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6_finite_element_problems.html

@@ -317,13 +317,13 @@ <h3><span class="section-number">6.2.4. </span>Sparse matrices<a class="headerli
 are avoided.</p>
 <div class="admonition hint">
 <p class="admonition-title">Hint</p>
-<p>The <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.html#module-scipy.sparse" title="(in SciPy v1.11.0.dev0+1409.6d43301)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse</span></code></a> package provides convenient interfaces
+<p>The <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.html#module-scipy.sparse" title="(in SciPy v1.11.0.dev0+1660.4a6f2d1)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse</span></code></a> package provides convenient interfaces
 which enable Python code to employ a variety of sparse matrix
 formats using essentially identical operations to the dense matrix
 case. The skeleton code already contains commands to construct
 empty sparse matrices and to solve the resulting linear system. You
 may, if you wish, experiment with choosing other sparse formats
-from <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.html#module-scipy.sparse" title="(in SciPy v1.11.0.dev0+1409.6d43301)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse</span></code></a>, but it is very strongly suggested that
+from <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.html#module-scipy.sparse" title="(in SciPy v1.11.0.dev0+1660.4a6f2d1)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse</span></code></a>, but it is very strongly suggested that
 you do <strong>not</strong> switch to a dense numpy array; unless, that is, you
 particularly enjoy running out of memory on your computer!</p>
 </div>

9_mixed_problems.html

@@ -380,7 +380,7 @@ <h3><span class="section-number">9.4.1. </span>Assembling block systems<a class=
 </div>
 <p>This means that one can first create a full vector of length <span class="math notranslate nohighlight">\(n+m\)</span> and then
 slice it to create subvectors that can be used for assembly.</p>
-<p>Conversely, <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.html#module-scipy.sparse" title="(in SciPy v1.11.0.dev0+1409.6d43301)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse</span></code></a> provides the <a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.sparse.bmat.html#scipy.sparse.bmat" title="(in SciPy v1.11.0.dev0+1409.6d43301)"><code class="xref py py-func docutils literal notranslate"><span class="pre">bmat()</span></code></a>
+<p>Conversely, <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.html#module-scipy.sparse" title="(in SciPy v1.11.0.dev0+1660.4a6f2d1)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse</span></code></a> provides the <a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.sparse.bmat.html#scipy.sparse.bmat" title="(in SciPy v1.11.0.dev0+1660.4a6f2d1)"><code class="xref py py-func docutils literal notranslate"><span class="pre">bmat()</span></code></a>
 function which will stitch together a larger sparse matrix from blocks. In
 order to have the full indexing options you are likely to want for imposing the
 boundary conditions, you will probably want to specify that the resulting
@@ -408,15 +408,15 @@ <h3><span class="section-number">9.4.3. </span>Solving the matrix system<a class
 <p>The block matrix system that you eventually produce will be larger than many of
 those we have previously encountered, and will have non-zero entries further
 from the diagonal. This can cause the matrix solver to become expensive in both
-time and memory. Fortunately, <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.linalg.html#module-scipy.sparse.linalg" title="(in SciPy v1.11.0.dev0+1409.6d43301)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse.linalg</span></code></a> now incorporates an
+time and memory. Fortunately, <a class="reference external" href="http://scipy.github.io/devdocs/reference/sparse.linalg.html#module-scipy.sparse.linalg" title="(in SciPy v1.11.0.dev0+1660.4a6f2d1)"><code class="xref py py-mod docutils literal notranslate"><span class="pre">scipy.sparse.linalg</span></code></a> now incorporates an
 interface to <a class="reference external" href="https://portal.nersc.gov/project/sparse/superlu/">SuperLU</a>,
 which is a high-performance direct sparse solver. The recommended solution
 strategy is therefore:</p>
 <ol class="arabic simple">
-<li><p>Convert your block matrix to <a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.sparse.csc_matrix.html#scipy.sparse.csc_matrix" title="(in SciPy v1.11.0.dev0+1409.6d43301)"><code class="xref py py-class docutils literal notranslate"><span class="pre">scipy.sparse.csc_matrix</span></code></a>, which is the
+<li><p>Convert your block matrix to <a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.sparse.csc_matrix.html#scipy.sparse.csc_matrix" title="(in SciPy v1.11.0.dev0+1660.4a6f2d1)"><code class="xref py py-class docutils literal notranslate"><span class="pre">scipy.sparse.csc_matrix</span></code></a>, which is the
 format that SuperLU requires.</p></li>
-<li><p>Factorise the matrix using <a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.sparse.linalg.splu.html#scipy.sparse.linalg.splu" title="(in SciPy v1.11.0.dev0+1409.6d43301)"><code class="xref py py-func docutils literal notranslate"><span class="pre">scipy.sparse.linalg.splu()</span></code></a>.</p></li>
-<li><p>Use the resulting <a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.sparse.linalg.SuperLU.html#scipy.sparse.linalg.SuperLU" title="(in SciPy v1.11.0.dev0+1409.6d43301)"><code class="xref py py-class docutils literal notranslate"><span class="pre">SuperLU</span></code></a> object to finally solve
+<li><p>Factorise the matrix using <a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.sparse.linalg.splu.html#scipy.sparse.linalg.splu" title="(in SciPy v1.11.0.dev0+1660.4a6f2d1)"><code class="xref py py-func docutils literal notranslate"><span class="pre">scipy.sparse.linalg.splu()</span></code></a>.</p></li>
+<li><p>Use the resulting <a class="reference external" href="http://scipy.github.io/devdocs/reference/generated/scipy.sparse.linalg.SuperLU.html#scipy.sparse.linalg.SuperLU" title="(in SciPy v1.11.0.dev0+1660.4a6f2d1)"><code class="xref py py-class docutils literal notranslate"><span class="pre">SuperLU</span></code></a> object to finally solve
 the system.</p></li>
 </ol>
 </section>

Finiteelementcourse.pdf

@@ -420,7 +420,7 @@ <h2><span class="section-number">4.1. </span>Finite element spaces and other Hil
 <p><span class="math notranslate nohighlight">\(L\)</span> bounded <span class="math notranslate nohighlight">\(\implies L(u)\leq C\|u\|_{H} \implies |L(u)-L(v)| = |L(u-v)| \leq C\|u-v\|_{H} \, \forall u,v\in H,\)</span> i.e. <span class="math notranslate nohighlight">\(L\)</span> is continuous.</p>
 <p>On the other hand, <span class="math notranslate nohighlight">\(L\)</span> continuous <span class="math notranslate nohighlight">\(\implies |L(u-v)| \leq C\|u-v\|_{H}
 \quad \forall u,v\in H\)</span>. Pick <span class="math notranslate nohighlight">\(v=0\)</span>, then <span class="math notranslate nohighlight">\(|L(u)| = |L(u-0)| \leq
-C|u-0|_H = C|u|_H\)</span>, i.e. <span class="math notranslate nohighlight">\(L\)</span> is bounded.</p>
+C\|u-0\|_H = C\|u\|_H\)</span>, i.e. <span class="math notranslate nohighlight">\(L\)</span> is bounded.</p>
 </div></div><p>We can also interpret bounded linear functionals as elements of a vector space.</p>
 <div class="proof proof-type-definition" id="id20">
 
@@ -462,7 +462,7 @@ <h2><span class="section-number">4.1. </span>Finite element spaces and other Hil
     </div><div class="proof-content">
 <p>For <span class="math notranslate nohighlight">\(v,w\in H\)</span>, <span class="math notranslate nohighlight">\(\alpha\in\mathbb{R}\)</span> we have</p>
 <div class="math notranslate nohighlight">
-\[L_u(v+\alpha w) = (u,v+\alpha w) = (u,v) + \alpha(u,v) = L_u(v)
+\[L_u(v+\alpha w) = (u,v+\alpha w) = (u,v) + \alpha(u,w) = L_u(v)
 + \alpha L_u(w).\]</div>
 <p>Hence <span class="math notranslate nohighlight">\(L_u\)</span> is linear.</p>
 <p>We see that <span class="math notranslate nohighlight">\(L_u\)</span> is bounded by Schwarz inequality,</p>

L4_feprobs.html

@@ -99,7 +99,7 @@ <h1>Source code for fe_utils.scripts.plot_interpolate_lagrange</h1><div class="h
         <span class="n">z</span> <span class="o">=</span> <span class="n">fe</span><span class="o">.</span><span class="n">tabulate</span><span class="p">(</span><span class="n">x</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">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</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">ax</span><span class="o">.</span><span class="n">plot_trisurf</span><span class="p">(</span><span class="n">x</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">x</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">coefs</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">RdBu</span><span class="p">,</span>
                         <span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>

_modules/fe_utils/scripts/plot_interpolate_lagrange.html

@@ -89,7 +89,7 @@ <h1>Source code for fe_utils.scripts.plot_lagrange_basis_functions</h1><div clas
         <span class="n">z</span> <span class="o">=</span> <span class="n">fe</span><span class="o">.</span><span class="n">tabulate</span><span class="p">(</span><span class="n">x</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">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">4</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">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</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">offsets</span> <span class="o">=</span> <span class="n">fe</span><span class="o">.</span><span class="n">nodes</span> <span class="o">*</span> <span class="n">fe</span><span class="o">.</span><span class="n">degree</span> <span class="o">*</span> <span class="mf">1.1</span>
 

_modules/fe_utils/scripts/plot_lagrange_basis_functions.html

@@ -323,7 +323,7 @@ and continuity.
 
    On the other hand, `L` continuous `\implies |L(u-v)| \leq C\|u-v\|_{H}
    \quad \forall u,v\in H`. Pick `v=0`, then `|L(u)| = |L(u-0)| \leq
-   C|u-0|_H = C|u|_H`, i.e. `L` is bounded.
+   C\|u-0\|_H = C\|u\|_H`, i.e. `L` is bounded.
 
 We can also interpret bounded linear functionals as elements of a vector space.
 
@@ -359,7 +359,7 @@ There is a simple mapping from `H` to `H'`.
 
    .. math::
 
-      L_u(v+\alpha w) = (u,v+\alpha w) = (u,v) + \alpha(u,v) = L_u(v)
+      L_u(v+\alpha w) = (u,v+\alpha w) = (u,v) + \alpha(u,w) = L_u(v)
       + \alpha L_u(w).
 
    Hence `L_u` is linear.