBilinearForms and DiscreteFunctionals rather than a property of spaces.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">reset_arrays</span></code>(*args)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteBilinearForm</span></code>(expr, kernel_expr, ...)</p></td>
+<td><p>Class that represents the concept of a discrete bi-linear form.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteFunctional</span></code>(expr, kernel_expr, ...[, ...])</p></td>
+<td><p>Class that represents the concept of a discrete functional form.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteLinearForm</span></code>(expr, kernel_expr, ...[, ...])</p></td>
+<td><p>Class that represents the concept of a discrete linear form.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteSesquilinearForm</span></code>(expr, kernel_expr, ...)</p></td>
+<td><p>Class that represents the concept of a discrete sesqui-linear form with the antilinearity on the first variable.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteSumForm</span></code>(a, kernel_expr, *args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteGltExpr</span></code>(expr, *args, **kwargs)</p></td>
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+  <section id="module-psydac.api.grid">
+<span id="psydac-api-grid"></span><h1>psydac.api.grid<a class="headerlink" href="#module-psydac.api.grid" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">create_collocation_basis</span></code>(glob_points, space)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_points_weights</span></code>(spaces, axis, e)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BasisValues</span></code>(V, nderiv[, trial, grid])</p></td>
+<td><p>Compute the basis values and the spans.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CollocationBasisValues</span></code>(points, V, nderiv)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">QuadratureGrid</span></code>(V[, axis, ext, trial_space])</p></td>
+<td><p></p></td>
+</tr>
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+    <h1>psydac.api.postprocessing</h1>
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+                  
+  <section id="module-psydac.api.postprocessing">
+<span id="psydac-api-postprocessing"></span><h1>psydac.api.postprocessing<a class="headerlink" href="#module-psydac.api.postprocessing" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_grid_lines_2d</span></code>(domain_h, V_h, *[, refine])</p></td>
+<td><p>Get the grid lines (i.e. element boundaries) of a 2D computational domain, which can be easily plotted with Matplotlib.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">OutputManager</span></code>(filename_space, filename_fields)</p></td>
+<td><p>A class meant to streamline the exportation of output data.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PostProcessManager</span></code>([geometry_file, domain, ...])</p></td>
+<td><p>A class to read saved information of a previous simulation and start post-processing from there.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<p class="rubric">Functions</p>
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+<td><p>Converts an expr to a string of Python code</p></td>
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+<span id="psydac-cad-cad"></span><h1>psydac.cad.cad<a class="headerlink" href="#module-psydac.cad.cad" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elevate</span></code>(mapping, axis, times)</p></td>
+<td><p>Elevate the mapping degree times time in the direction axis.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_string</span></code>(n)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">refine</span></code>(mapping, axis, values)</p></td>
+<td><p>Refine the mapping by inserting values in the direction axis.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">translate</span></code>(mapping, displ)</p></td>
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+  <section id="module-psydac.cad.gallery">
+<span id="psydac-cad-gallery"></span><h1>psydac.cad.gallery<a class="headerlink" href="#module-psydac.cad.gallery" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">annulus</span></code>([rmin, rmax, center])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">circle</span></code>([radius, center])</p></td>
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+  <section id="module-psydac.cad.geometry">
+<span id="psydac-cad-geometry"></span><h1>psydac.cad.geometry<a class="headerlink" href="#module-psydac.cad.geometry" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">export_nurbs_to_hdf5</span></code>(filename, nurbs[, ...])</p></td>
+<td><p>Export a single-patch igakit NURBS object to a Psydac geometry file in HDF5 format</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">import_geopdes_to_nurbs</span></code>(filename)</p></td>
+<td><p>This function reads a geopdes geometry file and convert it to igakit nurbs object</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">refine_knots</span></code>(knots, ncells, degree[, ...])</p></td>
+<td><p>This function refines the knot sequence.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">refine_nurbs</span></code>(nrb[, ncells, degree, ...])</p></td>
+<td><p>This function refines the nurbs object.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Geometry</span></code>([domain, ncells, periodic, ...])</p></td>
+<td><p>Distributed discrete geometry that works for single and multiple patches.</p></td>
+</tr>
+</tbody>
+</table>
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+  <section id="module-psydac.cad.multipatch">
+<span id="psydac-cad-multipatch"></span><h1>psydac.cad.multipatch<a class="headerlink" href="#module-psydac.cad.multipatch" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">export_multipatch_nurbs_to_hdf5</span></code>(filename, ...)</p></td>
+<td><p>Create the multipatch geometry file using one Igakit NURBS for each patch with the connectivity between the patches.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">export_analytical_mapping</span></code>(mapping, ncells, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">export_caid_geometry</span></code>(name, ncells, degree, ...)</p></td>
+<td><p></p></td>
+</tr>
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+    <h1>psydac.core.bsplines</h1>
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+  <section id="module-psydac.core.bsplines">
+<span id="psydac-core-bsplines"></span><h1>psydac.core.bsplines<a class="headerlink" href="#module-psydac.core.bsplines" title="Link to this heading">#</a></h1>
+<p>Basic module that provides the means for evaluating the B-Splines basis
+functions and their derivatives. In order to simplify automatic Fortran code
+generation with Pyccel, no object-oriented features are employed.</p>
+<p>References:</p>
+<blockquote>
+<div><ul class="simple">
+<li><p>[1] L. Piegl and W. Tiller. The NURBS Book, 2nd ed., Springer-Verlag Berlin Heidelberg GmbH, 1997.</p></li>
+<li><p>[2] SELALIB, Semi-Lagrangian Library. <a class="reference external" href="http://selalib.gforge.inria.fr">http://selalib.gforge.inria.fr</a></p></li>
+</ul>
+</div></blockquote>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_ders_on_irregular_grid</span></code>(knots, degree, ...)</p></td>
+<td><p>Evaluate B-Splines and their derivatives on an irregular_grid.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_ders_on_quad_grid</span></code>(knots, degree, ...)</p></td>
+<td><p>Evaluate B-Splines and their derivatives on the quadrature grid.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs</span></code>(knots, degree, x, span[, out])</p></td>
+<td><p>Compute the non-vanishing B-splines at a unique location.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_1st_der</span></code>(knots, degree, x, span[, out])</p></td>
+<td><p>Compute the first derivative of the non-vanishing B-splines at a location.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_all_ders</span></code>(knots, degree, x, span, n)</p></td>
+<td><p>Evaluate value and n derivatives at x of all basis functions with support in interval <span class="math notranslate nohighlight">\([x_{span-1}, x_{span}]\)</span>.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_array</span></code>(knots, degree, span, x[, out])</p></td>
+<td><p>Compute the non-vanishing B-splines at several locations.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_integrals</span></code>(knots, degree[, out])</p></td>
+<td><p>Return the integral of each B-spline basis function over the real line:</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">breakpoints</span></code>(knots, degree[, tol, out])</p></td>
+<td><p>Determine breakpoints' coordinates.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">cell_index</span></code>(breaks, i_grid[, tol, out])</p></td>
+<td><p>Computes in which cells a given array of locations belong.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">collocation_matrix</span></code>(knots, degree, periodic, ...)</p></td>
+<td><p>Computes the collocation matrix</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elements_spans</span></code>(knots, degree[, out])</p></td>
+<td><p>Compute the index of the last non-vanishing spline on each grid element (cell).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elevate_knots</span></code>(knots, degree, periodic[, ...])</p></td>
+<td><p>Given the knot sequence of a spline space S of degree p, compute the knot sequence of a spline space S_0 of degree p+1 such that u' is in S for all u in S_0.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_span</span></code>(knots, degree, x)</p></td>
+<td><p>Determine the knot span index at location x, given the B-Splines' knot sequence and polynomial degree.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_spans</span></code>(knots, degree, x[, out])</p></td>
+<td><p>Determine the knot span index at a set of locations x, given the B-Splines' knot sequence and polynomial degree.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">greville</span></code>(knots, degree, periodic[, out, ...])</p></td>
+<td><p>Compute coordinates of all Greville points.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">histopolation_matrix</span></code>(knots, degree, ...[, ...])</p></td>
+<td><p>Computes the histopolation matrix.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">hrefinement_matrix</span></code>(ts, p, knots)</p></td>
+<td><p>Computes the refinement matrix corresponding to the insertion of a given list of knots.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">make_knots</span></code>(breaks, degree, periodic[, ...])</p></td>
+<td><p>Create spline knots from breakpoints, with appropriate boundary conditions.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">quadrature_grid</span></code>(breaks, quad_rule_x, quad_rule_w)</p></td>
+<td><p>Compute the quadrature points and weights for performing integrals over each element (interval) of the 1D domain, given a certain Gaussian quadrature rule.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<ul class="nav bd-sidenav">
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+  <section id="module-psydac.core.bsplines_kernels">
+<span id="psydac-core-bsplines-kernels"></span><h1>psydac.core.bsplines_kernels<a class="headerlink" href="#module-psydac.core.bsplines_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_ders_on_irregular_grid_p</span></code></p></td>
+<td><p>Evaluate B-Splines and their derivatives on an irregular_grid.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_ders_on_quad_grid_p</span></code></p></td>
+<td><p>Evaluate B-Splines and their derivatives on the quadrature grid.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_1st_der_p</span></code></p></td>
+<td><p>Compute the first derivative of the non-vanishing B-splines at a location.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_all_ders_p</span></code></p></td>
+<td><p>Evaluate value and n derivatives at x of all basis functions with support in interval <span class="math notranslate nohighlight">\([x_{span-1}, x_{span}]\)</span>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_array_p</span></code></p></td>
+<td><p>Compute the non-vanishing B-splines at locations in x, given the knot sequence, polynomial degree and knot span.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_funs_p</span></code></p></td>
+<td><p>Compute the non-vanishing B-splines at a unique location.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">basis_integrals_p</span></code></p></td>
+<td><p>Return the integral of each B-spline basis function over the real line:</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">breakpoints_p</span></code></p></td>
+<td><p>Determine breakpoints' coordinates.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">cell_index_p</span></code></p></td>
+<td><p>Computes in which cells a given sorted array of locations belong.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">collocation_matrix_p</span></code></p></td>
+<td><p>Compute the collocation matrix <span class="math notranslate nohighlight">\(C_ij = B_j(x_i)\)</span>, which contains the values of each B-spline basis function <span class="math notranslate nohighlight">\(B_j\)</span> at all locations <span class="math notranslate nohighlight">\(x_i\)</span>.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elements_spans_p</span></code></p></td>
+<td><p>Compute the index of the last non-vanishing spline on each grid element (cell).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">elevate_knots_p</span></code></p></td>
+<td><p>Given the knot sequence of a spline space S of degree p, compute the knot sequence of a spline space S_0 of degree p+1 such that u' is in S for all u in S_0.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_span_p</span></code></p></td>
+<td><p>Determine the knot span index at location x, given the B-Splines' knot sequence and polynomial degree.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_spans_p</span></code></p></td>
+<td><p>Determine the knot span index at a set of locations x, given the B-Splines' knot sequence and polynomial degree.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">greville_p</span></code></p></td>
+<td><p>Compute coordinates of all Greville points.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">histopolation_matrix_p</span></code></p></td>
+<td><p>Computes the histopolation matrix.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">make_knots_p</span></code></p></td>
+<td><p>Create spline knots from breakpoints, with appropriate boundary conditions.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">merge_sort</span></code></p></td>
+<td><p>Performs a 'in place' merge sort of the input list</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">quadrature_grid_p</span></code></p></td>
+<td><p>Compute the quadrature points and weights for performing integrals over each element (interval) of the 1D domain, given a certain Gaussian quadrature rule.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+  <section id="module-psydac.core.field_evaluation_kernels">
+<span id="psydac-core-field-evaluation-kernels"></span><h1>psydac.core.field_evaluation_kernels<a class="headerlink" href="#module-psydac.core.field_evaluation_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_2d_irregular_no_weights</span></code></p></td>
+<td><p>Parameters
+
+np1: int
+
+Number of points in the X1 direction
+
+np2: int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_2d_irregular_weighted</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_2d_no_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_2d_weighted</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_3d_irregular_no_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_3d_irregular_weighted</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_3d_no_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_fields_3d_weighted</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_2d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_3d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_irregular_2d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_irregular_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_irregular_3d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jac_det_irregular_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_2d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_3d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_2d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_3d</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+nc1: int
+
+Number of cells in the X1 direction
+
+nc2: int
+
+Number of cells in the X2 direction
+
+nc3: int
+
+Number of cells in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_irregular_2d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_irregular_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_irregular_3d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_inv_irregular_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_irregular_2d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_irregular_2d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_irregular_3d</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">eval_jacobians_irregular_3d_weights</span></code></p></td>
+<td><p>Parameters
+
+np1
+
+int
+
+Number of points in the X1 direction
+
+np2
+
+int
+
+Number of points in the X2 direction
+
+np3
+
+int
+
+Number of points in the X3 direction</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_2d_hcurl</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (2, n_x1, n_x2, n_f) where:
+* 2 is the logical dimension of the problem (2 here)
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_f is the number of fields to push-forward in the Hcurl space.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_2d_hdiv</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (2, n_x1, n_x2, n_f) where:
+* 2 is the logical dimension of the problem (2 here)
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_f is the number of fields to push-forward in the Hdiv space.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_2d_l2</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (n_x1, n_x2, n_f) where:
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_f is the number of fields to push-forward in the L2 space.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_3d_hcurl</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (3, n_x1, n_x2, n_x3, n_f) where:
+* 3 is the logical dimension of the problem
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_x3 is the number of points in direction 3 of the implicit grid
+* n_f is the number of fields to push-forward in the Hcurl space.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_3d_hdiv</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (3, n_x1, n_x2, n_x3, n_f) where:
+* 3 is the logical dimension of the problem
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_x3 is the number of points in direction 3 of the implicit grid
+* n_f is the number of fields to push-forward in the Hdiv space.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pushforward_3d_l2</span></code></p></td>
+<td><p>Parameters
+
+fields_to_push: ndarray
+
+Field values to push forward on the mapping
+This array as shape (n_x1, n_x2, n_x3, n_f) where:
+* n_x1 is the number of points in direction 1 of the implicit grid.
+* n_x2 is the number of points in direction 2 of the implicit grid.
+* n_x3 is the number of points in direction 3 of the implicit grid.
+* n_f is the number of fields to push-forward in the L2 space.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+  <section id="module-psydac.ddm.blocking_data_exchanger">
+<span id="psydac-ddm-blocking-data-exchanger"></span><h1>psydac.ddm.blocking_data_exchanger<a class="headerlink" href="#module-psydac.ddm.blocking_data_exchanger" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockingCartDataExchanger</span></code>(cart, dtype, *[, ...])</p></td>
+<td><p>Type that takes care of updating the ghost regions (padding) of a multi-dimensional array distributed according to the given Cartesian decomposition of a tensor-product grid of coefficients.</p></td>
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+                <article class="bd-article">
+                  
+  <section id="module-psydac.ddm.cart">
+<span id="psydac-ddm-cart"></span><h1>psydac.ddm.cart<a class="headerlink" href="#module-psydac.ddm.cart" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">create_interfaces_cart</span></code>(domain_decomposition, ...)</p></td>
+<td><p>This function Connects the Cartesian grids when they share an interface.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">find_mpi_type</span></code>(dtype)</p></td>
+<td><p>Find correct MPI datatype that corresponds to user-provided datatype.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">CartDecomposition</span></code>(domain_decomposition, ...)</p></td>
+<td><p>Cartesian decomposition of a tensor-product grid of spline coefficients.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DomainDecomposition</span></code>(ncells, periods[, comm, ...])</p></td>
+<td><p>Cartesian decomposition of an N-Cube grid.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InterfaceCartDecomposition</span></code>(cart_minus, ...)</p></td>
+<td><p>The Cartesian decomposition of an interface constructed from the Cartesian decomposition of the patches that shares an interface.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MultiPatchDomainDecomposition</span></code>(ncells, periods)</p></td>
+<td><p>Cartesian decomposition of multiple N-Cube grids.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.ddm.interface_data_exchanger">
+<span id="psydac-ddm-interface-data-exchanger"></span><h1>psydac.ddm.interface_data_exchanger<a class="headerlink" href="#module-psydac.ddm.interface_data_exchanger" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InterfaceCartDataExchanger</span></code>(cart, dtype, *[, ...])</p></td>
+<td><p>This takes care of updating the ghost regions between two sides of an interface for a multi-dimensional array distributed according to the given Cartesian decomposition of a tensor-product grid of coefficients.</p></td>
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+  <section id="module-psydac.ddm.nonblocking_data_exchanger">
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+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NonBlockingCartDataExchanger</span></code>(cart, dtype, *)</p></td>
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+  <section id="module-psydac.ddm.partition">
+<span id="psydac-ddm-partition"></span><h1>psydac.ddm.partition<a class="headerlink" href="#module-psydac.ddm.partition" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_dims</span></code>(nnodes, gridsizes[, ...])</p></td>
+<td><p>With the aim of distributing a multi-dimensional array on a Cartesian topology, compute the number of processes along each dimension.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_dims_general</span></code>(mpi_size, npts[, ...])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_dims_uniform</span></code>(mpi_size, npts)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">partition_procs_per_patch</span></code>(npts, size)</p></td>
+<td><p>Compute the number of processes in each patch and assign to it an ascending range of processes.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.ddm.utilities">
+<span id="psydac-ddm-utilities"></span><h1>psydac.ddm.utilities<a class="headerlink" href="#module-psydac.ddm.utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_data_exchanger</span></code>(cart, dtype, *[, ...])</p></td>
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+    <h1>psydac.feec.derivatives</h1>
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+                  
+  <section id="module-psydac.feec.derivatives">
+<span id="psydac-feec-derivatives"></span><h1>psydac.feec.derivatives<a class="headerlink" href="#module-psydac.feec.derivatives" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">block_tostencil</span></code>(M)</p></td>
+<td><p>Convert a BlockLinearOperator that contains KroneckerStencilMatrix objects to a BlockLinearOperator that contains StencilMatrix objects</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Curl_3D</span></code>(Hcurl, Hdiv)</p></td>
+<td><p>Curl operator in 3D.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Derivative_1D</span></code>(H1, L2)</p></td>
+<td><p>1D derivative.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiffOperator</span></code>(domain, codomain, matrix)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DirectionalDerivativeOperator</span></code>(V, W, diffdir, *)</p></td>
+<td><p>Represents a matrix-free derivative operator in a specific cardinal direction.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Divergence_2D</span></code>(Hdiv, L2)</p></td>
+<td><p>Divergence operator in 2D.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Divergence_3D</span></code>(Hdiv, L2)</p></td>
+<td><p>Divergence operator in 3D.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Gradient_2D</span></code>(H1, Hcurl)</p></td>
+<td><p>Gradient operator in 2D.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Gradient_3D</span></code>(H1, Hcurl)</p></td>
+<td><p>Gradient operator in 3D.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ScalarCurl_2D</span></code>(Hcurl, L2)</p></td>
+<td><p>Scalar curl operator in 2D: computes a scalar field from a vector field.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorCurl_2D</span></code>(H1, Hdiv)</p></td>
+<td><p>Vector curl operator in 2D: computes a vector field from a scalar field.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.feec.global_projectors">
+<span id="psydac-feec-global-projectors"></span><h1>psydac.feec.global_projectors<a class="headerlink" href="#module-psydac.feec.global_projectors" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_1d_0form</span></code>(intp_x1, F, f)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_1d_1form</span></code>(quad_x1, quad_w1, F, f)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_2d_0form</span></code>(intp_x1, intp_x2, F, f)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_2d_1form_hcurl</span></code>(intp_x1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_2d_1form_hdiv</span></code>(intp_x1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_2d_2form</span></code>(quad_x1, quad_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_2d_vec</span></code>(intp_x1, intp_x2, F1, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_3d_0form</span></code>(intp_x1, intp_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_3d_1form</span></code>(intp_x1, intp_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_3d_2form</span></code>(intp_x1, intp_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_3d_3form</span></code>(quad_x1, quad_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">evaluate_dofs_3d_vec</span></code>(intp_x1, intp_x2, ...)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GlobalProjector</span></code>(space[, nquads])</p></td>
+<td><p>A global projector to some TensorFemSpace or VectorFemSpace object.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector_H1</span></code>(space[, nquads])</p></td>
+<td><p>Projector from H1 to an H1-conforming finite element space (i.e. a finite dimensional subspace of H1) constructed with tensor-product B-splines in 1, 2 or 3 dimensions.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector_H1vec</span></code>(space[, nquads])</p></td>
+<td><p>Projector from H1^3 = H1 x H1 x H1 to a conforming finite element space, i.e. a finite dimensional subspace of H1^3, constructed with tensor-product B-splines in 2 or 3 dimensions.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector_Hcurl</span></code>(space[, nquads])</p></td>
+<td><p>Projector from H(curl) to an H(curl)-conforming finite element space, i.e. a finite dimensional subspace of H(curl), constructed with tensor-product B- and M-splines in 2 or 3 dimensions.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector_Hdiv</span></code>(space[, nquads])</p></td>
+<td><p>Projector from H(div) to an H(div)-conforming finite element space, i.e. a finite dimensional subspace of H(div), constructed with tensor-product B- and M-splines in 2 or 3 dimensions.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Projector_L2</span></code>(space[, nquads])</p></td>
+<td><p>Projector from L2 to an L2-conforming finite element space (i.e. a finite dimensional subspace of L2) constructed with tensor-product M-splines in 1, 2 or 3 dimensions.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+  <section id="module-psydac.feec.multipatch.api">
+<span id="psydac-feec-multipatch-api"></span><h1>psydac.feec.multipatch.api<a class="headerlink" href="#module-psydac.feec.multipatch.api" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize</span></code>(expr, *args, **kwargs)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">discretize_derham_multipatch</span></code>(derham, ...)</p></td>
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+</tbody>
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+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DiscreteDerhamMultipatch</span></code>(*, mapping, ...[, ...])</p></td>
+<td><p>Represents the discrete De Rham sequence for multipatch domains.</p></td>
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+                  
+  <section id="module-psydac.feec.multipatch.fem_linear_operators">
+<span id="psydac-feec-multipatch-fem-linear-operators"></span><h1>psydac.feec.multipatch.fem_linear_operators<a class="headerlink" href="#module-psydac.feec.multipatch.fem_linear_operators" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComposedLinearOperator</span></code>(operators)</p></td>
+<td><p>operator L = L_1 .</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FemLinearOperator</span></code>([fem_domain, ...])</p></td>
+<td><p>Linear operators with an additional Fem layer</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IdLinearOperator</span></code>(V)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MultLinearOperator</span></code>(c, A)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SumLinearOperator</span></code>(B, A)</p></td>
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+  <section id="module-psydac.feec.multipatch.operators">
+<span id="psydac-feec-multipatch-operators"></span><h1>psydac.feec.multipatch.operators<a class="headerlink" href="#module-psydac.feec.multipatch.operators" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">allocate_interface_matrix</span></code>(corners, ...)</p></td>
+<td><p>Allocate the interface matrix for a vertex shared by two patches</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_K0_and_K0_inv</span></code>(V0h[, uniform_patches])</p></td>
+<td><p>Compute the change of basis matrices K0 and K0^{-1} in V0h.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_K1_and_K1_inv</span></code>(V1h[, uniform_patches])</p></td>
+<td><p>Compute the change of basis matrices K1 and K1^{-1} in Hcurl space V1h.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_interface_from_corners</span></code>(corner1, corner2, ...)</p></td>
+<td><p>Return the interface between two corners from two different patches that correspond to a single (physical) vertex.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_patch_index_from_face</span></code>(domain, face)</p></td>
+<td><p>Return the patch index of subdomain/boundary</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_row_col_index</span></code>(corner1, corner2, ...)</p></td>
+<td><p>Return the row and column index of a corner in the StencilInterfaceMatrix</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ortho_proj_Hcurl</span></code>(EE, V1h, domain_h, M1[, ...])</p></td>
+<td><p>return orthogonal projection of E on V1h, given M1 the mass matrix</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BrokenGradient_2D</span></code>(V0h, V1h)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BrokenScalarCurl_2D</span></code>(V1h, V2h)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BrokenTransposedGradient_2D</span></code>(V0h, V1h)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BrokenTransposedScalarCurl_2D</span></code>(V1h, V2h)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ConformingProjection_V0</span></code>(V0h, domain_h[, ...])</p></td>
+<td><p>Conforming projection from global broken V0 space to conforming global V0 space Defined by averaging of interface dofs</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ConformingProjection_V1</span></code>(V1h, domain_h[, ...])</p></td>
+<td><p>Conforming projection from global broken V1 space to conforming V1 global space</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">HodgeOperator</span></code>(Vh, domain_h[, ...])</p></td>
+<td><p>Change of basis operator: dual basis -&gt; primal basis</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Multipatch_Projector_H1</span></code>(V0h)</p></td>
+<td><p>to apply the H1 projection (2D) on every patch</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Multipatch_Projector_Hcurl</span></code>(V1h[, nquads])</p></td>
+<td><p>to apply the Hcurl projection (2D) on every patch</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Multipatch_Projector_L2</span></code>(V2h[, nquads])</p></td>
+<td><p>to apply the L2 projection (2D) on every patch</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.feec.multipatch.utilities">
+<span id="psydac-feec-multipatch-utilities"></span><h1>psydac.feec.multipatch.utilities<a class="headerlink" href="#module-psydac.feec.multipatch.utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">error_fn</span></code>([source_type, method, conf_proj, ...])</p></td>
+<td><p>Get the error filename based on the method used to solve the multpatch problem</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_fem_name</span></code>([method, k, DG_full, ...])</p></td>
+<td><p>Get Fem name used to solve the multipatch problem</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_load_dir</span></code>([method, DG_full, domain_name, ...])</p></td>
+<td><p>get load directory name based on the fem name</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_method_name</span></code>([method, k, conf_proj, ...])</p></td>
+<td><p>Get method name used to solve the multpatch problem</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">sol_ref_fn</span></code>(source_type, N_diag[, source_proj])</p></td>
+<td><p>Get the reference solution filename based on the source term type</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">source_name</span></code>([source_type, source_proj])</p></td>
+<td><p>Get the source term name</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">time_count</span></code>([t_stamp, msg])</p></td>
+<td><p></p></td>
+</tr>
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+</table>
+</section>
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+    <!-- Table of contents -->
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+  <section id="module-psydac.feec.pull_push">
+<span id="psydac-feec-pull-push"></span><h1>psydac.feec.pull_push<a class="headerlink" href="#module-psydac.feec.pull_push" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_1d_h1</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_1d_l2</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_2d_h1</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">pull_2d_h1vec</span></code>(f, F)</p></td>
+<td><p></p></td>
+</tr>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_2d_hdiv</span></code>(f1, f2, eta1, eta2, F)</p></td>
+<td><p></p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_2d_l2</span></code>(f, eta1, eta2, F)</p></td>
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+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_3d_hcurl</span></code>(f1, f2, f3, eta1, eta2, eta3, F)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_3d_hdiv</span></code>(f1, f2, f3, eta1, eta2, eta3, F)</p></td>
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+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">push_3d_l2</span></code>(f, eta1, eta2, eta3, F)</p></td>
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+  <section id="module-psydac.feec.pushforward">
+<span id="psydac-feec-pushforward"></span><h1>psydac.feec.pushforward<a class="headerlink" href="#module-psydac.feec.pushforward" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Pushforward</span></code>(grid[, mapping, npts_per_cell, ...])</p></td>
+<td><p>Class used to help push-forwarding several fields using the same mapping.</p></td>
+</tr>
+</tbody>
+</table>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<li class="toctree-l3"><a class="reference internal" href="../../../../../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../../../../../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
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+  <section id="module-psydac.fem.basic">
+<span id="psydac-fem-basic"></span><h1>psydac.fem.basic<a class="headerlink" href="#module-psydac.fem.basic" title="Link to this heading">#</a></h1>
+<p>In order to avoid multiple inheritence, we define the base objects for Finite
+Elements as abstract classes that contain a topological member. This member can
+be used to specify the used data structure for example.</p>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FemField</span></code>(space[, coeffs])</p></td>
+<td><p>Element of a finite element space V.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FemSpace</span></code>()</p></td>
+<td><p>Generic Finite Element space V.</p></td>
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+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">fem_context</span></code>(filename[, comm])</p></td>
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+  <section id="module-psydac.fem.grid">
+<span id="psydac-fem-grid"></span><h1>psydac.fem.grid<a class="headerlink" href="#module-psydac.fem.grid" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">FemAssemblyGrid</span></code>(space, start, end, *[, ...])</p></td>
+<td><p>Class that collects all 1D information local to process that are necessary for the correct assembly of l.h.s.</p></td>
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+  <section id="module-psydac.fem.partitioning">
+<span id="psydac-fem-partitioning"></span><h1>psydac.fem.partitioning<a class="headerlink" href="#module-psydac.fem.partitioning" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_connectivity</span></code>(domain)</p></td>
+<td><p>Compute the connectivity of the multipatch domain.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_interface_spaces</span></code>(...)</p></td>
+<td><p>Create the fem spaces for each interface in the domain given by the connectivity.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">construct_reduced_interface_spaces</span></code>(spaces, ...)</p></td>
+<td><p>Create the reduced spaces for the interface coefficients.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">create_cart</span></code>(domain_decomposition, spaces)</p></td>
+<td><p>Compute the cartesian decomposition of the coefficient space.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_minus_starts_ends</span></code>(plus_starts, ...)</p></td>
+<td><p>Compute the coefficients needed by the minus patch in a given interface.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">get_plus_starts_ends</span></code>(minus_starts, ...)</p></td>
+<td><p>Compute the coefficients needed by the plus patch in a given interface.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">partition_coefficients</span></code>(domain_decomposition, ...)</p></td>
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+  <section id="module-psydac.fem.projectors">
+<span id="psydac-fem-projectors"></span><h1>psydac.fem.projectors<a class="headerlink" href="#module-psydac.fem.projectors" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">knot_insertion_projection_operator</span></code>(domain, ...)</p></td>
+<td><p>Compute the projection operator based on the knot insertion technique.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">knots_to_insert</span></code>(coarse_grid, fine_grid[, tol])</p></td>
+<td><p>Compute the point difference between the fine grid and coarse grid.</p></td>
+</tr>
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+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
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+  <section id="module-psydac.fem.tensor">
+<span id="psydac-fem-tensor"></span><h1>psydac.fem.tensor<a class="headerlink" href="#module-psydac.fem.tensor" title="Link to this heading">#</a></h1>
+<p>We assume here that a tensor space is the product of fem spaces whom basis are
+of compact support</p>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">TensorFemSpace</span></code>(domain_decomposition, *spaces)</p></td>
+<td><p>Tensor-product Finite Element space V.</p></td>
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+<span id="psydac-fem-vector"></span><h1>psydac.fem.vector<a class="headerlink" href="#module-psydac.fem.vector" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ProductFemSpace</span></code>(*spaces[, connectivity])</p></td>
+<td><p>Product of FEM spaces this class is used to represent FEM spaces on a multi-patch domain.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorFemSpace</span></code>(*spaces)</p></td>
+<td><p>FEM space with a vector basis defined on a single patch this class is used to represent either spaces of vector-valued fem fields, or product spaces involved in systems of equations.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.linalg.basic">
+<span id="psydac-linalg-basic"></span><h1>psydac.linalg.basic<a class="headerlink" href="#module-psydac.linalg.basic" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ComposedLinearOperator</span></code>(domain, codomain, *args)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">IdentityOperator</span></code>(domain[, codomain])</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">InverseLinearOperator</span></code>(A, **kwargs)</p></td>
+<td><p>Abstract base class for the (approximate) inverse A_inv := A^{-1} of a square matrix A.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearOperator</span></code>()</p></td>
+<td><p>Linear operator acting between two (normed) vector spaces V (domain) and W (codomain).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LinearSolver</span></code>()</p></td>
+<td><p>Solver for square linear system Ax=b, where x and b belong to (normed) vector space V.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PowerLinearOperator</span></code>(domain, codomain, A, n)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ScaledLinearOperator</span></code>(domain, codomain, c, A)</p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SumLinearOperator</span></code>(domain, codomain, *args)</p></td>
+<td><p>A sum of linear operatos acting between the same (normed) vector spaces V (domain) and W (codomain).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">Vector</span></code>()</p></td>
+<td><p>Element of a (normed) vector space V.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">VectorSpace</span></code>()</p></td>
+<td><p>Finite-dimensional vector space V with a scalar (dot) product.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ZeroOperator</span></code>(domain[, codomain])</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.linalg.block">
+<span id="psydac-linalg-block"></span><h1>psydac.linalg.block<a class="headerlink" href="#module-psydac.linalg.block" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockLinearOperator</span></code>(V1, V2[, blocks])</p></td>
+<td><p>Linear operator that can be written as blocks of other Linear Operators.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockVector</span></code>(V[, blocks])</p></td>
+<td><p>Block of Vectors, which is an element of a BlockVectorSpace.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BlockVectorSpace</span></code>(*spaces[, connectivity])</p></td>
+<td><p>Product Vector Space V of two Vector Spaces (V1,V2) or more.</p></td>
+</tr>
+</tbody>
+</table>
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+  <section id="module-psydac.linalg.direct_solvers">
+<span id="psydac-linalg-direct-solvers"></span><h1>psydac.linalg.direct_solvers<a class="headerlink" href="#module-psydac.linalg.direct_solvers" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BandedSolver</span></code>(u, l, bmat[, transposed])</p></td>
+<td><p>Solve the equation Ax = b for x, assuming A is banded matrix.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">SparseSolver</span></code>(spmat[, transposed])</p></td>
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+  <section id="module-psydac.linalg.fft">
+<span id="psydac-linalg-fft"></span><h1>psydac.linalg.fft<a class="headerlink" href="#module-psydac.linalg.fft" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedDCT</span></code>(space[, norm, workers, ttype])</p></td>
+<td><p>Equals an n-dimensional DCT operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedDST</span></code>(space[, norm, workers, ttype])</p></td>
+<td><p>Equals an n-dimensional DST operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedFFT</span></code>(space[, norm, workers])</p></td>
+<td><p>Equals an n-dimensional FFT operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedFFTBase</span></code>(space, functions)</p></td>
+<td><p>A base class for the distributed FFT, DCT and DST.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedIDCT</span></code>(space[, norm, workers, ttype])</p></td>
+<td><p>Equals an n-dimensional IDCT operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedIDST</span></code>(space[, norm, workers, ttype])</p></td>
+<td><p>Equals an n-dimensional IDST operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DistributedIFFT</span></code>(space[, norm, workers])</p></td>
+<td><p>Equals an n-dimensional IFFT operation, except that it works on a distributed/parallel StencilVector.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.linalg.kernels.axpy_kernels">
+<span id="psydac-linalg-kernels-axpy-kernels"></span><h1>psydac.linalg.kernels.axpy_kernels<a class="headerlink" href="#module-psydac.linalg.kernels.axpy_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">axpy_1d</span></code></p></td>
+<td><p>Kernel for computing y = alpha * x + y.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">axpy_2d</span></code></p></td>
+<td><p>Kernel for computing y = alpha * x + y.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">axpy_3d</span></code></p></td>
+<td><p>Kernel for computing y = alpha * x + y.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+  <section id="module-psydac.linalg.kernels.inner_kernels">
+<span id="psydac-linalg-kernels-inner-kernels"></span><h1>psydac.linalg.kernels.inner_kernels<a class="headerlink" href="#module-psydac.linalg.kernels.inner_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">inner_1d</span></code></p></td>
+<td><p>Kernel for computing the inner product (case of two 1D vectors).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">inner_2d</span></code></p></td>
+<td><p>Kernel for computing the inner product (case of two 2D vectors).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">inner_3d</span></code></p></td>
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+                  
+  <section id="module-psydac.linalg.kernels.stencil2coo_kernels">
+<span id="psydac-linalg-kernels-stencil2coo-kernels"></span><h1>psydac.linalg.kernels.stencil2coo_kernels<a class="headerlink" href="#module-psydac.linalg.kernels.stencil2coo_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_1d_C</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_1d_F</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_2d_C</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_2d_F</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_3d_C</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">stencil2coo_3d_F</span></code></p></td>
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+  <section id="module-psydac.linalg.kernels.transpose_kernels">
+<span id="psydac-linalg-kernels-transpose-kernels"></span><h1>psydac.linalg.kernels.transpose_kernels<a class="headerlink" href="#module-psydac.linalg.kernels.transpose_kernels" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">interface_transpose_1d</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">interface_transpose_2d</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">interface_transpose_3d</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">transpose_1d</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">transpose_2d</span></code></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">transpose_3d</span></code></p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+                  
+  <section id="module-psydac.linalg.kron">
+<span id="psydac-linalg-kron"></span><h1>psydac.linalg.kron<a class="headerlink" href="#module-psydac.linalg.kron" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">kronecker_solve</span></code>(solvers, rhs[, out])</p></td>
+<td><p>Solve linear system Ax=b with A=kron( A_n, A_{n-1}, ..., A_2, A_1 ), given $n$ separate linear solvers $L_n$ for the 1D problems $A_n x_n = b_n$:</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">KroneckerDenseMatrix</span></code>(V, W, *args[, with_pads])</p></td>
+<td><p>Kronecker product of 1D dense matrices.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">KroneckerLinearSolver</span></code>(V, W, solvers)</p></td>
+<td><p>A solver for Ax=b, where A is a Kronecker matrix from arbirary dimension d, defined by d solvers.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">KroneckerStencilMatrix</span></code>(V, W, *args)</p></td>
+<td><p>Kronecker product of 1D stencil matrices.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.linalg.solvers">
+<span id="psydac-linalg-solvers"></span><h1>psydac.linalg.solvers<a class="headerlink" href="#module-psydac.linalg.solvers" title="Link to this heading">#</a></h1>
+<p>This module provides iterative solvers and preconditioners.</p>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">inverse</span></code>(A, solver, **kwargs)</p></td>
+<td><p>A function to create objects of all InverseLinearOperator subclasses.</p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BiConjugateGradient</span></code>(A, *[, x0, tol, ...])</p></td>
+<td><p>Biconjugate Gradient (BiCG).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">BiConjugateGradientStabilized</span></code>(A, *[, x0, ...])</p></td>
+<td><p>Biconjugate Gradient Stabilized (BiCGStab).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">ConjugateGradient</span></code>(A, *[, x0, tol, maxiter, ...])</p></td>
+<td><p>Conjugate Gradient (CG).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">GMRES</span></code>(A, *[, x0, tol, maxiter, verbose, recycle])</p></td>
+<td><p>Generalized Minimal Residual (GMRES).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">LSMR</span></code>(A, *[, x0, tol, atol, btol, maxiter, ...])</p></td>
+<td><p>Least Squares Minimal Residual (LSMR).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">MinimumResidual</span></code>(A, *[, x0, tol, maxiter, ...])</p></td>
+<td><p>Minimum Residual (MinRes).</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PBiConjugateGradientStabilized</span></code>(A, *[, pc, ...])</p></td>
+<td><p>Preconditioned Biconjugate Gradient Stabilized (PBiCGStab).</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">PConjugateGradient</span></code>(A, *[, pc, x0, tol, ...])</p></td>
+<td><p>Preconditioned Conjugate Gradient (PCG).</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+                  
+  <section id="module-psydac.linalg.stencil">
+<span id="psydac-linalg-stencil"></span><h1>psydac.linalg.stencil<a class="headerlink" href="#module-psydac.linalg.stencil" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">compute_diag_len</span></code>(pads, shifts_domain, ...[, ...])</p></td>
+<td><p>Compute the diagonal length and the padding of the stencil matrix for each direction, using the shifts of the domain and the codomain.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">flip_axis</span></code>(index, n)</p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilDiagonalMatrix</span></code>(V, W, data)</p></td>
+<td><p>Linear operator which operates between stencil vector spaces, and which can be represented by a matrix with non-zero entries only on its main diagonal.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilInterfaceMatrix</span></code>(V, W, s_d, s_c, ...)</p></td>
+<td><p>Matrix in n-dimensional stencil format for an interface.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilMatrix</span></code>(V, W[, pads, backend])</p></td>
+<td><p>Matrix in n-dimensional stencil format.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilVector</span></code>(V)</p></td>
+<td><p>Vector in n-dimensional stencil format.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">StencilVectorSpace</span></code>(cart[, dtype])</p></td>
+<td><p>Vector space for n-dimensional stencil format.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+  <section id="module-psydac.linalg.topetsc">
+<span id="psydac-linalg-topetsc"></span><h1>psydac.linalg.topetsc<a class="headerlink" href="#module-psydac.linalg.topetsc" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">flatten_vec</span></code>(vec)</p></td>
+<td><p>Return the flattened 1D array values and indices owned by the process of the given vector.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">mat_topetsc</span></code>(mat)</p></td>
+<td><p>Convert Psydac Matrix to PETSc data structure.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">vec_topetsc</span></code>(vec)</p></td>
+<td><p>Convert Psydac Vector to PETSc data structure.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+                  
+  <section id="module-psydac.linalg.utilities">
+<span id="psydac-linalg-utilities"></span><h1>psydac.linalg.utilities<a class="headerlink" href="#module-psydac.linalg.utilities" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">array_to_psydac</span></code>(x, Xh)</p></td>
+<td><p>converts a numpy array to StencilVector or BlockVector format</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">petsc_to_psydac</span></code>(vec, Xh)</p></td>
+<td><p>converts a petsc Vec object to a StencilVector or a BlockVector format.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
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+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">random_string</span></code>(n)</p></td>
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">NurbsMapping</span></code>(*components[, name])</p></td>
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+</tr>
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+<span id="psydac-polar-c1-cart"></span><h1>psydac.polar.c1_cart<a class="headerlink" href="#module-psydac.polar.c1_cart" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">C1_Cart</span></code>(cart[, radial_dim])</p></td>
+<td><p>A CartDecomposition subclass whose objects are generated from an existing CartDecomposition object by removing all degrees of freedom corresponding to i_d = 0, 1 with d being the radial dimension.</p></td>
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+  <section id="module-psydac.polar.c1_spaces">
+<span id="psydac-polar-c1-spaces"></span><h1>psydac.polar.c1_spaces<a class="headerlink" href="#module-psydac.polar.c1_spaces" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">new_c1_vector_space</span></code>(V[, radial_dim, angle_dim])</p></td>
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+<span id="psydac-polar-dense"></span><h1>psydac.polar.dense<a class="headerlink" href="#module-psydac.polar.dense" title="Link to this heading">#</a></h1>
+<p class="rubric">Classes</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DenseMatrix</span></code>(V, W, data)</p></td>
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+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">DenseVector</span></code>(V, data)</p></td>
+<td><p></p></td>
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+  <section id="module-psydac.utilities.quadratures">
+<span id="psydac-utilities-quadratures"></span><h1>psydac.utilities.quadratures<a class="headerlink" href="#module-psydac.utilities.quadratures" title="Link to this heading">#</a></h1>
+<p>This module contains some routines to generate quadrature points in 1D
+it has also a routine uniform, which generates uniform points
+with weights equal to 1</p>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">gauss_legendre</span></code>(ordergl[, tol])</p></td>
+<td><p>Returns nodal abscissas {x} and weights {A} of Gauss-Legendre m-point quadrature.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">gauss_lobatto</span></code>(k)</p></td>
+<td><p>Returns nodal abscissas {x} and weights {A} of Gauss-Legendre m-point quadrature.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">quadrature</span></code>(a, k[, method])</p></td>
+<td><p>this routine generates a quad pts on the grid linspace(a,b,N)</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+  <section id="module-psydac.utilities.utils">
+<span id="psydac-utilities-utils"></span><h1>psydac.utilities.utils<a class="headerlink" href="#module-psydac.utilities.utils" title="Link to this heading">#</a></h1>
+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">animate_field</span></code>(fields, domain, mapping[, ...])</p></td>
+<td><p>Animate a sequence of scalar fields over a geometry.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">is_real</span></code>(x)</p></td>
+<td><p>Determine whether the given input represents a real number.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">refine_array_1d</span></code>(x, n[, remove_duplicates])</p></td>
+<td><p>Refines a 1D array by subdividing each interval (x[i], x[i+1]) into n identical parts.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">roll_edges</span></code>(domain, points)</p></td>
+<td><p>If necessary, &quot;roll&quot; back intervals that cross boundary of periodic domain.</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">split_field</span></code>(uh, spaces[, out])</p></td>
+<td><p>Split a field into a list of fields that corresponds to the spaces.</p></td>
+</tr>
+<tr class="row-even"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">split_space</span></code>(Xh)</p></td>
+<td><p>Split the flattened fem spaces into</p></td>
+</tr>
+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">unroll_edges</span></code>(domain, xgrid)</p></td>
+<td><p>If necessary, &quot;unroll&quot; intervals that cross boundary of periodic domain.</p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+<p class="rubric">Functions</p>
+<table class="autosummary longtable table autosummary">
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+<tr class="row-odd"><td><p><code class="xref py py-obj docutils literal notranslate"><span class="pre">writeParallelVTKUnstructuredGrid</span></code>(path, ...)</p></td>
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+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.api.ast.basic.html#module-psydac.api.ast.basic" title="psydac.api.ast.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ast.basic</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.api.ast.evaluation.html#module-psydac.api.ast.evaluation" title="psydac.api.ast.evaluation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ast.evaluation</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.api.ast.expr.html#module-psydac.api.ast.expr" title="psydac.api.ast.expr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ast.expr</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.api.ast.fem.html#module-psydac.api.ast.fem" title="psydac.api.ast.fem"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ast.fem</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.api.ast.glt.html#module-psydac.api.ast.glt" title="psydac.api.ast.glt"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ast.glt</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.api.ast.linalg.html#module-psydac.api.ast.linalg" title="psydac.api.ast.linalg"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ast.linalg</span></code></a></p></td>
+<td><p></p></td>
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+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.api.ast.parser.html#module-psydac.api.ast.parser" title="psydac.api.ast.parser"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ast.parser</span></code></a></p></td>
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+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.api.ast.utilities.html#module-psydac.api.ast.utilities" title="psydac.api.ast.utilities"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ast.utilities</span></code></a></p></td>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>api</h1>
+    <!-- Table of contents -->
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+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#api-submodules">api submodules</a></li>
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+</div>
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+              
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+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section id="api">
+<h1>api<a class="headerlink" href="#api" title="Link to this heading">#</a></h1>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.api.basic.html#module-psydac.api.basic" title="psydac.api.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">api.basic</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.api.discretization.html#module-psydac.api.discretization" title="psydac.api.discretization"><code class="xref py py-obj docutils literal notranslate"><span class="pre">api.discretization</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.api.equation.html#module-psydac.api.equation" title="psydac.api.equation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">api.equation</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.api.essential_bc.html#module-psydac.api.essential_bc" title="psydac.api.essential_bc"><code class="xref py py-obj docutils literal notranslate"><span class="pre">api.essential_bc</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.api.expr.html#module-psydac.api.expr" title="psydac.api.expr"><code class="xref py py-obj docutils literal notranslate"><span class="pre">api.expr</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.api.feec.html#module-psydac.api.feec" title="psydac.api.feec"><code class="xref py py-obj docutils literal notranslate"><span class="pre">api.feec</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.api.fem.html#module-psydac.api.fem" title="psydac.api.fem"><code class="xref py py-obj docutils literal notranslate"><span class="pre">api.fem</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.api.glt.html#module-psydac.api.glt" title="psydac.api.glt"><code class="xref py py-obj docutils literal notranslate"><span class="pre">api.glt</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.api.grid.html#module-psydac.api.grid" title="psydac.api.grid"><code class="xref py py-obj docutils literal notranslate"><span class="pre">api.grid</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.api.postprocessing.html#module-psydac.api.postprocessing" title="psydac.api.postprocessing"><code class="xref py py-obj docutils literal notranslate"><span class="pre">api.postprocessing</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<section id="api-submodules">
+<h2>api submodules<a class="headerlink" href="#api-submodules" title="Link to this heading">#</a></h2>
+<div class="toctree-wrapper compound">
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="api.ast.html">api.ast</a></li>
+<li class="toctree-l1"><a class="reference internal" href="api.printing.html">api.printing</a></li>
+</ul>
+</div>
+</section>
+</section>
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diff --git a/psydac/docs/source/modules/api.printing.html b/psydac/docs/source/modules/api.printing.html
new file mode 100644
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@@ -0,0 +1,937 @@
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+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.core.bsplines_kernels.html#module-psydac.core.bsplines_kernels" title="psydac.core.bsplines_kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">core.bsplines_kernels</span></code></a></p></td>
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+<td><p>Basic module that provides the means for evaluating the B-Splines basis functions and their derivatives.</p></td>
+</tr>
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+                <article class="bd-article">
+                  
+  <section id="ddm">
+<h1>ddm<a class="headerlink" href="#ddm" title="Link to this heading">#</a></h1>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.ddm.basic.html#module-psydac.ddm.basic" title="psydac.ddm.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ddm.basic</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.ddm.blocking_data_exchanger.html#module-psydac.ddm.blocking_data_exchanger" title="psydac.ddm.blocking_data_exchanger"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ddm.blocking_data_exchanger</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.ddm.cart.html#module-psydac.ddm.cart" title="psydac.ddm.cart"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ddm.cart</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.ddm.interface_data_exchanger.html#module-psydac.ddm.interface_data_exchanger" title="psydac.ddm.interface_data_exchanger"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ddm.interface_data_exchanger</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.ddm.nonblocking_data_exchanger.html#module-psydac.ddm.nonblocking_data_exchanger" title="psydac.ddm.nonblocking_data_exchanger"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ddm.nonblocking_data_exchanger</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.ddm.partition.html#module-psydac.ddm.partition" title="psydac.ddm.partition"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ddm.partition</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.ddm.petsc.html#module-psydac.ddm.petsc" title="psydac.ddm.petsc"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ddm.petsc</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.ddm.utilities.html#module-psydac.ddm.utilities" title="psydac.ddm.utilities"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ddm.utilities</span></code></a></p></td>
+<td><p></p></td>
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+<div id="searchbox"></div>
+                <article class="bd-article">
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+  <section id="feec">
+<h1>feec<a class="headerlink" href="#feec" title="Link to this heading">#</a></h1>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.feec.derivatives.html#module-psydac.feec.derivatives" title="psydac.feec.derivatives"><code class="xref py py-obj docutils literal notranslate"><span class="pre">feec.derivatives</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.feec.global_projectors.html#module-psydac.feec.global_projectors" title="psydac.feec.global_projectors"><code class="xref py py-obj docutils literal notranslate"><span class="pre">feec.global_projectors</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.feec.pull_push.html#module-psydac.feec.pull_push" title="psydac.feec.pull_push"><code class="xref py py-obj docutils literal notranslate"><span class="pre">feec.pull_push</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.feec.pushforward.html#module-psydac.feec.pushforward" title="psydac.feec.pushforward"><code class="xref py py-obj docutils literal notranslate"><span class="pre">feec.pushforward</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<section id="feec-submodules">
+<h2>feec submodules<a class="headerlink" href="#feec-submodules" title="Link to this heading">#</a></h2>
+<div class="toctree-wrapper compound">
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="feec.multipatch.html">feec.multipatch</a></li>
+</ul>
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+  <section id="feec-multipatch">
+<h1>feec.multipatch<a class="headerlink" href="#feec-multipatch" title="Link to this heading">#</a></h1>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.feec.multipatch.api.html#module-psydac.feec.multipatch.api" title="psydac.feec.multipatch.api"><code class="xref py py-obj docutils literal notranslate"><span class="pre">multipatch.api</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.feec.multipatch.fem_linear_operators.html#module-psydac.feec.multipatch.fem_linear_operators" title="psydac.feec.multipatch.fem_linear_operators"><code class="xref py py-obj docutils literal notranslate"><span class="pre">multipatch.fem_linear_operators</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.feec.multipatch.operators.html#module-psydac.feec.multipatch.operators" title="psydac.feec.multipatch.operators"><code class="xref py py-obj docutils literal notranslate"><span class="pre">multipatch.operators</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.feec.multipatch.utilities.html#module-psydac.feec.multipatch.utilities" title="psydac.feec.multipatch.utilities"><code class="xref py py-obj docutils literal notranslate"><span class="pre">multipatch.utilities</span></code></a></p></td>
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+<li class="toctree-l2"><a class="reference internal" href="../../../../chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../../../../chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="../../../../chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="../../../../chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="../../../../chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
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+    <h1>fem</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+        </div>
+    </div>
+</div>
+
+              
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+                <article class="bd-article">
+                  
+  <section id="fem">
+<h1>fem<a class="headerlink" href="#fem" title="Link to this heading">#</a></h1>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.fem.basic.html#module-psydac.fem.basic" title="psydac.fem.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fem.basic</span></code></a></p></td>
+<td><p>In order to avoid multiple inheritence, we define the base objects for Finite Elements as abstract classes that contain a topological member.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.fem.context.html#module-psydac.fem.context" title="psydac.fem.context"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fem.context</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.fem.grid.html#module-psydac.fem.grid" title="psydac.fem.grid"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fem.grid</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.fem.partitioning.html#module-psydac.fem.partitioning" title="psydac.fem.partitioning"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fem.partitioning</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.fem.projectors.html#module-psydac.fem.projectors" title="psydac.fem.projectors"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fem.projectors</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.fem.splines.html#module-psydac.fem.splines" title="psydac.fem.splines"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fem.splines</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.fem.tensor.html#module-psydac.fem.tensor" title="psydac.fem.tensor"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fem.tensor</span></code></a></p></td>
+<td><p>We assume here that a tensor space is the product of fem spaces whom basis are of compact support</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.fem.vector.html#module-psydac.fem.vector" title="psydac.fem.vector"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fem.vector</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+</section>
+
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+                <article class="bd-article">
+                  
+  <section id="linalg">
+<h1>linalg<a class="headerlink" href="#linalg" title="Link to this heading">#</a></h1>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.basic.html#module-psydac.linalg.basic" title="psydac.linalg.basic"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.basic</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.block.html#module-psydac.linalg.block" title="psydac.linalg.block"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.block</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.direct_solvers.html#module-psydac.linalg.direct_solvers" title="psydac.linalg.direct_solvers"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.direct_solvers</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.fft.html#module-psydac.linalg.fft" title="psydac.linalg.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.fft</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.kernels.html#module-psydac.linalg.kernels" title="psydac.linalg.kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.kernels</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.kron.html#module-psydac.linalg.kron" title="psydac.linalg.kron"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.kron</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.solvers.html#module-psydac.linalg.solvers" title="psydac.linalg.solvers"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.solvers</span></code></a></p></td>
+<td><p>This module provides iterative solvers and preconditioners.</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.stencil.html#module-psydac.linalg.stencil" title="psydac.linalg.stencil"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.stencil</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.topetsc.html#module-psydac.linalg.topetsc" title="psydac.linalg.topetsc"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.topetsc</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.utilities.html#module-psydac.linalg.utilities" title="psydac.linalg.utilities"><code class="xref py py-obj docutils literal notranslate"><span class="pre">linalg.utilities</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+</tbody>
+</table>
+<section id="linalg-submodules">
+<h2>linalg submodules<a class="headerlink" href="#linalg-submodules" title="Link to this heading">#</a></h2>
+<div class="toctree-wrapper compound">
+<ul>
+<li class="toctree-l1"><a class="reference internal" href="linalg.kernels.html">linalg.kernels</a></li>
+</ul>
+</div>
+</section>
+</section>
+
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diff --git a/psydac/docs/source/modules/linalg.kernels.html b/psydac/docs/source/modules/linalg.kernels.html
new file mode 100644
index 0000000..495d2d2
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+  <section id="linalg-kernels">
+<h1>linalg.kernels<a class="headerlink" href="#linalg-kernels" title="Link to this heading">#</a></h1>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.kernels.axpy_kernels.html#module-psydac.linalg.kernels.axpy_kernels" title="psydac.linalg.kernels.axpy_kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">kernels.axpy_kernels</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.kernels.inner_kernels.html#module-psydac.linalg.kernels.inner_kernels" title="psydac.linalg.kernels.inner_kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">kernels.inner_kernels</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.kernels.matvec_kernels.html#module-psydac.linalg.kernels.matvec_kernels" title="psydac.linalg.kernels.matvec_kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">kernels.matvec_kernels</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.kernels.stencil2coo_kernels.html#module-psydac.linalg.kernels.stencil2coo_kernels" title="psydac.linalg.kernels.stencil2coo_kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">kernels.stencil2coo_kernels</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.linalg.kernels.transpose_kernels.html#module-psydac.linalg.kernels.transpose_kernels" title="psydac.linalg.kernels.transpose_kernels"><code class="xref py py-obj docutils literal notranslate"><span class="pre">kernels.transpose_kernels</span></code></a></p></td>
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+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.polar.c1_cart.html#module-psydac.polar.c1_cart" title="psydac.polar.c1_cart"><code class="xref py py-obj docutils literal notranslate"><span class="pre">polar.c1_cart</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.polar.c1_linops.html#module-psydac.polar.c1_linops" title="psydac.polar.c1_linops"><code class="xref py py-obj docutils literal notranslate"><span class="pre">polar.c1_linops</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.polar.c1_projections.html#module-psydac.polar.c1_projections" title="psydac.polar.c1_projections"><code class="xref py py-obj docutils literal notranslate"><span class="pre">polar.c1_projections</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.polar.c1_spaces.html#module-psydac.polar.c1_spaces" title="psydac.polar.c1_spaces"><code class="xref py py-obj docutils literal notranslate"><span class="pre">polar.c1_spaces</span></code></a></p></td>
+<td><p></p></td>
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+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.polar.dense.html#module-psydac.polar.dense" title="psydac.polar.dense"><code class="xref py py-obj docutils literal notranslate"><span class="pre">polar.dense</span></code></a></p></td>
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+  <section id="utilities">
+<h1>utilities<a class="headerlink" href="#utilities" title="Link to this heading">#</a></h1>
+<table class="autosummary longtable table autosummary">
+<tbody>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.utilities.quadratures.html#module-psydac.utilities.quadratures" title="psydac.utilities.quadratures"><code class="xref py py-obj docutils literal notranslate"><span class="pre">utilities.quadratures</span></code></a></p></td>
+<td><p>This module contains some routines to generate quadrature points in 1D it has also a routine uniform, which generates uniform points with weights equal to 1</p></td>
+</tr>
+<tr class="row-even"><td><p><a class="reference internal" href="STUBDIR/psydac.utilities.utils.html#module-psydac.utilities.utils" title="psydac.utilities.utils"><code class="xref py py-obj docutils literal notranslate"><span class="pre">utilities.utils</span></code></a></p></td>
+<td><p></p></td>
+</tr>
+<tr class="row-odd"><td><p><a class="reference internal" href="STUBDIR/psydac.utilities.vtk.html#module-psydac.utilities.vtk" title="psydac.utilities.vtk"><code class="xref py py-obj docutils literal notranslate"><span class="pre">utilities.vtk</span></code></a></p></td>
+<td><p></p></td>
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Solving Helmholtz’s equation on non-periodic topological domains</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-building-the-domain">Step 1 : Building the domain</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-defining-the-abstract-pde-model-using-sympde">Step 2: Defining the Abstract PDE model using SymPDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-discretizing-the-domain-spaces-and-equations">Step 3: Discretizing the domain, spaces and equations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-solving-the-equation-and-computing-the-error-norms">Step 4: Solving the equation and computing the error norms</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-saving-the-results">Step 5: Saving the results</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-6-exporting-the-results-to-vtk-and-visualizing-them-with-paraview">Step 6: Exporting the results to VTK and visualizing them with Paraview</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#example-results-on-the-square-domain">Example: Results on the square domain</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="solving-helmholtz-s-equation-on-non-periodic-topological-domains">
+<h1>Solving Helmholtz’s equation on non-periodic topological domains<a class="headerlink" href="#solving-helmholtz-s-equation-on-non-periodic-topological-domains" title="Link to this heading">#</a></h1>
+<p>In this notebook, we will show how to solve Helmholtz’s equation on single patch and multipatch domains. As we are running this in a notebook, everything will be run in serial and hence we are limiting ourselves to a fairly coarse discretization to avoid taking too much time. However, Psydac allows for hybrid MPI + OpenMP parallelization with barely any changed to the code. The lines that are impacted by that will be preceeded by their MPI equivalent</p>
+<section id="step-1-building-the-domain">
+<h2>Step 1 : Building the domain<a class="headerlink" href="#step-1-building-the-domain" title="Link to this heading">#</a></h2>
+<p>Psydac uses the powerful topological tools of SymPDE to build a large variety of domains. Here we show how to create a Square.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">Square</span>
+<span class="n">Omega</span><span class="o">=</span><span class="n">Square</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="step-2-defining-the-abstract-pde-model-using-sympde">
+<h2>Step 2: Defining the Abstract PDE model using SymPDE<a class="headerlink" href="#step-2-defining-the-abstract-pde-model-using-sympde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">minus</span><span class="p">,</span> <span class="n">plus</span>
+
+<span class="kn">from</span> <span class="nn">sympde.expr.expr</span>          <span class="kn">import</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">BilinearForm</span>
+<span class="kn">from</span> <span class="nn">sympde.expr.expr</span>          <span class="kn">import</span> <span class="n">integral</span>              
+<span class="kn">from</span> <span class="nn">sympde.expr.expr</span>          <span class="kn">import</span> <span class="n">Norm</span>                       
+<span class="kn">from</span> <span class="nn">sympde.expr</span>               <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span>
+
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">ScalarFunctionSpace</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">elements_of</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">NormalVector</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">Union</span>
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">cos</span><span class="p">,</span> <span class="n">sin</span><span class="p">,</span> <span class="n">symbols</span><span class="p">,</span> <span class="n">conjugate</span><span class="p">,</span> <span class="n">exp</span><span class="p">,</span> <span class="n">sqrt</span>
+
+<span class="c1"># Define the abstract model to solve Helmholtz&#39;s equation using the manufactured solution method</span>
+
+<span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">Omega</span><span class="o">.</span><span class="n">coordinates</span>
+<span class="n">kappa</span> <span class="o">=</span> <span class="mi">2</span><span class="o">*</span><span class="n">pi</span>
+<span class="n">solution</span> <span class="o">=</span> <span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">kappa</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">sin</span><span class="p">(</span><span class="n">kappa</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span>
+<span class="n">e_w_0</span> <span class="o">=</span> <span class="n">sin</span><span class="p">(</span><span class="n">kappa</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span> <span class="c1"># value of incoming wave at x=0, forall y</span>
+<span class="n">dx_e_w_0</span> <span class="o">=</span> <span class="mi">1</span><span class="n">j</span><span class="o">*</span><span class="n">kappa</span><span class="o">*</span><span class="n">sin</span><span class="p">(</span><span class="n">kappa</span> <span class="o">*</span> <span class="n">y</span><span class="p">)</span> <span class="c1"># derivative wrt. x of incoming wave at x=0, forall y</span>
+
+<span class="n">V</span>   <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">Omega</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+<span class="n">V</span><span class="o">.</span><span class="n">codomain_type</span><span class="o">=</span><span class="s1">&#39;complex&#39;</span>
+
+<span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">elements_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="s1">&#39;u, v&#39;</span><span class="p">)</span>
+
+<span class="n">error</span>  <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="n">solution</span>
+
+<span class="n">expr</span>   <span class="o">=</span> <span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">),</span><span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">))</span> <span class="o">-</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">kappa</span> <span class="o">**</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">u</span> <span class="o">*</span> <span class="n">v</span>
+
+<span class="c1">#we impose an incoming wave from the left and absorbing boundary conditions at the right.</span>
+<span class="n">boundary_expr</span> <span class="o">=</span> <span class="o">-</span> <span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">kappa</span> <span class="o">*</span> <span class="n">u</span> <span class="o">*</span> <span class="n">v</span>
+<span class="n">x_boundary</span> <span class="o">=</span> <span class="n">Union</span><span class="p">(</span><span class="n">Omega</span><span class="o">.</span><span class="n">get_boundary</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">ext</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span> <span class="n">Omega</span><span class="o">.</span><span class="n">get_boundary</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">ext</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
+<span class="n">boundary_source_expr</span> <span class="o">=</span> <span class="o">-</span> <span class="n">dx_e_w_0</span> <span class="o">*</span> <span class="n">v</span> <span class="o">-</span> <span class="mi">1</span><span class="n">j</span> <span class="o">*</span> <span class="n">kappa</span> <span class="o">*</span> <span class="n">e_w_0</span> <span class="o">*</span> <span class="n">v</span>    
+
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">),</span> <span class="n">integral</span><span class="p">(</span><span class="n">Omega</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span> <span class="o">+</span> <span class="n">integral</span><span class="p">(</span><span class="n">x_boundary</span><span class="p">,</span> <span class="n">boundary_expr</span><span class="p">))</span>
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">Omega</span><span class="o">.</span><span class="n">get_boundary</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">ext</span><span class="o">=-</span><span class="mi">1</span><span class="p">),</span> <span class="n">boundary_source_expr</span><span class="p">))</span>
+
+<span class="n">equation</span> <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
+
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">error</span><span class="p">,</span> <span class="n">Omega</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">error</span><span class="p">,</span> <span class="n">Omega</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="step-3-discretizing-the-domain-spaces-and-equations">
+<h2>Step 3: Discretizing the domain, spaces and equations<a class="headerlink" href="#step-3-discretizing-the-domain-spaces-and-equations" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Discretize the geometry and equation</span>
+<span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+<span class="kn">from</span> <span class="nn">psydac.api.settings</span> <span class="kn">import</span> <span class="n">PSYDAC_BACKENDS</span>
+
+<span class="c1"># The backends are as follows and can all be run with OpenMP by setting backend[&#39;omp&#39;] = True.</span>
+<span class="c1"># For all but the Python backend, compilation flags can be accessed and changed using backend[&#39;flags&#39;].</span>
+<span class="c1"># - Python backend: Generate and runs python files. Accessed via PSYDAC_BACKENDS[&#39;python&#39;]</span>
+<span class="c1"># - Pyccel GCC backend: Accessed via PSYDAC_BACKENDS[&#39;pyccel-gcc&#39;]</span>
+<span class="c1"># - Pyccel Intel backend: Accessed via PSYDAC_BACKENDS[&#39;pyccel-intel&#39;]</span>
+<span class="c1"># - Pyccel PGI backend: Accessed via PSYDAC_BACKENDS[&#39;pyccel-pgi&#39;]</span>
+<span class="c1"># - Pyccel Numba backend: Accessed via PSYDAC_BACKENDS[&#39;numba&#39;]</span>
+
+<span class="n">backend</span> <span class="o">=</span> <span class="n">PSYDAC_BACKENDS</span><span class="p">[</span><span class="s1">&#39;python&#39;</span><span class="p">]</span>
+
+<span class="c1"># Uncomment to use OpenMp</span>
+<span class="c1"># import os</span>
+<span class="c1"># os.environ[&#39;OMP_NUM_THREADS&#39;] = &quot;4&quot;</span>
+<span class="c1"># backend[&#39;omp&#39;] = True</span>
+
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">30</span><span class="p">,</span> <span class="mi">30</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">]</span>
+<span class="n">periodic</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">,</span> <span class="kc">True</span><span class="p">]</span>
+
+<span class="c1"># MPI version</span>
+<span class="c1"># from mpi4py import MPI</span>
+<span class="c1"># comm = MPI.COMM_WORLD</span>
+<span class="c1"># Omega_h = discretize(Omega, ncells=ncells, comm=comm)</span>
+<span class="n">Omega_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">Omega</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">periodic</span><span class="o">=</span><span class="n">periodic</span><span class="p">)</span>
+
+<span class="n">Vh</span>         <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">Omega_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">Omega_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">],</span> <span class="n">backend</span><span class="o">=</span><span class="n">backend</span><span class="p">)</span>
+<span class="n">l2norm_h</span>   <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm</span><span class="p">,</span> <span class="n">Omega_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">,</span> <span class="n">backend</span><span class="o">=</span><span class="n">backend</span><span class="p">)</span>
+<span class="n">h1norm_h</span>   <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">Omega_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">,</span> <span class="n">backend</span><span class="o">=</span><span class="n">backend</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="step-4-solving-the-equation-and-computing-the-error-norms">
+<h2>Step 4: Solving the equation and computing the error norms<a class="headerlink" href="#step-4-solving-the-equation-and-computing-the-error-norms" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set the solver parameters</span>
+<span class="c1"># &#39;cbig&#39; -&gt; Biconjugate gradient method</span>
+<span class="n">equation_h</span><span class="o">.</span><span class="n">set_solver</span><span class="p">(</span><span class="s1">&#39;bicg&#39;</span><span class="p">,</span> <span class="n">info</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-14</span><span class="p">)</span>
+
+<span class="kn">import</span> <span class="nn">time</span>
+
+<span class="n">t0_s</span>     <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+<span class="n">uh</span><span class="p">,</span> <span class="n">info</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+<span class="n">t1_s</span>     <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+
+
+<span class="n">t0_d</span>     <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+<span class="n">l2_error</span> <span class="o">=</span> <span class="n">l2norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+<span class="n">t1_d</span>     <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+
+<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;&gt; CG info       :: &#39;</span><span class="p">,</span><span class="n">info</span> <span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;&gt; L2 error      :: </span><span class="si">{:.2e}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">l2_error</span><span class="p">))</span>
+<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;&gt; H1 error      :: </span><span class="si">{:.2e}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">h1_error</span><span class="p">))</span>
+<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;&gt; Solution time :: </span><span class="si">{:.2e}</span><span class="s1">s&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">t1_s</span> <span class="o">-</span> <span class="n">t0_s</span><span class="p">))</span>
+<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;&gt; Evaluat. time :: </span><span class="si">{:.2e}</span><span class="s1">s &#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">t1_d</span> <span class="o">-</span> <span class="n">t0_d</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&gt; CG info       ::  {&#39;niter&#39;: 45, &#39;success&#39;: True, &#39;res_norm&#39;: 9.329589911751023e-15}
+&gt; L2 error      :: 2.07e-07
+&gt; H1 error      :: 3.80e-05
+&gt; Solution time :: 7.80e+00s
+&gt; Evaluat. time :: 7.59e-01s 
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="step-5-saving-the-results">
+<h2>Step 5: Saving the results<a class="headerlink" href="#step-5-saving-the-results" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Save the results using OutputManager</span>
+<span class="kn">from</span> <span class="nn">psydac.api.postprocessing</span> <span class="kn">import</span> <span class="n">OutputManager</span>
+<span class="kn">import</span> <span class="nn">os</span>
+
+<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="s1">&#39;results_Helmholtz&#39;</span><span class="p">,</span> <span class="n">exist_ok</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="n">Om</span> <span class="o">=</span> <span class="n">OutputManager</span><span class="p">(</span>
+    <span class="sa">f</span><span class="s1">&#39;results_Helmholtz/space_info_</span><span class="si">{</span><span class="n">Omega</span><span class="o">.</span><span class="n">name</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">,</span>
+    <span class="sa">f</span><span class="s1">&#39;results_Helmholtz/field_info_</span><span class="si">{</span><span class="n">Omega</span><span class="o">.</span><span class="n">name</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">,</span>
+<span class="c1"># MPI version</span>
+<span class="c1">#   comm=comm,    </span>
+<span class="p">)</span>
+
+<span class="n">Om</span><span class="o">.</span><span class="n">add_spaces</span><span class="p">(</span><span class="n">V</span><span class="o">=</span><span class="n">Vh</span><span class="p">)</span>
+<span class="n">Om</span><span class="o">.</span><span class="n">export_space_info</span><span class="p">()</span>
+
+<span class="n">Om</span><span class="o">.</span><span class="n">set_static</span><span class="p">()</span>
+<span class="n">Om</span><span class="o">.</span><span class="n">export_fields</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="n">Om</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="step-6-exporting-the-results-to-vtk-and-visualizing-them-with-paraview">
+<h2>Step 6: Exporting the results to VTK and visualizing them with Paraview<a class="headerlink" href="#step-6-exporting-the-results-to-vtk-and-visualizing-them-with-paraview" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Export the results to VTK using PostProcessManager</span>
+<span class="kn">from</span> <span class="nn">psydac.api.postprocessing</span> <span class="kn">import</span> <span class="n">PostProcessManager</span>
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">lambdify</span>
+
+<span class="n">Pm</span> <span class="o">=</span> <span class="n">PostProcessManager</span><span class="p">(</span>
+    <span class="n">domain</span><span class="o">=</span><span class="n">Omega</span><span class="p">,</span>
+    <span class="n">space_file</span><span class="o">=</span><span class="sa">f</span><span class="s1">&#39;results_Helmholtz/space_info_</span><span class="si">{</span><span class="n">Omega</span><span class="o">.</span><span class="n">name</span><span class="si">}</span><span class="s1">.yaml&#39;</span><span class="p">,</span>
+    <span class="n">fields_file</span><span class="o">=</span><span class="sa">f</span><span class="s1">&#39;results_Helmholtz/field_info_</span><span class="si">{</span><span class="n">Omega</span><span class="o">.</span><span class="n">name</span><span class="si">}</span><span class="s1">.h5&#39;</span><span class="p">,</span>
+<span class="c1"># MPI version</span>
+<span class="c1">#   comm=comm,    </span>
+<span class="p">)</span>
+
+<span class="c1"># The complex fields will be exported in two field: a real and an imaginary field</span>
+<span class="n">Pm</span><span class="o">.</span><span class="n">export_to_vtk</span><span class="p">(</span>
+    <span class="sa">f</span><span class="s1">&#39;results_Helmholtz/visu_</span><span class="si">{</span><span class="n">Omega</span><span class="o">.</span><span class="n">name</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">,</span>
+    <span class="n">grid</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
+    <span class="n">npts_per_cell</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span>
+    <span class="n">fields</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span>
+    <span class="n">additional_physical_functions</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;exact_solution&#39;</span><span class="p">:</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">Omega</span><span class="o">.</span><span class="n">coordinates</span><span class="p">,</span> <span class="n">solution</span><span class="p">,</span> <span class="n">modules</span><span class="o">=</span><span class="s1">&#39;numpy&#39;</span><span class="p">)}</span>
+<span class="p">)</span>
+
+<span class="n">Pm</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<section id="example-results-on-the-square-domain">
+<h3>Example: Results on the square domain<a class="headerlink" href="#example-results-on-the-square-domain" title="Link to this heading">#</a></h3>
+<p float="left">
+<table><tr>
+  <td><img src="./paraview_images/Helmholtz_Square_Real_Computed.png" width="100%" /> 
+  <td><img src="./paraview_images/Helmholtz_Square_Imag_Computed.png" width="100%" />
+</tr></table>
+</p>
+<p float="left">
+<table><tr>
+  <td><img src="./paraview_images/Helmholtz_Square_Imag_exact.png" width="90%" /> 
+  <td><img src="./paraview_images/Helmholtz_Square_Real_exact.png" width="100%" /> 
+</tr></table> 
+</p>
+<p float="left">
+<table><tr>
+  <td><img src="./paraview_images/Helmholtz_Square_Real_error.png" width="100%" />
+  <td><img src="./paraview_images/Helmholtz_Square_Imag_error.png" width="100%" />
+</tr></table> 
+</p></section>
+</section>
+</section>
+
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-building-the-domain">Step 1 : Building the domain</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-defining-the-abstract-pde-model-using-sympde">Step 2: Defining the Abstract PDE model using SymPDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-discretizing-the-domain-spaces-and-equations">Step 3: Discretizing the domain, spaces and equations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-solving-the-equation-and-computing-the-error-norms">Step 4: Solving the equation and computing the error norms</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-saving-the-results">Step 5: Saving the results</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-6-exporting-the-results-to-vtk-and-visualizing-them-with-paraview">Step 6: Exporting the results to VTK and visualizing them with Paraview</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#example-results-on-the-square-domain">Example: Results on the square domain</a></li>
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diff --git a/psydac/examples/notebooks/Poisson_non_periodic.html b/psydac/examples/notebooks/Poisson_non_periodic.html
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-building-the-domain">Step 1 : Building the domain</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-defining-the-abstract-pde-model-using-sympde">Step 2: Defining the Abstract PDE model using SymPDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-discretizing-the-domain-spaces-and-equations">Step 3: Discretizing the domain, spaces and equations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-solving-the-equation-and-computing-the-error-norms">Step 4: Solving the equation and computing the error norms</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-saving-the-results">Step 5: Saving the results</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-6-exporting-the-results-to-vtk-and-visualizing-them-with-paraview">Step 6: Exporting the results to VTK and visualizing them with Paraview</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#example-results-on-the-pretzel-domain">Example: Results on the pretzel domain</a></li>
+</ul>
+</li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="solving-poisson-s-equation-on-non-periodic-topological-domains">
+<h1>Solving Poisson’s equation on non-periodic topological domains<a class="headerlink" href="#solving-poisson-s-equation-on-non-periodic-topological-domains" title="Link to this heading">#</a></h1>
+<p>In this notebook, we will show how to solve poisson’s equation on single patch and multipatch domains. As we are running this in a notebook, everything will be run in serial and hence we are limiting ourselves to a fairly coarse discretization to avoid taking too much time. However, Psydac allows for hybrid MPI + OpenMP parallelization with barely any changed to the code. The lines that are impacted by that will be preceeded by their MPI equivalent</p>
+<section id="step-1-building-the-domain">
+<h2>Step 1 : Building the domain<a class="headerlink" href="#step-1-building-the-domain" title="Link to this heading">#</a></h2>
+<p>Psydac uses the powerful topological tools of SymPDE to build a large variety of domains. Here we show how to do a quarter annulus using a Square and a polar mapping and how to join two annulus to create a multipatch domain. Left commented is an example of a complex multipatch domain built using SymPDE.</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">Square</span><span class="p">,</span> <span class="n">PolarMapping</span>
+<span class="kn">from</span> <span class="nn">sympde.utilities.utils</span> <span class="kn">import</span> <span class="n">plot_domain</span>
+
+<span class="kn">from</span> <span class="nn">psydac.api.tests.build_domain</span> <span class="kn">import</span> <span class="n">build_pretzel</span>
+
+<span class="c1"># Define the topological geometry for each patch</span>
+<span class="n">rmin</span><span class="p">,</span> <span class="n">rmax</span> <span class="o">=</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">1.</span>
+
+<span class="c1"># First quarter annulus</span>
+<span class="n">domain_log_1</span> <span class="o">=</span> <span class="n">Square</span><span class="p">(</span><span class="s1">&#39;A_1&#39;</span><span class="p">,</span> <span class="n">bounds1</span><span class="o">=</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">),</span> <span class="n">bounds2</span><span class="o">=</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">2</span><span class="p">))</span>
+<span class="n">F_1</span> <span class="o">=</span> <span class="n">PolarMapping</span><span class="p">(</span><span class="s1">&#39;F_1&#39;</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">c1</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span> <span class="n">c2</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span> <span class="n">rmin</span><span class="o">=</span><span class="n">rmin</span><span class="p">,</span> <span class="n">rmax</span><span class="o">=</span><span class="n">rmax</span><span class="p">)</span>
+<span class="n">Omega_1</span> <span class="o">=</span> <span class="n">F_1</span><span class="p">(</span><span class="n">domain_log_1</span><span class="p">)</span>
+
+<span class="c1"># Second quarter annulus</span>
+<span class="n">domain_log_2</span> <span class="o">=</span> <span class="n">Square</span><span class="p">(</span><span class="s1">&#39;A_2&#39;</span><span class="p">,</span> <span class="n">bounds1</span><span class="o">=</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">),</span> <span class="n">bounds2</span><span class="o">=</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">3</span><span class="o">/</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">))</span>
+<span class="n">F_2</span> <span class="o">=</span> <span class="n">PolarMapping</span><span class="p">(</span><span class="s1">&#39;F_1&#39;</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">c1</span><span class="o">=</span><span class="n">rmin</span><span class="o">+</span><span class="n">rmax</span><span class="p">,</span> <span class="n">c2</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span> <span class="n">rmin</span><span class="o">=</span><span class="n">rmin</span><span class="p">,</span> <span class="n">rmax</span><span class="o">=</span><span class="n">rmax</span><span class="p">)</span>
+<span class="n">Omega_2</span> <span class="o">=</span> <span class="n">F_2</span><span class="p">(</span><span class="n">domain_log_2</span><span class="p">)</span>
+
+<span class="c1"># Join the patches</span>
+
+<span class="kn">from</span> <span class="nn">sympde</span> <span class="kn">import</span> <span class="n">Domain</span>
+<span class="n">connectivity</span> <span class="o">=</span> <span class="p">[((</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="o">-</span><span class="mi">1</span><span class="p">))]</span>
+<span class="n">patches</span> <span class="o">=</span> <span class="p">[</span><span class="n">Omega_1</span><span class="p">,</span><span class="n">Omega_2</span><span class="p">]</span>
+<span class="n">Omega</span> <span class="o">=</span> <span class="n">Domain</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">patches</span><span class="p">,</span> <span class="n">connectivity</span><span class="p">,</span> <span class="s1">&#39;domain&#39;</span><span class="p">)</span>
+
+<span class="c1"># Example of a complex multi-patch domain</span>
+<span class="c1"># Omega = build_pretzel()</span>
+
+<span class="c1"># Simple visualization of the topological domain</span>
+<span class="n">plot_domain</span><span class="p">(</span><span class="n">Omega</span><span class="p">,</span> <span class="n">draw</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">isolines</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../../../_images/caf5f27042898fa083be9e14a63a368f98c6a8bc5f505c4c9af2090d3a4c1d47.png" src="../../../_images/caf5f27042898fa083be9e14a63a368f98c6a8bc5f505c4c9af2090d3a4c1d47.png" />
+</div>
+</div>
+</section>
+<section id="step-2-defining-the-abstract-pde-model-using-sympde">
+<h2>Step 2: Defining the Abstract PDE model using SymPDE<a class="headerlink" href="#step-2-defining-the-abstract-pde-model-using-sympde" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">grad</span><span class="p">,</span> <span class="n">dot</span>
+<span class="kn">from</span> <span class="nn">sympde.calculus</span> <span class="kn">import</span> <span class="n">minus</span><span class="p">,</span> <span class="n">plus</span>
+
+<span class="kn">from</span> <span class="nn">sympde.expr.expr</span>          <span class="kn">import</span> <span class="n">LinearForm</span><span class="p">,</span> <span class="n">BilinearForm</span>
+<span class="kn">from</span> <span class="nn">sympde.expr.expr</span>          <span class="kn">import</span> <span class="n">integral</span>              
+<span class="kn">from</span> <span class="nn">sympde.expr.expr</span>          <span class="kn">import</span> <span class="n">Norm</span>                       
+<span class="kn">from</span> <span class="nn">sympde.expr</span>               <span class="kn">import</span> <span class="n">find</span><span class="p">,</span> <span class="n">EssentialBC</span>
+
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">ScalarFunctionSpace</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">elements_of</span>
+<span class="kn">from</span> <span class="nn">sympde.topology</span> <span class="kn">import</span> <span class="n">NormalVector</span>
+
+<span class="c1"># Define the abstract model to solve Poisson&#39;s equation using the manufactured solution method</span>
+<span class="n">x</span><span class="p">,</span><span class="n">y</span>       <span class="o">=</span> <span class="n">Omega</span><span class="o">.</span><span class="n">coordinates</span>
+<span class="n">solution</span>  <span class="o">=</span> <span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">+</span> <span class="n">y</span><span class="o">**</span><span class="mi">2</span>
+<span class="n">f</span>         <span class="o">=</span> <span class="o">-</span><span class="mi">4</span>
+
+<span class="n">V</span>   <span class="o">=</span> <span class="n">ScalarFunctionSpace</span><span class="p">(</span><span class="s1">&#39;V&#39;</span><span class="p">,</span> <span class="n">Omega</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
+
+<span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="o">=</span> <span class="n">elements_of</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">names</span><span class="o">=</span><span class="s1">&#39;u, v&#39;</span><span class="p">)</span>
+<span class="n">nn</span>   <span class="o">=</span> <span class="n">NormalVector</span><span class="p">(</span><span class="s1">&#39;nn&#39;</span><span class="p">)</span>
+
+<span class="n">bc</span>   <span class="o">=</span> <span class="n">EssentialBC</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">solution</span><span class="p">,</span> <span class="n">Omega</span><span class="o">.</span><span class="n">boundary</span><span class="p">)</span>
+
+<span class="n">error</span>  <span class="o">=</span> <span class="n">u</span> <span class="o">-</span> <span class="n">solution</span>
+
+<span class="n">I</span> <span class="o">=</span> <span class="n">Omega</span><span class="o">.</span><span class="n">interfaces</span>
+
+<span class="n">kappa</span>  <span class="o">=</span> <span class="mi">10</span><span class="o">**</span><span class="mi">3</span>
+
+<span class="n">expr_I</span> <span class="o">=-</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">plus</span><span class="p">(</span><span class="n">u</span><span class="p">)),</span><span class="n">nn</span><span class="p">)</span><span class="o">*</span><span class="n">minus</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>  <span class="o">+</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">minus</span><span class="p">(</span><span class="n">v</span><span class="p">)),</span><span class="n">nn</span><span class="p">)</span><span class="o">*</span><span class="n">plus</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>  <span class="o">-</span> <span class="n">kappa</span><span class="o">*</span><span class="n">plus</span><span class="p">(</span><span class="n">u</span><span class="p">)</span><span class="o">*</span><span class="n">minus</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>\
+        <span class="o">+</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">minus</span><span class="p">(</span><span class="n">u</span><span class="p">)),</span><span class="n">nn</span><span class="p">)</span><span class="o">*</span><span class="n">plus</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>  <span class="o">-</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">plus</span><span class="p">(</span><span class="n">v</span><span class="p">)),</span><span class="n">nn</span><span class="p">)</span><span class="o">*</span><span class="n">minus</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>  <span class="o">-</span> <span class="n">kappa</span><span class="o">*</span><span class="n">plus</span><span class="p">(</span><span class="n">v</span><span class="p">)</span><span class="o">*</span><span class="n">minus</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>\
+        <span class="o">-</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">minus</span><span class="p">(</span><span class="n">v</span><span class="p">)),</span><span class="n">nn</span><span class="p">)</span><span class="o">*</span><span class="n">minus</span><span class="p">(</span><span class="n">u</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">minus</span><span class="p">(</span><span class="n">u</span><span class="p">)),</span><span class="n">nn</span><span class="p">)</span><span class="o">*</span><span class="n">minus</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="o">+</span> <span class="n">kappa</span><span class="o">*</span><span class="n">minus</span><span class="p">(</span><span class="n">u</span><span class="p">)</span><span class="o">*</span><span class="n">minus</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>\
+        <span class="o">+</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">plus</span><span class="p">(</span><span class="n">v</span><span class="p">)),</span><span class="n">nn</span><span class="p">)</span><span class="o">*</span><span class="n">plus</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>   <span class="o">+</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">plus</span><span class="p">(</span><span class="n">u</span><span class="p">)),</span><span class="n">nn</span><span class="p">)</span><span class="o">*</span><span class="n">plus</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>   <span class="o">+</span> <span class="n">kappa</span><span class="o">*</span><span class="n">plus</span><span class="p">(</span><span class="n">u</span><span class="p">)</span><span class="o">*</span><span class="n">plus</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
+
+<span class="n">expr</span>   <span class="o">=</span> <span class="n">dot</span><span class="p">(</span><span class="n">grad</span><span class="p">(</span><span class="n">u</span><span class="p">),</span><span class="n">grad</span><span class="p">(</span><span class="n">v</span><span class="p">))</span>
+
+<span class="n">a</span> <span class="o">=</span> <span class="n">BilinearForm</span><span class="p">((</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span>  <span class="n">integral</span><span class="p">(</span><span class="n">Omega</span><span class="p">,</span> <span class="n">expr</span><span class="p">)</span> <span class="o">+</span> <span class="n">integral</span><span class="p">(</span><span class="n">I</span><span class="p">,</span> <span class="n">expr_I</span><span class="p">))</span>
+<span class="n">l</span> <span class="o">=</span> <span class="n">LinearForm</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">integral</span><span class="p">(</span><span class="n">Omega</span><span class="p">,</span> <span class="n">f</span><span class="o">*</span><span class="n">v</span><span class="p">))</span>
+
+<span class="n">equation</span> <span class="o">=</span> <span class="n">find</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">forall</span><span class="o">=</span><span class="n">v</span><span class="p">,</span> <span class="n">lhs</span><span class="o">=</span><span class="n">a</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">v</span><span class="p">),</span> <span class="n">rhs</span><span class="o">=</span><span class="n">l</span><span class="p">(</span><span class="n">v</span><span class="p">),</span> <span class="n">bc</span><span class="o">=</span><span class="n">bc</span><span class="p">)</span>
+
+<span class="n">l2norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">error</span><span class="p">,</span> <span class="n">Omega</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;l2&#39;</span><span class="p">)</span>
+<span class="n">h1norm</span> <span class="o">=</span> <span class="n">Norm</span><span class="p">(</span><span class="n">error</span><span class="p">,</span> <span class="n">Omega</span><span class="p">,</span> <span class="n">kind</span><span class="o">=</span><span class="s1">&#39;h1&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="step-3-discretizing-the-domain-spaces-and-equations">
+<h2>Step 3: Discretizing the domain, spaces and equations<a class="headerlink" href="#step-3-discretizing-the-domain-spaces-and-equations" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Discretize the geometry and equation</span>
+<span class="kn">from</span> <span class="nn">psydac.api.discretization</span> <span class="kn">import</span> <span class="n">discretize</span>
+<span class="kn">from</span> <span class="nn">psydac.api.settings</span> <span class="kn">import</span> <span class="n">PSYDAC_BACKENDS</span>
+
+<span class="c1"># The backends are as follows and can all be run with OpenMP by setting backend[&#39;omp&#39;] = True.</span>
+<span class="c1"># For all but the Python backend, compilation flags can be accessed and changed using backend[&#39;flags&#39;].</span>
+<span class="c1"># - Python backend: Generate and runs python files. Accessed via PSYDAC_BACKENDS[&#39;python&#39;]</span>
+<span class="c1"># - Pyccel GCC backend: Accessed via PSYDAC_BACKENDS[&#39;pyccel-gcc&#39;]</span>
+<span class="c1"># - Pyccel Intel backend: Accessed via PSYDAC_BACKENDS[&#39;pyccel-intel&#39;]</span>
+<span class="c1"># - Pyccel PGI backend: Accessed via PSYDAC_BACKENDS[&#39;pyccel-pgi&#39;]</span>
+<span class="c1"># - Pyccel Numba backend: Accessed via PSYDAC_BACKENDS[&#39;numba&#39;]</span>
+
+<span class="n">backend</span> <span class="o">=</span> <span class="n">PSYDAC_BACKENDS</span><span class="p">[</span><span class="s1">&#39;python&#39;</span><span class="p">]</span>
+
+<span class="c1"># Uncomment to use OpenMp</span>
+<span class="c1"># import os</span>
+<span class="c1"># os.environ[&#39;OMP_NUM_THREADS&#39;] = &quot;4&quot;</span>
+<span class="c1"># backend[&#39;omp&#39;] = True</span>
+
+<span class="n">ncells</span> <span class="o">=</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">]</span>
+<span class="n">degree</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
+<span class="n">periodic</span> <span class="o">=</span> <span class="p">[</span><span class="kc">False</span><span class="p">,</span> <span class="kc">False</span><span class="p">]</span>
+
+<span class="c1"># MPI version</span>
+<span class="c1"># from mpi4py import MPI</span>
+<span class="c1"># comm = MPI.COMM_WORLD</span>
+<span class="c1"># Omega_h = discretize(Omega, ncells=ncells, comm=comm)</span>
+<span class="n">Omega_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">Omega</span><span class="p">,</span> <span class="n">ncells</span><span class="o">=</span><span class="n">ncells</span><span class="p">,</span> <span class="n">periodic</span><span class="o">=</span><span class="n">periodic</span><span class="p">)</span>
+
+<span class="n">Vh</span>         <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">V</span><span class="p">,</span> <span class="n">Omega_h</span><span class="p">,</span> <span class="n">degree</span><span class="o">=</span><span class="n">degree</span><span class="p">)</span>
+<span class="n">equation_h</span> <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">equation</span><span class="p">,</span> <span class="n">Omega_h</span><span class="p">,</span> <span class="p">[</span><span class="n">Vh</span><span class="p">,</span> <span class="n">Vh</span><span class="p">],</span> <span class="n">backend</span><span class="o">=</span><span class="n">backend</span><span class="p">)</span>
+<span class="n">l2norm_h</span>   <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">l2norm</span><span class="p">,</span> <span class="n">Omega_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">,</span> <span class="n">backend</span><span class="o">=</span><span class="n">backend</span><span class="p">)</span>
+<span class="n">h1norm_h</span>   <span class="o">=</span> <span class="n">discretize</span><span class="p">(</span><span class="n">h1norm</span><span class="p">,</span> <span class="n">Omega_h</span><span class="p">,</span> <span class="n">Vh</span><span class="p">,</span> <span class="n">backend</span><span class="o">=</span><span class="n">backend</span><span class="p">)</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympy/matrices/repmatrix.py:85: SymPyDeprecationWarning: 
+
+non-Expr objects in a Matrix has been deprecated since SymPy 1.9. Use
+list of lists, TableForm or some other data structure instead. See
+https://github.com/sympy/sympy/issues/21497 for more info.
+
+  SymPyDeprecationWarning(
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="step-4-solving-the-equation-and-computing-the-error-norms">
+<h2>Step 4: Solving the equation and computing the error norms<a class="headerlink" href="#step-4-solving-the-equation-and-computing-the-error-norms" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set the solver parameters</span>
+<span class="c1"># &#39;cg&#39; -&gt; Conjugate gradient method</span>
+<span class="n">equation_h</span><span class="o">.</span><span class="n">set_solver</span><span class="p">(</span><span class="s1">&#39;cg&#39;</span><span class="p">,</span> <span class="n">info</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">tol</span><span class="o">=</span><span class="mf">1e-14</span><span class="p">)</span>
+
+<span class="kn">import</span> <span class="nn">time</span>
+
+<span class="n">t0_s</span>     <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+<span class="n">uh</span><span class="p">,</span> <span class="n">info</span> <span class="o">=</span> <span class="n">equation_h</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
+<span class="n">t1_s</span>     <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+
+
+<span class="n">t0_d</span>     <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+<span class="n">l2_error</span> <span class="o">=</span> <span class="n">l2norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+<span class="n">h1_error</span> <span class="o">=</span> <span class="n">h1norm_h</span><span class="o">.</span><span class="n">assemble</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+<span class="n">t1_d</span>     <span class="o">=</span> <span class="n">time</span><span class="o">.</span><span class="n">time</span><span class="p">()</span>
+
+<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;&gt; CG info       :: &#39;</span><span class="p">,</span><span class="n">info</span> <span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;&gt; L2 error      :: </span><span class="si">{:.2e}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">l2_error</span><span class="p">))</span>
+<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;&gt; H1 error      :: </span><span class="si">{:.2e}</span><span class="s1">&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">h1_error</span><span class="p">))</span>
+<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;&gt; Solution time :: </span><span class="si">{:.2e}</span><span class="s1">s&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">t1_s</span> <span class="o">-</span> <span class="n">t0_s</span><span class="p">))</span>
+<span class="nb">print</span><span class="p">(</span> <span class="s1">&#39;&gt; Evaluat. time :: </span><span class="si">{:.2e}</span><span class="s1">s &#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">t1_d</span> <span class="o">-</span> <span class="n">t0_d</span><span class="p">))</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>&gt; CG info       ::  {&#39;niter&#39;: 144, &#39;success&#39;: True, &#39;res_norm&#39;: 7.489105000677988e-15}
+&gt; L2 error      :: 4.72e-01
+&gt; H1 error      :: 5.11e+00
+&gt; Solution time :: 1.62e+00s
+&gt; Evaluat. time :: 6.61e-02s 
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="step-5-saving-the-results">
+<h2>Step 5: Saving the results<a class="headerlink" href="#step-5-saving-the-results" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Save the results using OutputManager</span>
+<span class="kn">from</span> <span class="nn">psydac.api.postprocessing</span> <span class="kn">import</span> <span class="n">OutputManager</span>
+<span class="kn">import</span> <span class="nn">os</span>
+
+<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="s1">&#39;results_poisson&#39;</span><span class="p">,</span> <span class="n">exist_ok</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
+
+<span class="n">Om</span> <span class="o">=</span> <span class="n">OutputManager</span><span class="p">(</span>
+    <span class="sa">f</span><span class="s1">&#39;results_poisson/space_info_</span><span class="si">{</span><span class="n">Omega</span><span class="o">.</span><span class="n">name</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">,</span>
+    <span class="sa">f</span><span class="s1">&#39;results_poisson/field_info_</span><span class="si">{</span><span class="n">Omega</span><span class="o">.</span><span class="n">name</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">,</span>
+<span class="c1"># MPI version</span>
+<span class="c1">#   comm=comm,    </span>
+<span class="p">)</span>
+
+<span class="n">Om</span><span class="o">.</span><span class="n">add_spaces</span><span class="p">(</span><span class="n">V</span><span class="o">=</span><span class="n">Vh</span><span class="p">)</span>
+<span class="n">Om</span><span class="o">.</span><span class="n">export_space_info</span><span class="p">()</span>
+
+<span class="n">Om</span><span class="o">.</span><span class="n">set_static</span><span class="p">()</span>
+<span class="n">Om</span><span class="o">.</span><span class="n">export_fields</span><span class="p">(</span><span class="n">u</span><span class="o">=</span><span class="n">uh</span><span class="p">)</span>
+
+<span class="n">Om</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+</section>
+<section id="step-6-exporting-the-results-to-vtk-and-visualizing-them-with-paraview">
+<h2>Step 6: Exporting the results to VTK and visualizing them with Paraview<a class="headerlink" href="#step-6-exporting-the-results-to-vtk-and-visualizing-them-with-paraview" title="Link to this heading">#</a></h2>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Export the results to VTK using PostProcessManager</span>
+<span class="kn">from</span> <span class="nn">psydac.api.postprocessing</span> <span class="kn">import</span> <span class="n">PostProcessManager</span>
+<span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="n">lambdify</span>
+
+<span class="n">Pm</span> <span class="o">=</span> <span class="n">PostProcessManager</span><span class="p">(</span>
+    <span class="n">domain</span><span class="o">=</span><span class="n">Omega</span><span class="p">,</span>
+    <span class="n">space_file</span><span class="o">=</span><span class="sa">f</span><span class="s1">&#39;results_poisson/space_info_</span><span class="si">{</span><span class="n">Omega</span><span class="o">.</span><span class="n">name</span><span class="si">}</span><span class="s1">.yaml&#39;</span><span class="p">,</span>
+    <span class="n">fields_file</span><span class="o">=</span><span class="sa">f</span><span class="s1">&#39;results_poisson/field_info_</span><span class="si">{</span><span class="n">Omega</span><span class="o">.</span><span class="n">name</span><span class="si">}</span><span class="s1">.h5&#39;</span><span class="p">,</span>
+<span class="c1"># MPI version</span>
+<span class="c1">#   comm=comm,    </span>
+<span class="p">)</span>
+
+<span class="n">Pm</span><span class="o">.</span><span class="n">export_to_vtk</span><span class="p">(</span>
+    <span class="sa">f</span><span class="s1">&#39;results_poisson/visu_</span><span class="si">{</span><span class="n">Omega</span><span class="o">.</span><span class="n">name</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">,</span>
+    <span class="n">grid</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
+    <span class="n">npts_per_cell</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span>
+    <span class="n">fields</span><span class="o">=</span><span class="s1">&#39;u&#39;</span><span class="p">,</span>
+    <span class="n">additional_physical_functions</span><span class="o">=</span><span class="p">{</span><span class="s1">&#39;exact_solution&#39;</span><span class="p">:</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">Omega</span><span class="o">.</span><span class="n">coordinates</span><span class="p">,</span> <span class="n">solution</span><span class="p">,</span> <span class="n">modules</span><span class="o">=</span><span class="s1">&#39;numpy&#39;</span><span class="p">)}</span>
+<span class="p">)</span>
+
+<span class="n">Pm</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+</div>
+<section id="example-results-on-the-pretzel-domain">
+<h3>Example: Results on the pretzel domain<a class="headerlink" href="#example-results-on-the-pretzel-domain" title="Link to this heading">#</a></h3>
+<p float="left">
+  <img src="./paraview_images/poisson_pretzel_u.png" width="32%" />
+  <img src="./paraview_images/poisson_pretzel_exact.png" width="32%" />
+  <img src="./paraview_images/poisson_pretzel_error.png" width="32%" />
+</p></section>
+</section>
+</section>
+
+    <script type="text/x-thebe-config">
+    {
+        requestKernel: true,
+        binderOptions: {
+            repo: "binder-examples/jupyter-stacks-datascience",
+            ref: "master",
+        },
+        codeMirrorConfig: {
+            theme: "abcdef",
+            mode: "v_psydac"
+        },
+        kernelOptions: {
+            name: "v_psydac",
+            path: "./psydac/examples/notebooks"
+        },
+        predefinedOutput: true
+    }
+    </script>
+    <script>kernelName = 'v_psydac'</script>
+
+                </article>
+              
+
+              
+              
+              
+              
+                <footer class="prev-next-footer">
+                  
+<div class="prev-next-area">
+</div>
+                </footer>
+              
+            </div>
+            
+            
+              
+                <div class="bd-sidebar-secondary bd-toc"><div class="sidebar-secondary-items sidebar-secondary__inner">
+
+
+  <div class="sidebar-secondary-item">
+  <div class="page-toc tocsection onthispage">
+    <i class="fa-solid fa-list"></i> Contents
+  </div>
+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-1-building-the-domain">Step 1 : Building the domain</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-2-defining-the-abstract-pde-model-using-sympde">Step 2: Defining the Abstract PDE model using SymPDE</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-3-discretizing-the-domain-spaces-and-equations">Step 3: Discretizing the domain, spaces and equations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-4-solving-the-equation-and-computing-the-error-norms">Step 4: Solving the equation and computing the error norms</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-5-saving-the-results">Step 5: Saving the results</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#step-6-exporting-the-results-to-vtk-and-visualizing-them-with-paraview">Step 6: Exporting the results to VTK and visualizing them with Paraview</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#example-results-on-the-pretzel-domain">Example: Results on the pretzel domain</a></li>
+</ul>
+</li>
+</ul>
+  </nav></div>
+
+</div></div>
+              
+            
+          </div>
+          <footer class="bd-footer-content">
+            
+<div class="bd-footer-content__inner container">
+  
+  <div class="footer-item">
+    
+<p class="component-author">
+By Ahmed Ratnani
+</p>
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+  
+  <div class="footer-item">
+    
+
+  <p class="copyright">
+    
+      © Copyright 2023.
+      <br/>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Linear Problems</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter3/burgers.html">1D Burgers equation</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="../chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="../chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="../chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="../chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
+</li>
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+</li>
+<li class="toctree-l1"><a class="reference internal" href="../chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter5/mhd.html">MHD</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="../chapter5/multiphysics.html">Multiphysics</a></li>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Documentation</span></p>
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+    <h1>Psydac’s outputs</h1>
+    <!-- Table of contents -->
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+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#structure">Structure</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#usage-of-class-outputmanager">Usage of class <code class="docutils literal notranslate"><span class="pre">OutputManager</span></code></a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#usage-of-class-postprocessmanager">Usage of class <code class="docutils literal notranslate"><span class="pre">PostProcessManager</span></code></a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#further-examples">Further Examples</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="psydac-s-outputs">
+<h1>Psydac’s outputs<a class="headerlink" href="#psydac-s-outputs" title="Link to this heading">#</a></h1>
+<section id="structure">
+<h2>Structure<a class="headerlink" href="#structure" title="Link to this heading">#</a></h2>
+<p>Psydac has a class meant to take care of outputing simulation results. This class, named <code class="docutils literal notranslate"><span class="pre">OuputManager</span></code> is located in <code class="docutils literal notranslate"><span class="pre">psydac/api/postprocessing.py</span></code>.
+It writes <code class="docutils literal notranslate"><span class="pre">FemSpace</span></code> related information in the Yaml syntax. The file looks like this:</p>
+<div class="highlight-yaml notranslate"><div class="highlight"><pre><span></span><span class="nt">ndim</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">2</span>
+<span class="nt">fields</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">file.h5</span><span class="w"> </span><span class="c1"># Name of the fields file</span>
+<span class="nt">patches</span><span class="p">:</span>
+<span class="p p-Indicator">-</span><span class="w"> </span><span class="nt">name</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">patch_0</span>
+<span class="w">  </span><span class="nt">breakpoints</span><span class="p">:</span>
+<span class="w">  </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">]</span>
+<span class="w">  </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">]</span>
+<span class="w">  </span><span class="nt">scalar_spaces</span><span class="p">:</span>
+<span class="w">  </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="nt">name</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">Scalar_space</span>
+<span class="w">    </span><span class="nt">ldim</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">2</span>
+<span class="w">    </span><span class="nt">kind</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">l2</span>
+<span class="w">    </span><span class="nt">dtype</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">&lt;class &#39;float&#39;&gt;</span>
+<span class="w">    </span><span class="nt">rational</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">false</span>
+<span class="w">    </span><span class="nt">periodic</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">false</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">false</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">degree</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">1</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">multiplicity</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">1</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">basis</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">B</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">B</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">knots</span><span class="p">:</span>
+<span class="w">    </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">]</span>
+<span class="w">  </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="nl">&amp;id001</span>
+<span class="w">    </span><span class="nt">name</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">Vector_space[0]</span>
+<span class="w">    </span><span class="nt">ldim</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">2</span>
+<span class="w">    </span><span class="nt">kind</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">undefined</span>
+<span class="w">    </span><span class="nt">dtype</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">&lt;class &#39;float&#39;&gt;</span>
+<span class="w">    </span><span class="nt">rational</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">false</span>
+<span class="w">    </span><span class="nt">periodic</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">false</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">false</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">degree</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">1</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">2</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">multiplicity</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">1</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">basis</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">B</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">B</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">knots</span><span class="p">:</span>
+<span class="w">    </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">]</span>
+<span class="w">  </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="nl">&amp;id002</span>
+<span class="w">    </span><span class="nt">name</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">Vector_space[1]</span>
+<span class="w">    </span><span class="nt">ldim</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">2</span>
+<span class="w">    </span><span class="nt">kind</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">undefined</span>
+<span class="w">    </span><span class="nt">dtype</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">&lt;class &#39;float&#39;&gt;</span>
+<span class="w">    </span><span class="nt">rational</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">false</span>
+<span class="w">    </span><span class="nt">periodic</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">false</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">false</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">degree</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">2</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">multiplicity</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">1</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">basis</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">B</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">B</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">knots</span><span class="p">:</span>
+<span class="w">    </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">]</span>
+<span class="w">  </span><span class="nt">vector_spaces</span><span class="p">:</span>
+<span class="w">  </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="nt">name</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">Vector_space</span>
+<span class="w">    </span><span class="nt">kind</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">hcurl</span>
+<span class="w">    </span><span class="nt">components</span><span class="p">:</span>
+<span class="w">    </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="nv">*id001</span>
+<span class="w">    </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="nv">*id002</span>
+<span class="p p-Indicator">-</span><span class="w"> </span><span class="nt">name</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">patch_1</span>
+<span class="w">  </span><span class="nt">breakpoints</span><span class="p">:</span>
+<span class="w">  </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">2.0</span><span class="p p-Indicator">]</span>
+<span class="w">  </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">]</span>
+<span class="w">  </span><span class="nt">scalar_spaces</span><span class="p">:</span>
+<span class="w">  </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="nt">name</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">Scalar_space</span>
+<span class="w">    </span><span class="nt">ldim</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">2</span>
+<span class="w">    </span><span class="nt">kind</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">l2</span>
+<span class="w">    </span><span class="nt">dtype</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">&lt;class &#39;float&#39;&gt;</span>
+<span class="w">    </span><span class="nt">rational</span><span class="p">:</span><span class="w"> </span><span class="l l-Scalar l-Scalar-Plain">false</span>
+<span class="w">    </span><span class="nt">periodic</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">false</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">false</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">degree</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">1</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">multiplicity</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">1</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">basis</span><span class="p">:</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">B</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">B</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="nt">knots</span><span class="p">:</span>
+<span class="w">    </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">2.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">2.0</span><span class="p p-Indicator">]</span>
+<span class="w">    </span><span class="p p-Indicator">-</span><span class="w"> </span><span class="p p-Indicator">[</span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.25</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.5</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">0.75</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">,</span><span class="w"> </span><span class="nv">1.0</span><span class="p p-Indicator">]</span>
+</pre></div>
+</div>
+<p>The field coefficients are saved to the <code class="docutils literal notranslate"><span class="pre">HDF5</span></code> format in the following manner :</p>
+<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span>file.h5
+<span class="w">    </span>attribute:<span class="w"> </span>spaces<span class="w"> </span><span class="c1"># name of the aforementioned Yaml file</span>
+<span class="w">    </span>static/
+<span class="w">        </span>scalar_space_1/
+<span class="w">            </span>field_s1_1
+<span class="w">            </span>field_s1_2
+<span class="w">            </span>....
+<span class="w">            </span>field_s1_n
+<span class="w">        </span>vector_space_1_<span class="o">[</span><span class="m">0</span><span class="o">]</span>/
+<span class="w">            </span>attribute:<span class="w"> </span>parent_space<span class="w"> </span><span class="c1"># &#39;vector_space_1&#39;</span>
+<span class="w">            </span>field_v1_1_<span class="o">[</span><span class="m">0</span><span class="o">]</span>
+<span class="w">                </span>attribute:<span class="w"> </span>parent_field<span class="w"> </span><span class="c1"># &#39;field_v1_1&#39;</span>
+<span class="w">        </span>vector_space_1_<span class="o">[</span><span class="m">1</span><span class="o">]</span>/
+<span class="w">            </span>attribute:<span class="w"> </span>parent_space<span class="w"> </span><span class="c1"># &#39;vector_space_1&#39;</span>
+<span class="w">            </span>field_v1_1_<span class="o">[</span><span class="m">1</span><span class="o">]</span>
+<span class="w">                </span>attribute:<span class="w"> </span>parent_field<span class="w"> </span><span class="c1"># &#39;field_v1_1&#39;</span>
+<span class="w">        </span>...
+<span class="w">    </span>snapshot_1/
+<span class="w">        </span>attribute:<span class="w"> </span>t
+<span class="w">        </span>attribute:<span class="w"> </span>ts
+<span class="w">        </span>space_1/
+<span class="w">        </span>...
+<span class="w">        </span>space_n/
+<span class="w">    </span>...
+<span class="w">    </span>snapshot_n/
+</pre></div>
+</div>
+<p>In addition to that, Psydac also features the <code class="docutils literal notranslate"><span class="pre">PostProcessManager</span></code> class to read those files, recreate all the <code class="docutils literal notranslate"><span class="pre">FemSpace</span></code> and <code class="docutils literal notranslate"><span class="pre">FemField</span></code> objects and export them to <code class="docutils literal notranslate"><span class="pre">VTK</span></code>.</p>
+</section>
+<section id="usage-of-class-outputmanager">
+<h2>Usage of class <code class="docutils literal notranslate"><span class="pre">OutputManager</span></code><a class="headerlink" href="#usage-of-class-outputmanager" title="Link to this heading">#</a></h2>
+<p>An instance of the <code class="docutils literal notranslate"><span class="pre">OutputManager</span></code> class is created at the beginning of the simulation, by specifying the following:</p>
+<ol class="arabic simple">
+<li><p>The name of the YAML file (e.g. <code class="docutils literal notranslate"><span class="pre">spaces.yaml</span></code>) where the information about all FEM spaces will be written, and</p></li>
+<li><p>The name of the HDF5 file (e.g. <code class="docutils literal notranslate"><span class="pre">fields.h5</span></code>) where the coefficients of all FEM fields will be written.</p></li>
+</ol>
+<p>References to the available FEM spaces are given to the OutputManager object through the <code class="docutils literal notranslate"><span class="pre">add_spaces(**kwargs)</span></code> method, and the corresponding YAML file is created upon calling the method <code class="docutils literal notranslate"><span class="pre">export_space_info()</span></code>. In order to inform the OutputManager object that the next fields to be exported are time-independent, the user should call the <code class="docutils literal notranslate"><span class="pre">set_static()</span></code> method. In the case of time-dependent fields, the user should prepare a time snapshot (which is defined for a specific integer time step <code class="docutils literal notranslate"><span class="pre">ts</span></code> and time value <code class="docutils literal notranslate"><span class="pre">t</span></code>) by calling the method <code class="docutils literal notranslate"><span class="pre">add_snapshot(t,</span> <span class="pre">ts)</span></code>. In both cases the fields are exported to the HDF5 file through a call to the method <code class="docutils literal notranslate"><span class="pre">export_fields(**kwargs)</span></code>. Here is a usage example:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># SymPDE Layer</span>
+<span class="c1"># Discretization</span>
+<span class="c1"># V0h and V1h are discretized SymPDE Space</span>
+<span class="c1"># u0 and u1 are FemFields belonging to either of those spaces</span>
+<span class="n">output_m</span> <span class="o">=</span> <span class="n">OutputManager</span><span class="p">(</span><span class="s1">&#39;spaces.yml&#39;</span><span class="p">,</span> <span class="s1">&#39;fields.h5&#39;</span><span class="p">)</span>
+
+<span class="n">output_m</span><span class="o">.</span><span class="n">add_spaces</span><span class="p">(</span><span class="n">V0</span><span class="o">=</span><span class="n">V0h</span><span class="p">,</span> <span class="n">V1</span><span class="o">=</span><span class="n">V1h</span><span class="p">)</span>
+<span class="n">output_m</span><span class="o">.</span><span class="n">export_space_info</span><span class="p">()</span> <span class="c1"># Writes the space information to Yaml</span>
+
+<span class="n">output_m</span><span class="o">.</span><span class="n">set_static</span><span class="p">()</span> <span class="c1"># Tells the object to save in /static/</span>
+<span class="n">output_m</span><span class="o">.</span><span class="n">export_fields</span><span class="p">(</span><span class="n">u0_static</span><span class="o">=</span><span class="n">u0</span><span class="p">,</span> <span class="n">u1_static</span><span class="o">=</span><span class="n">u1</span><span class="p">)</span> <span class="c1"># Actually does the saving</span>
+
+<span class="n">output_m</span><span class="o">.</span><span class="n">add_snapshot</span><span class="p">(</span><span class="n">t</span><span class="o">=</span><span class="mf">0.</span><span class="p">,</span> <span class="n">ts</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
+<span class="c1"># The line above tells the object to:</span>
+<span class="c1"># 1. create the group snapshot_x with attribute t and ts</span>
+<span class="c1"># 2. save in this snapshot</span>
+<span class="n">output_m</span><span class="o">.</span><span class="n">export_fields</span><span class="p">(</span><span class="n">u0</span><span class="o">=</span><span class="n">u0</span><span class="p">,</span> <span class="n">u1</span><span class="o">=</span><span class="n">u1</span><span class="p">)</span>
+</pre></div>
+</div>
+</section>
+<section id="usage-of-class-postprocessmanager">
+<h2>Usage of class <code class="docutils literal notranslate"><span class="pre">PostProcessManager</span></code><a class="headerlink" href="#usage-of-class-postprocessmanager" title="Link to this heading">#</a></h2>
+<p>Typically the <code class="docutils literal notranslate"><span class="pre">PostProcessManager</span></code> class is used in a separate post-processing script, which is run after the simulation has finished. In essence it evaluates the FEM fields over a uniform grid (applying the appropriate push-forward operations) and exports the values to a VTK file (or a sequence of files in the case of a time series). An instance of the <code class="docutils literal notranslate"><span class="pre">PostProcessManager</span></code> class is created by specifying the following:</p>
+<ol class="arabic simple">
+<li><p>The name of the geometry file (in HDF5 format) which defines the geometry or the topological domain from which the geometry is derived.</p></li>
+<li><p>The name of the YAML file that contains the information about the FEM spaces</p></li>
+<li><p>The name of the HDF5 file that contains the coefficients of all the FEM fields</p></li>
+</ol>
+<p>In order to export the fields to a VTK file, the user needs to call the method <code class="docutils literal notranslate"><span class="pre">export_to_vtk(base_name,</span> <span class="pre">grid,</span> <span class="pre">npts_per_cell,</span> <span class="pre">snapshots,</span> <span class="pre">fields)</span></code>, where:</p>
+<ol class="arabic simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">base_name</span></code> is the base name for the VTK output files.</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">grid</span></code> is either a user specified evaluation grid or <code class="docutils literal notranslate"><span class="pre">None</span></code>.</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">npts_per_cell</span></code> specifies the refinement in the case of a uniform grid.</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">snapshots</span></code> specifies which time snapshots should be extracted from the HDF5 file (<code class="docutils literal notranslate"><span class="pre">none</span></code> in the case of static fields)</p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">fields</span></code> is a tuple  of <code class="docutils literal notranslate"><span class="pre">h5_field_name</span></code>.</p></li>
+</ol>
+<p>Here is a usage example:</p>
+<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="c1"># geometry.h5 is where the domain comes from. See PostProcessManager&#39;s docstring for more information</span>
+<span class="n">post</span> <span class="o">=</span> <span class="n">PostProcessManager</span><span class="p">(</span><span class="n">geometry_file</span><span class="o">=</span><span class="s1">&#39;geometry.h5&#39;</span><span class="p">,</span> <span class="n">space_file</span><span class="o">=</span><span class="s1">&#39;spaces.yml&#39;</span><span class="p">,</span> <span class="n">fields_file</span><span class="o">=</span><span class="s1">&#39;fields.h5&#39;</span><span class="p">)</span>
+
+<span class="c1"># See PostProcessManager.export_to_vtk&#39;s and TensorFemSpace.eval_fields&#39; docstrings for more information</span>
+<span class="n">post</span><span class="o">.</span><span class="n">export_to_vtk</span><span class="p">(</span><span class="s1">&#39;filename_vtk&#39;</span><span class="p">,</span> <span class="n">grid</span><span class="o">=</span><span class="n">grid</span><span class="p">,</span> <span class="n">npts_per_cell</span><span class="o">=</span><span class="n">npts_per_cell</span><span class="p">,</span> <span class="n">snapshots</span><span class="o">=</span><span class="s1">&#39;all&#39;</span><span class="p">,</span> <span class="n">fields</span> <span class="o">=</span> <span class="p">(</span><span class="s1">&#39;u0&#39;</span><span class="p">,</span> <span class="s1">&#39;u1&#39;</span><span class="p">))</span>
+</pre></div>
+</div>
+</section>
+<section id="further-examples">
+<h2>Further Examples<a class="headerlink" href="#further-examples" title="Link to this heading">#</a></h2>
+<p>Further examples are present in the following files:</p>
+<ul class="simple">
+<li><p><code class="docutils literal notranslate"><span class="pre">examples/poisson_3d_target_torus.py</span></code></p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">examples/sample_multipatch_parallel.py</span></code></p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">examples/notebooks/Poisson_non_periodic.ipynb</span></code></p></li>
+<li><p><code class="docutils literal notranslate"><span class="pre">psydac/api/tests/test_postprocessing.py</span></code></p></li>
+</ul>
+</section>
+</section>
+
+    <script type="text/x-thebe-config">
+    {
+        requestKernel: true,
+        binderOptions: {
+            repo: "binder-examples/jupyter-stacks-datascience",
+            ref: "master",
+        },
+        codeMirrorConfig: {
+            theme: "abcdef",
+            mode: "python"
+        },
+        kernelOptions: {
+            name: "python3",
+            path: "./psydac"
+        },
+        predefinedOutput: true
+    }
+    </script>
+    <script>kernelName = 'python3'</script>
+
+                </article>
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+    <i class="fa-solid fa-list"></i> Contents
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+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#structure">Structure</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#usage-of-class-outputmanager">Usage of class <code class="docutils literal notranslate"><span class="pre">OutputManager</span></code></a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#usage-of-class-postprocessmanager">Usage of class <code class="docutils literal notranslate"><span class="pre">PostProcessManager</span></code></a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#further-examples">Further Examples</a></li>
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+<ul class="nav bd-sidenav">
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+<li class="toctree-l1"><a class="reference internal" href="chapter2/biharmonic.html">The Biharmonic problem</a></li>
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+<li class="toctree-l2"><a class="reference internal" href="chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
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+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
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+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
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+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l2"><a class="reference internal" href="chapter5/compressible_flow_in_a_nozzle.html">Compressible Flow in a Nozzle</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/convection-diffusion_equation_in_a_channel.html">Convection-Diffusion Equation in a Channel</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/free_surface_flow.html">Free Surface Flow (Navier-Stokes with Free Surface)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/heat_conduction_in_a_solid.html">Heat Conduction in a Solid</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/incompressible_flow_past_a_cylinder.html">Incompressible Flow Past a Cylinder</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/magnetohydrodynamics_flow.html">Magnetohydrodynamics (MHD) Flow</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/particle-laden_flow.html">Particle-Laden Flow (Lagrangian-Eulerian Approach)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/stokes_flow_in_a_lid-driven_cavity.html">Stokes Flow in a Lid-Driven Cavity</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/two-phase_flow.html">Two-Phase Flow (Immersed Boundary Method)</a></li>
+<li class="toctree-l2 has-children"><a class="reference internal" href="chapter5/non-newtonian-fluids.html">Non-Newtonian Fluids</a><input class="toctree-checkbox" id="toctree-checkbox-12" name="toctree-checkbox-12" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-12"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+<li class="toctree-l3"><a class="reference internal" href="chapter5/bingham_plastic_flow_in_a_pipe.html">Bingham Plastic Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/oldroyd-b_fluid_flow_in_a_channel.html">Oldroyd-B Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/herschel-bulkley_fluid_flow_in_a_pipe.html">Herschel-Bulkley Fluid Flow in a Pipe</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/cross_power_law_fluid_flow_in_a_channel.html">Cross Power Law Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/casson_fluid_flow_in_a_channel.html">Casson Fluid Flow in a Channel</a></li>
+<li class="toctree-l3"><a class="reference internal" href="chapter5/papanastasiou_fluid_flow_in_a_channel.html">Papanastasiou Fluid Flow in a Channel</a></li>
+</ul>
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+<li class="toctree-l1"><a class="reference internal" href="chapter5/fsi.html">Fluid-Structure Interaction</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="chapter5/mhd.html">MHD</a></li>
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+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/psydac.utilities.vtk.html">psydac.utilities.vtk</a></li>
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+     <tr>
+       <td><img src="_static/minus.png" class="toggler"
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+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.cart.html#module-psydac.ddm.cart"><code class="xref">psydac.ddm.cart</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.interface_data_exchanger.html#module-psydac.ddm.interface_data_exchanger"><code class="xref">psydac.ddm.interface_data_exchanger</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.nonblocking_data_exchanger.html#module-psydac.ddm.nonblocking_data_exchanger"><code class="xref">psydac.ddm.nonblocking_data_exchanger</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.partition.html#module-psydac.ddm.partition"><code class="xref">psydac.ddm.partition</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.petsc.html#module-psydac.ddm.petsc"><code class="xref">psydac.ddm.petsc</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.ddm.tests.html#module-psydac.ddm.tests"><code class="xref">psydac.ddm.tests</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.ddm.tests.test_cart_1d.html#module-psydac.ddm.tests.test_cart_1d"><code class="xref">psydac.ddm.tests.test_cart_1d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.ddm.tests.test_cart_2d.html#module-psydac.ddm.tests.test_cart_2d"><code class="xref">psydac.ddm.tests.test_cart_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.ddm.tests.test_cart_3d.html#module-psydac.ddm.tests.test_cart_3d"><code class="xref">psydac.ddm.tests.test_cart_3d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.ddm.tests.test_multicart_2d.html#module-psydac.ddm.tests.test_multicart_2d"><code class="xref">psydac.ddm.tests.test_multicart_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.ddm.tests.test_partition.html#module-psydac.ddm.tests.test_partition"><code class="xref">psydac.ddm.tests.test_partition</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.ddm.utilities.html#module-psydac.ddm.utilities"><code class="xref">psydac.ddm.utilities</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.html#module-psydac.feec"><code class="xref">psydac.feec</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.derivatives.html#module-psydac.feec.derivatives"><code class="xref">psydac.feec.derivatives</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.global_projectors.html#module-psydac.feec.global_projectors"><code class="xref">psydac.feec.global_projectors</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.multipatch.html#module-psydac.feec.multipatch"><code class="xref">psydac.feec.multipatch</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.api.html#module-psydac.feec.multipatch.api"><code class="xref">psydac.feec.multipatch.api</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.multipatch.examples.html#module-psydac.feec.multipatch.examples"><code class="xref">psydac.feec.multipatch.examples</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.multipatch.examples.h1_source_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.h1_source_pbms_conga_2d"><code class="xref">psydac.feec.multipatch.examples.h1_source_pbms_conga_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d"><code class="xref">psydac.feec.multipatch.examples.hcurl_eigen_pbms_conga_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d"><code class="xref">psydac.feec.multipatch.examples.hcurl_source_pbms_conga_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d.html#module-psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d"><code class="xref">psydac.feec.multipatch.examples.mixed_source_pbms_conga_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.multipatch.examples.ppc_test_cases.html#module-psydac.feec.multipatch.examples.ppc_test_cases"><code class="xref">psydac.feec.multipatch.examples.ppc_test_cases</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.fem_linear_operators.html#module-psydac.feec.multipatch.fem_linear_operators"><code class="xref">psydac.feec.multipatch.fem_linear_operators</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.multipatch.multipatch_domain_utilities.html#module-psydac.feec.multipatch.multipatch_domain_utilities"><code class="xref">psydac.feec.multipatch.multipatch_domain_utilities</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.operators.html#module-psydac.feec.multipatch.operators"><code class="xref">psydac.feec.multipatch.operators</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.multipatch.plotting_utilities.html#module-psydac.feec.multipatch.plotting_utilities"><code class="xref">psydac.feec.multipatch.plotting_utilities</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.multipatch.utilities.html#module-psydac.feec.multipatch.utilities"><code class="xref">psydac.feec.multipatch.utilities</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.multipatch.utils_conga_2d.html#module-psydac.feec.multipatch.utils_conga_2d"><code class="xref">psydac.feec.multipatch.utils_conga_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.pull_push.html#module-psydac.feec.pull_push"><code class="xref">psydac.feec.pull_push</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.feec.pushforward.html#module-psydac.feec.pushforward"><code class="xref">psydac.feec.pushforward</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.tests.html#module-psydac.feec.tests"><code class="xref">psydac.feec.tests</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.tests.test_axis_projection.html#module-psydac.feec.tests.test_axis_projection"><code class="xref">psydac.feec.tests.test_axis_projection</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.tests.test_commuting_projections.html#module-psydac.feec.tests.test_commuting_projections"><code class="xref">psydac.feec.tests.test_commuting_projections</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.tests.test_commuting_projections_dual.html#module-psydac.feec.tests.test_commuting_projections_dual"><code class="xref">psydac.feec.tests.test_commuting_projections_dual</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.tests.test_differentiation_matrices.html#module-psydac.feec.tests.test_differentiation_matrices"><code class="xref">psydac.feec.tests.test_differentiation_matrices</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.tests.test_global_projectors.html#module-psydac.feec.tests.test_global_projectors"><code class="xref">psydac.feec.tests.test_global_projectors</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.feec.tests.test_projections_parallel.html#module-psydac.feec.tests.test_projections_parallel"><code class="xref">psydac.feec.tests.test_projections_parallel</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.html#module-psydac.fem"><code class="xref">psydac.fem</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.basic.html#module-psydac.fem.basic"><code class="xref">psydac.fem.basic</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.context.html#module-psydac.fem.context"><code class="xref">psydac.fem.context</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.grid.html#module-psydac.fem.grid"><code class="xref">psydac.fem.grid</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.partitioning.html#module-psydac.fem.partitioning"><code class="xref">psydac.fem.partitioning</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.projectors.html#module-psydac.fem.projectors"><code class="xref">psydac.fem.projectors</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.splines.html#module-psydac.fem.splines"><code class="xref">psydac.fem.splines</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.tensor.html#module-psydac.fem.tensor"><code class="xref">psydac.fem.tensor</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.tests.html#module-psydac.fem.tests"><code class="xref">psydac.fem.tests</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.tests.analytical_profiles_1d.html#module-psydac.fem.tests.analytical_profiles_1d"><code class="xref">psydac.fem.tests.analytical_profiles_1d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.tests.analytical_profiles_base.html#module-psydac.fem.tests.analytical_profiles_base"><code class="xref">psydac.fem.tests.analytical_profiles_base</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.tests.splines_error_bounds.html#module-psydac.fem.tests.splines_error_bounds"><code class="xref">psydac.fem.tests.splines_error_bounds</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.tests.test_eval_fields_parallel.html#module-psydac.fem.tests.test_eval_fields_parallel"><code class="xref">psydac.fem.tests.test_eval_fields_parallel</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.tests.test_product.html#module-psydac.fem.tests.test_product"><code class="xref">psydac.fem.tests.test_product</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.tests.test_spline_histopolation.html#module-psydac.fem.tests.test_spline_histopolation"><code class="xref">psydac.fem.tests.test_spline_histopolation</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.tests.test_spline_interpolation.html#module-psydac.fem.tests.test_spline_interpolation"><code class="xref">psydac.fem.tests.test_spline_interpolation</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.tests.test_splines.html#module-psydac.fem.tests.test_splines"><code class="xref">psydac.fem.tests.test_splines</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.tests.test_splines_par.html#module-psydac.fem.tests.test_splines_par"><code class="xref">psydac.fem.tests.test_splines_par</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.fem.tests.utilities.html#module-psydac.fem.tests.utilities"><code class="xref">psydac.fem.tests.utilities</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.fem.vector.html#module-psydac.fem.vector"><code class="xref">psydac.fem.vector</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.linalg.html#module-psydac.linalg"><code class="xref">psydac.linalg</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.basic.html#module-psydac.linalg.basic"><code class="xref">psydac.linalg.basic</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.block.html#module-psydac.linalg.block"><code class="xref">psydac.linalg.block</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.direct_solvers.html#module-psydac.linalg.direct_solvers"><code class="xref">psydac.linalg.direct_solvers</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.fft.html#module-psydac.linalg.fft"><code class="xref">psydac.linalg.fft</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.linalg.kernels.html#module-psydac.linalg.kernels"><code class="xref">psydac.linalg.kernels</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
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+       <td>&#160;&#160;&#160;
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+       <td></td>
+       <td>&#160;&#160;&#160;
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+     <tr class="cg-1">
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+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
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+       <em></em></td></tr>
+     <tr class="cg-1">
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+       <td>&#160;&#160;&#160;
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+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.polar.html#module-psydac.polar"><code class="xref">psydac.polar</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.polar.c1_cart.html#module-psydac.polar.c1_cart"><code class="xref">psydac.polar.c1_cart</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.polar.c1_linops.html#module-psydac.polar.c1_linops"><code class="xref">psydac.polar.c1_linops</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.polar.c1_projections.html#module-psydac.polar.c1_projections"><code class="xref">psydac.polar.c1_projections</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.polar.c1_spaces.html#module-psydac.polar.c1_spaces"><code class="xref">psydac.polar.c1_spaces</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.polar.dense.html#module-psydac.polar.dense"><code class="xref">psydac.polar.dense</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.polar.tests.html#module-psydac.polar.tests"><code class="xref">psydac.polar.tests</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.polar.tests.test_c1_linops.html#module-psydac.polar.tests.test_c1_linops"><code class="xref">psydac.polar.tests.test_c1_linops</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.polar.tests.test_c1_projections.html#module-psydac.polar.tests.test_c1_projections"><code class="xref">psydac.polar.tests.test_c1_projections</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.html#module-psydac.pyccel"><code class="xref">psydac.pyccel</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.html#module-psydac.pyccel.ast"><code class="xref">psydac.pyccel.ast</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.basic.html#module-psydac.pyccel.ast.basic"><code class="xref">psydac.pyccel.ast.basic</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.builtins.html#module-psydac.pyccel.ast.builtins"><code class="xref">psydac.pyccel.ast.builtins</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.core.html#module-psydac.pyccel.ast.core"><code class="xref">psydac.pyccel.ast.core</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.datatypes.html#module-psydac.pyccel.ast.datatypes"><code class="xref">psydac.pyccel.ast.datatypes</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.functionalexpr.html#module-psydac.pyccel.ast.functionalexpr"><code class="xref">psydac.pyccel.ast.functionalexpr</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.mathext.html#module-psydac.pyccel.ast.mathext"><code class="xref">psydac.pyccel.ast.mathext</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.numbers.html#module-psydac.pyccel.ast.numbers"><code class="xref">psydac.pyccel.ast.numbers</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.numpyext.html#module-psydac.pyccel.ast.numpyext"><code class="xref">psydac.pyccel.ast.numpyext</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.singleton.html#module-psydac.pyccel.ast.singleton"><code class="xref">psydac.pyccel.ast.singleton</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.type_inference.html#module-psydac.pyccel.ast.type_inference"><code class="xref">psydac.pyccel.ast.type_inference</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.ast.utilities.html#module-psydac.pyccel.ast.utilities"><code class="xref">psydac.pyccel.ast.utilities</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.codegen.html#module-psydac.pyccel.codegen"><code class="xref">psydac.pyccel.codegen</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.codegen.printing.html#module-psydac.pyccel.codegen.printing"><code class="xref">psydac.pyccel.codegen.printing</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.codegen.printing.pycode.html#module-psydac.pyccel.codegen.printing.pycode"><code class="xref">psydac.pyccel.codegen.printing.pycode</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.errors.html#module-psydac.pyccel.errors"><code class="xref">psydac.pyccel.errors</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.errors.errors.html#module-psydac.pyccel.errors.errors"><code class="xref">psydac.pyccel.errors.errors</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.errors.messages.html#module-psydac.pyccel.errors.messages"><code class="xref">psydac.pyccel.errors.messages</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.symbolic.html#module-psydac.pyccel.symbolic"><code class="xref">psydac.pyccel.symbolic</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.pyccel.symbolic.lambdify.html#module-psydac.pyccel.symbolic.lambdify"><code class="xref">psydac.pyccel.symbolic.lambdify</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/psydac.utilities.html#module-psydac.utilities"><code class="xref">psydac.utilities</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.utilities.quadratures.html#module-psydac.utilities.quadratures"><code class="xref">psydac.utilities.quadratures</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.utilities.utils.html#module-psydac.utilities.utils"><code class="xref">psydac.utilities.utils</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-1">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="psydac/docs/source/modules/STUBDIR/psydac.utilities.vtk.html#module-psydac.utilities.vtk"><code class="xref">psydac.utilities.vtk</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="pcap"><td></td><td>&#160;</td><td></td></tr>
+     <tr class="cap" id="cap-s"><td></td><td>
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+     <tr>
+       <td><img src="_static/minus.png" class="toggler"
+              id="toggle-2" style="display: none" alt="-" /></td>
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+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.calculus.html#module-sympde.calculus"><code class="xref">sympde.calculus</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.core.html#module-sympde.core"><code class="xref">sympde.core</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.core.algebra.html#module-sympde.core.algebra"><code class="xref">sympde.core.algebra</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.core.basic.html#module-sympde.core.basic"><code class="xref">sympde.core.basic</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.core.tests.html#module-sympde.core.tests"><code class="xref">sympde.core.tests</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.core.tests.test_utils.html#module-sympde.core.tests.test_utils"><code class="xref">sympde.core.tests.test_utils</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.core.utils.html#module-sympde.core.utils"><code class="xref">sympde.core.utils</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.html#module-sympde.expr"><code class="xref">sympde.expr</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.basic.html#module-sympde.expr.basic"><code class="xref">sympde.expr.basic</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.equation.html#module-sympde.expr.equation"><code class="xref">sympde.expr.equation</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.errors.html#module-sympde.expr.errors"><code class="xref">sympde.expr.errors</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.evaluation.html#module-sympde.expr.evaluation"><code class="xref">sympde.expr.evaluation</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.expr.html#module-sympde.expr.expr"><code class="xref">sympde.expr.expr</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.tests.html#module-sympde.expr.tests"><code class="xref">sympde.expr.tests</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.tests.test_equation_2d.html#module-sympde.expr.tests.test_equation_2d"><code class="xref">sympde.expr.tests.test_equation_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.tests.test_essential_bc.html#module-sympde.expr.tests.test_essential_bc"><code class="xref">sympde.expr.tests.test_essential_bc</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.tests.test_expr.html#module-sympde.expr.tests.test_expr"><code class="xref">sympde.expr.tests.test_expr</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.tests.test_find_2d.html#module-sympde.expr.tests.test_find_2d"><code class="xref">sympde.expr.tests.test_find_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.tests.test_newton_2d.html#module-sympde.expr.tests.test_newton_2d"><code class="xref">sympde.expr.tests.test_newton_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.expr.tests.test_tensor_2d.html#module-sympde.expr.tests.test_tensor_2d"><code class="xref">sympde.expr.tests.test_tensor_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.exterior.html#module-sympde.exterior"><code class="xref">sympde.exterior</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.exterior.calculus.html#module-sympde.exterior.calculus"><code class="xref">sympde.exterior.calculus</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.exterior.compiler.html#module-sympde.exterior.compiler"><code class="xref">sympde.exterior.compiler</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.exterior.datatype.html#module-sympde.exterior.datatype"><code class="xref">sympde.exterior.datatype</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.exterior.form.html#module-sympde.exterior.form"><code class="xref">sympde.exterior.form</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.exterior.inference.html#module-sympde.exterior.inference"><code class="xref">sympde.exterior.inference</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.exterior.tests.html#module-sympde.exterior.tests"><code class="xref">sympde.exterior.tests</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.exterior.tests.test_compiler.html#module-sympde.exterior.tests.test_compiler"><code class="xref">sympde.exterior.tests.test_compiler</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.exterior.tests.test_datatype.html#module-sympde.exterior.tests.test_datatype"><code class="xref">sympde.exterior.tests.test_datatype</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.exterior.tests.test_exterior.html#module-sympde.exterior.tests.test_exterior"><code class="xref">sympde.exterior.tests.test_exterior</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.exterior.tests.test_inference.html#module-sympde.exterior.tests.test_inference"><code class="xref">sympde.exterior.tests.test_inference</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.printing.html#module-sympde.printing"><code class="xref">sympde.printing</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.printing.tests.html#module-sympde.printing.tests"><code class="xref">sympde.printing.tests</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.printing.tests.test_latex.html#module-sympde.printing.tests.test_latex"><code class="xref">sympde.printing.tests.test_latex</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.html#module-sympde.topology"><code class="xref">sympde.topology</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.analytical_mapping.html#module-sympde.topology.analytical_mapping"><code class="xref">sympde.topology.analytical_mapping</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.basic.html#module-sympde.topology.basic"><code class="xref">sympde.topology.basic</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.callable_mapping.html#module-sympde.topology.callable_mapping"><code class="xref">sympde.topology.callable_mapping</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.datatype.html#module-sympde.topology.datatype"><code class="xref">sympde.topology.datatype</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.derivatives.html#module-sympde.topology.derivatives"><code class="xref">sympde.topology.derivatives</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.domain.html#module-sympde.topology.domain"><code class="xref">sympde.topology.domain</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.mapping.html#module-sympde.topology.mapping"><code class="xref">sympde.topology.mapping</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.measure.html#module-sympde.topology.measure"><code class="xref">sympde.topology.measure</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.space.html#module-sympde.topology.space"><code class="xref">sympde.topology.space</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.tests.html#module-sympde.topology.tests"><code class="xref">sympde.topology.tests</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.tests.test_callable_mapping.html#module-sympde.topology.tests.test_callable_mapping"><code class="xref">sympde.topology.tests.test_callable_mapping</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.tests.test_derivatives.html#module-sympde.topology.tests.test_derivatives"><code class="xref">sympde.topology.tests.test_derivatives</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.tests.test_field_2d.html#module-sympde.topology.tests.test_field_2d"><code class="xref">sympde.topology.tests.test_field_2d</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.tests.test_gallery.html#module-sympde.topology.tests.test_gallery"><code class="xref">sympde.topology.tests.test_gallery</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.tests.test_logical_expr.html#module-sympde.topology.tests.test_logical_expr"><code class="xref">sympde.topology.tests.test_logical_expr</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.tests.test_mapping.html#module-sympde.topology.tests.test_mapping"><code class="xref">sympde.topology.tests.test_mapping</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.tests.test_space.html#module-sympde.topology.tests.test_space"><code class="xref">sympde.topology.tests.test_space</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.tests.test_symbolic_expr.html#module-sympde.topology.tests.test_symbolic_expr"><code class="xref">sympde.topology.tests.test_symbolic_expr</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.topology.tests.test_topology.html#module-sympde.topology.tests.test_topology"><code class="xref">sympde.topology.tests.test_topology</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.utilities.html#module-sympde.utilities"><code class="xref">sympde.utilities</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.utilities.tests.html#module-sympde.utilities.tests"><code class="xref">sympde.utilities.tests</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.utilities.tests.test_utilities.html#module-sympde.utilities.tests.test_utilities"><code class="xref">sympde.utilities.tests.test_utilities</code></a></td><td>
+       <em></em></td></tr>
+     <tr class="cg-2">
+       <td></td>
+       <td>&#160;&#160;&#160;
+       <a href="documentation/_autosummary/sympde.utilities.utils.html#module-sympde.utilities.utils"><code class="xref">sympde.utilities.utils</code></a></td><td>
+       <em></em></td></tr>
+   </table>
+
+
+                </article>
+              
+
+              
+              
+              
+              
+                <footer class="prev-next-footer">
+                  
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+          </div>
+          <footer class="bd-footer-content">
+            
+<div class="bd-footer-content__inner container">
+  
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+    
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+</p>
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+    
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+    
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\ No newline at end of file
diff --git a/reports/chapter2/elliptic-general-form.err.log b/reports/chapter2/elliptic-general-form.err.log
new file mode 100644
index 0000000..f105e34
--- /dev/null
+++ b/reports/chapter2/elliptic-general-form.err.log
@@ -0,0 +1,56 @@
+Traceback (most recent call last):
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/jupyter_cache/executors/utils.py", line 58, in single_nb_execution
+    executenb(
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 1314, in execute
+    return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/jupyter_core/utils/__init__.py", line 165, in wrapped
+    return loop.run_until_complete(inner)
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/asyncio/base_events.py", line 647, in run_until_complete
+    return future.result()
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 709, in async_execute
+    await self.async_execute_cell(
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 1062, in async_execute_cell
+    await self._check_raise_for_error(cell, cell_index, exec_reply)
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 918, in _check_raise_for_error
+    raise CellExecutionError.from_cell_and_msg(cell, exec_reply_content)
+nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
+------------------
+from sympde.expr import BilinearForm, LinearForm, integral
+from sympde.expr     import find, EssentialBC, Norm, SemiNorm
+from sympde.topology import ScalarFunctionSpace, Square, element_of
+from sympde.calculus import grad, dot, div
+from sympde.core import Vector, Matrix
+
+from sympy import pi, sin
+
+from psydac.api.discretization import discretize
+
+domain = Square()
+
+V = ScalarFunctionSpace('V', domain)
+
+x,y = domain.coordinates
+
+u,v = [element_of(V, name=i) for i in ['u', 'v']]
+
+c = x*y
+b = Vector([1e-2, 1e-1], name='b')
+A = Matrix([[1,1], [0,1]], name='A')
+
+# bilinear form
+expr = dot(grad(v), A * grad(u)) + dot(b, grad(u))*v + c*u*v
+a = BilinearForm((u,v), integral(domain, expr))
+------------------
+
+
+---------------------------------------------------------------------------
+ImportError                               Traceback (most recent call last)
+Cell In[1], line 5
+      3 from sympde.topology import ScalarFunctionSpace, Square, element_of
+      4 from sympde.calculus import grad, dot, div
+----> 5 from sympde.core import Vector, Matrix
+      7 from sympy import pi, sin
+      9 from psydac.api.discretization import discretize
+
+ImportError: cannot import name 'Vector' from 'sympde.core' (/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympde/core/__init__.py)
+
diff --git a/reports/chapter2/linear-elasticity.err.log b/reports/chapter2/linear-elasticity.err.log
new file mode 100644
index 0000000..efcbe69
--- /dev/null
+++ b/reports/chapter2/linear-elasticity.err.log
@@ -0,0 +1,54 @@
+Traceback (most recent call last):
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/jupyter_cache/executors/utils.py", line 58, in single_nb_execution
+    executenb(
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 1314, in execute
+    return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/jupyter_core/utils/__init__.py", line 165, in wrapped
+    return loop.run_until_complete(inner)
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/asyncio/base_events.py", line 647, in run_until_complete
+    return future.result()
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 709, in async_execute
+    await self.async_execute_cell(
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 1062, in async_execute_cell
+    await self._check_raise_for_error(cell, cell_index, exec_reply)
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 918, in _check_raise_for_error
+    raise CellExecutionError.from_cell_and_msg(cell, exec_reply_content)
+nbclient.exceptions.CellExecutionError: An error occurred while executing the following cell:
+------------------
+from sympde.expr import BilinearForm, LinearForm, integral
+from sympde.expr     import find, EssentialBC, Norm, SemiNorm
+from sympde.topology import VectorFunctionSpace, Cube, element_of
+from sympde.calculus import grad, dot, inner, outer, cross, div
+from sympde.core import Constant
+from sympde.core import Matrix, Vector, Transpose
+
+domain = Cube()
+
+I = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]], name='I')
+
+kappa = Constant('kappa', is_real=True)
+mu    = Constant('mu',    is_real=True)
+rho   = Constant('rho',   is_real=True)
+
+epsilon = lambda w: 0.5*(grad(w) + Transpose(grad(w)))
+sigma   = lambda w: kappa * div(w) * I + 2 * mu * epsilon(w)
+
+V = VectorFunctionSpace('V', domain)
+
+x,y,z = domain.coordinates
+
+u,v = [element_of(V, name=i) for i in ['u', 'v']]
+------------------
+
+
+---------------------------------------------------------------------------
+ImportError                               Traceback (most recent call last)
+Cell In[1], line 6
+      4 from sympde.calculus import grad, dot, inner, outer, cross, div
+      5 from sympde.core import Constant
+----> 6 from sympde.core import Matrix, Vector, Transpose
+      8 domain = Cube()
+     10 I = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]], name='I')
+
+ImportError: cannot import name 'Vector' from 'sympde.core' (/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/sympde/core/__init__.py)
+
diff --git a/reports/chapter2/vector-poisson.err.log b/reports/chapter2/vector-poisson.err.log
new file mode 100644
index 0000000..8e20c2f
--- /dev/null
+++ b/reports/chapter2/vector-poisson.err.log
@@ -0,0 +1,35 @@
+Traceback (most recent call last):
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 782, in _async_poll_for_reply
+    msg = await ensure_async(self.kc.shell_channel.get_msg(timeout=new_timeout))
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/jupyter_core/utils/__init__.py", line 198, in ensure_async
+    result = await obj
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/jupyter_client/channels.py", line 315, in get_msg
+    raise Empty
+_queue.Empty
+
+During handling of the above exception, another exception occurred:
+
+Traceback (most recent call last):
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/jupyter_cache/executors/utils.py", line 58, in single_nb_execution
+    executenb(
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 1314, in execute
+    return NotebookClient(nb=nb, resources=resources, km=km, **kwargs).execute()
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/jupyter_core/utils/__init__.py", line 165, in wrapped
+    return loop.run_until_complete(inner)
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/asyncio/base_events.py", line 647, in run_until_complete
+    return future.result()
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 709, in async_execute
+    await self.async_execute_cell(
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 1005, in async_execute_cell
+    exec_reply = await self.task_poll_for_reply
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 806, in _async_poll_for_reply
+    error_on_timeout_execute_reply = await self._async_handle_timeout(timeout, cell)
+  File "/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/nbclient/client.py", line 856, in _async_handle_timeout
+    raise CellTimeoutError.error_from_timeout_and_cell(
+nbclient.exceptions.CellTimeoutError: A cell timed out while it was being executed, after 30 seconds.
+The message was: Cell execution timed out.
+Here is a preview of the cell contents:
+-------------------
+uh = equation_h.solve()
+-------------------
+
diff --git a/search.html b/search.html
new file mode 100644
index 0000000..9ced498
--- /dev/null
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@@ -0,0 +1,879 @@
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+                    IGA-Python
+                </a>
+            </li>
+        </ul>
+        <p aria-level="2" class="caption" role="heading"><span class="caption-text">Introduction</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="chapter0/howto.html">What to expect from IGA-Python</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter0/cad.html">Computer Aided Design</a><input class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-1"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2 has-children"><a class="reference internal" href="chapter0/bezier.html">Bernstein polynomials</a><input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-2"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l3"><a class="reference internal" href="chapter0/bezier-curves.html">Bézier curves</a></li>
+</ul>
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+<li class="toctree-l3"><a class="reference internal" href="chapter0/bsplines-operations.html">Fundamental geometric operations for B-Splines</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter0/iga.html">Isogeometric Analysis</a><input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-4"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+
+<li class="toctree-l2"><a class="reference internal" href="chapter0/poisson-2d.html">B-splines FEM solver for Poisson equation (2D)</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter0/performance-acceleration.html">Performance and Acceleration</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Getting Started</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="chapter1/install.html">Installation</a></li>
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+<li class="toctree-l1"><a class="reference internal" href="chapter2/vector-poisson.html">Vector Poisson equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter2/advection-diffusion.html">Advection-diffusion equation</a><input class="toctree-checkbox" id="toctree-checkbox-6" name="toctree-checkbox-6" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-6"><i class="fa-solid fa-chevron-down"></i></label><ul>
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+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter2/poisson-mixed.html">Mixed FEM for the Poisson problem</a><input class="toctree-checkbox" id="toctree-checkbox-7" name="toctree-checkbox-7" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-7"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter2/poisson-mixed-v1.html">First mixed formulation of the Poisson problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter2/poisson-mixed-v2.html">Second mixed formulation of the Poisson problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter2/stokes.html">Mixed FEM for the Stokes problem</a><input class="toctree-checkbox" id="toctree-checkbox-8" name="toctree-checkbox-8" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-8"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter2/stokes-v1.html">First mixed formulation of the Stokes problem</a></li>
+<li class="toctree-l2"><a class="reference internal" href="chapter2/stokes-v2.html">Second mixed formulation of the Stokes problem</a></li>
+</ul>
+</li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/elliptic-curl.html">Elliptic-curl Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/elliptic-div.html">Elliptic-div Problem</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter2/linear-elasticity.html">Linear Elasticity Problem</a></li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Nonlinear Problems</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1"><a class="reference internal" href="chapter3/poisson.html">Nonlinear Poisson in 2D</a></li>
+<li class="toctree-l1"><a class="reference internal" href="chapter3/burgers.html">1D Burgers equation</a></li>
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter3/navier-stokes-steady.html">the steady-state Navier-Stokes equations for incompressible fluids</a><input class="toctree-checkbox" id="toctree-checkbox-9" name="toctree-checkbox-9" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-9"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter3/navier-stokes-steady-streamfunction-velocity.html">The streamfunction-velocity formulation of the steady-state Navier-Stokes equations for incompressible fluids</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced subjects</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter4/subdomains.html">Subdomains</a><input class="toctree-checkbox" id="toctree-checkbox-10" name="toctree-checkbox-10" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-10"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter4/poisson-nitsche.html">The Poisson problem using Nitsche method</a></li>
+</ul>
+</li>
+</ul>
+<p aria-level="2" class="caption" role="heading"><span class="caption-text">Applications</span></p>
+<ul class="nav bd-sidenav">
+<li class="toctree-l1 has-children"><a class="reference internal" href="chapter5/cfd.html">Computational Fluid Dynamics</a><input class="toctree-checkbox" id="toctree-checkbox-11" name="toctree-checkbox-11" type="checkbox"/><label class="toctree-toggle" for="toctree-checkbox-11"><i class="fa-solid fa-chevron-down"></i></label><ul>
+<li class="toctree-l2"><a class="reference internal" href="chapter5/buoyancy-driven_natural_convection.html">Buoyancy-Driven Natural Convection</a></li>
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+</li>
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+</li>
+<li class="toctree-l3"><a class="reference internal" href="documentation/_autosummary/psydac.api.settings.html">psydac.api.settings</a></li>
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(Derivatives of B-Splines)": [[2, "proposition-23"]], "Evaluation of B-Splines": [[2, "evaluation-of-b-splines"]], "Knots vector families": [[2, "knots-vector-families"], [357, "knots-vector-families"]], "Clamped knots (open knots vector)": [[2, "clamped-knots-open-knots-vector"]], "uniform": [[2, "uniform"], [2, "id1"]], "non-uniform": [[2, "non-uniform"], [2, "id2"]], "Unclamped knots": [[2, "unclamped-knots"]], "B-Splines curves": [[3, "b-splines-curves"]], " (B-Spline curve)": [[3, "definition-0"]], "Properties of B-Splines curves": [[3, "properties-of-b-splines-curves"]], "Derivative of a B-spline curve": [[3, "derivative-of-a-b-spline-curve"]], "Rational B-Splines (NURBS) curves": [[3, "rational-b-splines-nurbs-curves"]], " (Quarter circle)": [[3, "nurbs-curves-ex-1"]], " (Circular arc of 120\u00b0)": [[3, "nurbs-curves-ex-2"]], " (half circle)": [[3, "nurbs-curves-ex-3"]], " (Circle as 4 arcs)": [[3, "nurbs-curves-ex-4"]], " (Circle as 3 arcs)": [[3, "nurbs-curves-ex-5"]], "Fundamental geometric operations for B-Splines": [[4, "fundamental-geometric-operations-for-b-splines"]], "Knot insertion": [[4, "knot-insertion"], [357, "knot-insertion"]], "Order elevation": [[4, "order-elevation"], [357, "order-elevation"]], "Translation": [[4, "translation"], [357, "translation"]], "Rotation": [[4, "rotation"], [357, "rotation"]], "Homothetic transformation": [[4, "homothetic-transformation"]], "B-Splines surfaces": [[5, "b-splines-surfaces"]], "Computer Aided Design": [[6, "computer-aided-design"]], "Data Structure": [[7, "data-structure"]], "1D Case": [[7, "d-case"], [11, "d-case"]], "2D Case": [[7, "id1"]], "Matrix profile": [[7, "matrix-profile"], [12, "matrix-profile"]], "Introduction to B-Splines FEM": [[8, "introduction-to-b-splines-fem"]], "Discretization of the Poisson problem": [[8, "discretization-of-the-poisson-problem"]], "Expressions": [[8, "expressions"]], "Notations": [[8, "notations"]], "Assembly procedures": [[8, "assembly-procedures"]], "Pre-process": [[8, "pre-process"]], "Evaluation of a discrete field on a given element": [[8, "evaluation-of-a-discrete-field-on-a-given-element"]], "Kernel for an expression with arity = 1": [[8, "kernel-for-an-expression-with-arity-1"]], "Computing contributions of test basis functions (1D)": [[8, "computing-contributions-of-test-basis-functions-1d"]], "Computing contributions of test basis functions (2D)": [[8, "computing-contributions-of-test-basis-functions-2d"]], "Kernel for an expression with arity = 2": [[8, "kernel-for-an-expression-with-arity-2"]], "Computing contributions between test and trial basis functions (1D)": [[8, "computing-contributions-between-test-and-trial-basis-functions-1d"]], "Computing contributions between test and trial basis functions (2D)": [[8, "computing-contributions-between-test-and-trial-basis-functions-2d"]], "What to expect from IGA-Python": [[9, "what-to-expect-from-iga-python"]], "Isogeometric Analysis": [[10, "isogeometric-analysis"]], "Performance and Acceleration": [[11, "performance-and-acceleration"]], "Pure Python": [[11, "pure-python"], [11, "id2"]], "Timing using pure Python": [[11, "timing-using-pure-python"]], "Using Pyccel": [[11, "using-pyccel"], [11, "id4"]], "Timing": [[11, "timing"], [11, "id3"], [11, "id5"]], "2D case": [[11, "id1"]], "Exercises": [[11, "exercises"], [12, "exercises"]], "B-splines FEM solver for Poisson equation (1D)": [[12, "b-splines-fem-solver-for-poisson-equation-1d"]], "Analytical solution": [[12, "analytical-solution"]], "The Spline Space class": [[12, "the-spline-space-class"]], "Next step": [[12, "next-step"]], "Exercise 1.": [[12, "exercise-1"]], "Exercise 2.": [[12, "exercise-2"]], "Exercise 3.": [[12, "exercise-3"]], "B-splines FEM solver for Poisson equation (2D)": [[13, "b-splines-fem-solver-for-poisson-equation-2d"]], "Create the Finite Elements Space": [[13, "create-the-finite-elements-space"]], "Assemble the Stiffness Matrix": [[13, "assemble-the-stiffness-matrix"]], "Assemble the rhs": [[13, "assemble-the-rhs"]], "Imposing boundary conditions": [[13, "imposing-boundary-conditions"]], "Installation": [[14, "installation"]], "Your first code using SymPDE & PsyDAC": [[15, "your-first-code-using-sympde-psydac"]], "Variational Formulation": [[15, "variational-formulation"], [25, "variational-formulation"], [27, "variational-formulation"]], "Formal Model": [[15, "formal-model"], [20, "formal-model"], [21, "formal-model"], [22, "formal-model"], [23, "formal-model"], [24, "formal-model"], [25, "formal-model"], [26, "formal-model"], [27, "formal-model"], [29, "formal-model"], [30, "formal-model"], [31, "formal-model"], [33, "formal-model"], [34, "formal-model"], [35, "formal-model"]], "Discretization": [[15, "discretization"], [20, "discretization"], [21, "discretization"], [22, "discretization"], [23, "discretization"], [24, "discretization"], [25, "discretization"], [26, "discretization"], [27, "discretization"], [29, "discretization"], [30, "discretization"], [31, "discretization"], [33, "discretization"], [34, "discretization"], [35, "discretization"]], "Solving the PDE": [[15, "solving-the-pde"], [20, "solving-the-pde"], [21, "solving-the-pde"], [22, "solving-the-pde"], [23, "solving-the-pde"], [24, "solving-the-pde"], [25, "solving-the-pde"], [26, "solving-the-pde"], [27, "solving-the-pde"], [29, "solving-the-pde"], [30, "solving-the-pde"], [31, "solving-the-pde"], [33, "solving-the-pde"], [34, "solving-the-pde"], [35, "solving-the-pde"]], "Computing the error norm": [[15, "computing-the-error-norm"], [20, "computing-the-error-norm"], [21, "computing-the-error-norm"], [22, "computing-the-error-norm"], [23, "computing-the-error-norm"], [24, "computing-the-error-norm"], [25, "computing-the-error-norm"], [27, "computing-the-error-norm"], [29, "computing-the-error-norm"], [30, "computing-the-error-norm"], [33, "computing-the-error-norm"], [34, "computing-the-error-norm"], [35, "computing-the-error-norm"]], "Computing the 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Newton iteration": [[39, "newton-iteration"]], "Computing F^{\\prime} the derivative of F": [[39, "computing-f-prime-the-derivative-of-f"]], "The Poisson problem using Nitsche method": [[40, "the-poisson-problem-using-nitsche-method"]], "Subdomains": [[41, "subdomains"]], "Bingham Plastic Flow in a Pipe": [[42, "bingham-plastic-flow-in-a-pipe"]], "Mathematical Model": [[42, "mathematical-model"], [43, "mathematical-model"], [44, "mathematical-model"], [46, "mathematical-model"], [47, "mathematical-model"], [48, "mathematical-model"], [50, "mathematical-model"], [52, "mathematical-model"], [53, "mathematical-model"], [55, "mathematical-model"], [60, "mathematical-model"], [61, "mathematical-model"], [62, "mathematical-model"], [63, "mathematical-model"], [64, "mathematical-model"], [65, "mathematical-model"]], "Boundary Conditions": [[42, "boundary-conditions"], [43, "boundary-conditions"], [44, "boundary-conditions"], [46, "boundary-conditions"], [47, "boundary-conditions"], [48, 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"sympde.printing": [[313, "module-sympde.printing"]], "sympde.printing.latex": [[314, "sympde-printing-latex"]], "sympde.printing.tests": [[315, "module-sympde.printing.tests"]], "sympde.printing.tests.test_latex": [[316, "module-sympde.printing.tests.test_latex"]], "sympde.topology": [[317, "module-sympde.topology"]], "sympde.topology.analytical_mapping": [[318, "module-sympde.topology.analytical_mapping"]], "sympde.topology.basic": [[319, "module-sympde.topology.basic"]], "sympde.topology.callable_mapping": [[320, "module-sympde.topology.callable_mapping"]], "sympde.topology.datatype": [[321, "module-sympde.topology.datatype"]], "sympde.topology.derivatives": [[322, "module-sympde.topology.derivatives"]], "sympde.topology.domain": [[323, "module-sympde.topology.domain"]], "sympde.topology.mapping": [[324, "module-sympde.topology.mapping"]], "sympde.topology.measure": [[325, "module-sympde.topology.measure"]], "sympde.topology.space": [[326, "module-sympde.topology.space"]], "sympde.topology.tests": [[327, "module-sympde.topology.tests"]], "sympde.topology.tests.test_callable_mapping": [[328, "module-sympde.topology.tests.test_callable_mapping"]], "sympde.topology.tests.test_derivatives": [[329, "module-sympde.topology.tests.test_derivatives"]], "sympde.topology.tests.test_field_2d": [[330, "module-sympde.topology.tests.test_field_2d"]], "sympde.topology.tests.test_gallery": [[331, "module-sympde.topology.tests.test_gallery"]], "sympde.topology.tests.test_logical_expr": [[332, "module-sympde.topology.tests.test_logical_expr"]], "sympde.topology.tests.test_mapping": [[333, "module-sympde.topology.tests.test_mapping"]], "sympde.topology.tests.test_space": [[334, "module-sympde.topology.tests.test_space"]], "sympde.topology.tests.test_symbolic_expr": [[335, "module-sympde.topology.tests.test_symbolic_expr"]], "sympde.topology.tests.test_topology": [[336, "module-sympde.topology.tests.test_topology"]], "sympde.utilities": [[337, "module-sympde.utilities"]], "sympde.utilities.tests": [[338, "module-sympde.utilities.tests"]], "sympde.utilities.tests.test_utilities": [[339, "module-sympde.utilities.tests.test_utilities"]], "sympde.utilities.utils": [[340, "module-sympde.utilities.utils"]], "Psydac API Reference": [[341, "psydac-api-reference"]], "SymPDE API Reference": [[342, "sympde-api-reference"]], "IGA-Python": [[343, "iga-python"]], "TODO": [[343, "todo"], [345, "todo"]], "Introduction": [[343, null], [350, "introduction"]], "Getting Started": [[343, null]], "Linear Problems": [[343, null]], "Nonlinear Problems": [[343, null]], "Advanced subjects": [[343, null]], "Applications": [[343, null]], "Documentation": [[343, null]], "Welcome to PSYDAC": [[344, "welcome-to-psydac"]], "Table of contents": [[344, "table-of-contents"]], "Requirements": [[344, "requirements"]], "Linux Debian-Ubuntu-Mint": [[344, "linux-debian-ubuntu-mint"]], "macOS": [[344, "macos"]], "Other operating systems": [[344, "other-operating-systems"]], "Python setup and project download": [[344, "python-setup-and-project-download"]], "Installing the library": [[344, "installing-the-library"]], "Uninstall": [[344, "uninstall"]], "Running tests": [[344, "running-tests"]], "Speeding up Psydac\u2019s core": [[344, "speeding-up-psydac-s-core"]], "User documentation": [[344, "user-documentation"]], "Code documentation": [[344, "code-documentation"]], "Mesh Generation": [[344, "mesh-generation"]], "Example of usage": [[344, "example-of-usage"]], "Core": [[345, "core"]], "API": [[345, "api"], [347, "api"]], "Fortran API": [[347, "fortran-api"]], "Python API": [[347, "python-api"]], "Dive into PSYDAC": [[349, "dive-into-psydac"]], "Contents": [[349, "contents"], [361, "contents"]], "Input and Output": [[351, "input-and-output"]], "Welcome to spl\u2019s documentation!": [[352, "welcome-to-spl-s-documentation"]], "Contents:": [[352, null], [356, null]], "Indices and tables": [[352, "indices-and-tables"], [361, "indices-and-tables"]], "DeRham sequences": [[353, "derham-sequences"]], "Pullbacks": [[353, "pullbacks"]], "Discrete Spaces": [[353, "discrete-spaces"]], "Projectors": [[353, "projectors"]], "Discrete DeRham sequence for B-Splines": [[353, "discrete-derham-sequence-for-b-splines"]], "1d case": [[353, "d-case"]], "Discrete derivatives:": [[353, null], [353, null], [353, null]], "2d case": [[353, "id2"]], "3d case": [[353, "id3"]], "Algebraic identities": [[353, "algebraic-identities"]], "Proposition:": [[353, null]], "Exterior Algebra": [[354, "exterior-algebra"]], "Definition, Alternating algebraic forms:": [[354, null]], "Definition, Exterior product:": [[354, null]], "Definition, Grassmann Algebra:": [[354, null]], "Basis": [[354, "basis"]], "Definition, Interior product:": [[354, null]], "Definition, Inner product:": [[354, null]], "Orientation and Volume form": [[354, "orientation-and-volume-form"]], "Definition, Pullback:": [[354, null]], "Definition, The tangential and normal parts:": [[354, null]], "Definition, The Hodge star operation:": [[354, null]], "Exterior Calculus on manifolds and Differential forms": [[354, "exterior-calculus-on-manifolds-and-differential-forms"]], "Exterior product:": [[354, null]], "Exterior differentiation:": [[354, null]], "Integration:": [[354, null]], "Theorem, Stokes theorem:": [[354, null]], "Theorem, Integration by parts:": [[354, null]], "Sobolev spaces of differential forms": [[354, "sobolev-spaces-of-differential-forms"]], "Cohomology and De Rham Complex": [[354, "cohomology-and-de-rham-complex"]], "Cohomology with boundary conditions": [[354, "cohomology-with-boundary-conditions"]], "Definition, Harmonic forms:": [[354, null]], "Proposition, Poincar\u00e9 duality:": [[354, null]], "Homological Algebra and Hilbert complexes": [[354, "homological-algebra-and-hilbert-complexes"]], "Homological Algebra": [[354, "homological-algebra"]], "Cycles and boundaries of the De Rham complex": [[354, "cycles-and-boundaries-of-the-de-rham-complex"]], "The Hodge decomposition": [[354, "the-hodge-decomposition"]], "Summary": [[354, "summary"]], "GLT": [[355, "glt"]], "Where do the GLTs come from?": [[355, "where-do-the-glts-come-from"]], "Spectral preliminaries": [[355, "spectral-preliminaries"]], "B-Splines and NURBS": [[357, "b-splines-and-nurbs"]], "B-Splines series": [[357, "b-splines-series"]], "B-Spline curve": [[357, "b-spline-curve"]], "Multivariate tensor product splines": [[357, "multivariate-tensor-product-splines"]], "Deriving a B-spline curve": [[357, "deriving-a-b-spline-curve"]], "proposition": [[357, "proposition"]], "Scaling": [[357, "scaling"]], "First Steps with PSYDAC": [[358, "first-steps-with-psydac"]], "Install PSYDAC": [[358, "install-psydac"]], "Details": [[359, "details"]], "Modules": [[362, "modules"]], "api": [[441, "api"]], "api submodules": [[441, "api-submodules"]], "api.ast": [[442, "api-ast"]], "api.printing": [[443, "api-printing"]], "cad": [[444, "cad"]], "cmd": [[445, "cmd"]], "core": [[446, "core"]], "ddm": [[447, "ddm"]], "feec": [[448, "feec"]], "feec submodules": [[448, "feec-submodules"]], "feec.multipatch": [[449, "feec-multipatch"]], "fem": [[450, "fem"]], "linalg": [[451, "linalg"]], "linalg submodules": [[451, "linalg-submodules"]], "linalg.kernels": [[452, "linalg-kernels"]], "mapping": [[453, "mapping"]], "polar": [[454, "polar"]], "utilities": [[455, "utilities"]], "Solving Helmholtz\u2019s equation on non-periodic topological domains": [[456, "solving-helmholtz-s-equation-on-non-periodic-topological-domains"]], "Step 1 : Building the domain": [[456, "step-1-building-the-domain"], [457, "step-1-building-the-domain"]], "Step 2: Defining the Abstract PDE model using SymPDE": [[456, "step-2-defining-the-abstract-pde-model-using-sympde"], [457, "step-2-defining-the-abstract-pde-model-using-sympde"]], "Step 3: Discretizing the domain, spaces and equations": [[456, "step-3-discretizing-the-domain-spaces-and-equations"], [457, "step-3-discretizing-the-domain-spaces-and-equations"]], "Step 4: Solving the equation and computing the error norms": [[456, "step-4-solving-the-equation-and-computing-the-error-norms"], [457, "step-4-solving-the-equation-and-computing-the-error-norms"]], "Step 5: Saving the results": [[456, "step-5-saving-the-results"], [457, "step-5-saving-the-results"]], "Step 6: Exporting the results to VTK and visualizing them with Paraview": [[456, "step-6-exporting-the-results-to-vtk-and-visualizing-them-with-paraview"], [457, "step-6-exporting-the-results-to-vtk-and-visualizing-them-with-paraview"]], "Example: Results on the square domain": [[456, "example-results-on-the-square-domain"]], "Solving Poisson\u2019s equation on non-periodic topological domains": [[457, "solving-poisson-s-equation-on-non-periodic-topological-domains"]], "Example: Results on the pretzel domain": [[457, "example-results-on-the-pretzel-domain"]], "Psydac\u2019s outputs": [[458, "psydac-s-outputs"]], "Structure": [[458, "structure"]], "Usage of class OutputManager": [[458, "usage-of-class-outputmanager"]], "Usage of class PostProcessManager": [[458, "usage-of-class-postprocessmanager"]], "Further Examples": [[458, "further-examples"]]}, "indexentries": {"module": [[66, "module-psydac.api"], [67, "module-psydac.api.ast"], [68, "module-psydac.api.ast.basic"], [69, "module-psydac.api.ast.evaluation"], [70, "module-psydac.api.ast.expr"], [71, "module-psydac.api.ast.fem"], [72, "module-psydac.api.ast.glt"], [73, "module-psydac.api.ast.linalg"], [74, "module-psydac.api.ast.nodes"], [75, "module-psydac.api.ast.parser"], [76, "module-psydac.api.ast.tests"], [77, "module-psydac.api.ast.tests.boundary"], [78, "module-psydac.api.ast.tests.system_1"], [79, "module-psydac.api.ast.tests.system_2"], [80, "module-psydac.api.ast.tests.system_3"], [81, "module-psydac.api.ast.tests.test_nodes"], [82, "module-psydac.api.ast.utilities"], [83, "module-psydac.api.basic"], [84, "module-psydac.api.discretization"], [85, "module-psydac.api.equation"], [86, 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\ No newline at end of file