Update documentation

_sources/aposteriori/aposteriori.ipynb

@@ -8,7 +8,7 @@
     "# A posteriori error estimates\n",
     "\n",
     "We will derive methods to estimate the error of the computed finite\n",
-    "element approximation. Such {\\em a posteriori} error estimates may use the finite element solution $u_h$, and input data such as the source term $f$. \n",
+    "element approximation. Such **a posteriori** error estimates may use the finite element solution $u_h$, and input data such as the source term $f$. \n",
     "\n",
     "$$\n",
     "\\eta(u_h, f)\n",
@@ -93,7 +93,7 @@
     "$$\n",
     "\\| p - \\tilde p_h \\|_{L_2} \\leq \\alpha \\, \\| p - p_h \\|_{L_2}\n",
     "$$\n",
-    "holds with a small constant $\\alpha \\ll 1$. This property is known as *em super-convergence*.It is indeed true on (locally) uniform meshes, and smoothness assumptions onto the source term $f$. \n",
+    "holds with a small constant $\\alpha \\ll 1$. This property is known as *super-convergence*.It is indeed true on (locally) uniform meshes, and smoothness assumptions onto the source term $f$. \n",
     "\n",
     "The ZZ error estimator replaces the true gradient in the error $p-p_h$ by the good approximation $\\tilde p_h$:\n",
     "\n",
@@ -117,7 +117,7 @@
     "It is also efficient, a similar short application of the triangle inequality.\n",
     "\n",
     "There is a rigorous analysis of the ZZ error estimator, e.g., by showing equivalence\n",
-    "to the following residual error estimator."
+    "to the residual error estimator below."
    ]
   },
   {
@@ -348,7 +348,7 @@
    "id": "a5b25c26-17d4-49a9-aa5e-cb6b14bda367",
    "metadata": {},
    "source": [
-    "We observe a very strong refinement along the material interface. This is unnecessary, since the solution is smooth on both sides, we only expect singularities at the corners. The problem is that the flux-averaging did an averaging of the full flux vector. The tangential component of it is not supposed to be continuous, and this is wrongly measured as error:"
+    "We observe a very strong refinement along the material interface. This is unnecessary, since the solution is smooth on both sides, we only expect singularities at the corners. The problem is that the flux-averaging did an averaging of the full flux vector. The tangential component of it is not supposed to be continuous, and this it highly overestimates the error:"
    ]
   },
   {
@@ -392,7 +392,7 @@
    "source": [
     "### Flux recovery in $H(\\operatorname{div})$\n",
     "\n",
-    "The over-refinement along the edges can be overcome by averaging only the normal component of the flux, since only this is supposed to be continuous. This we obtain by recovering the flux in an $H(\\operatorname{div})$ finite element spaces "
+    "The over-refinement along the edges can be overcome by averaging only the normal component of the flux, since only this is supposed to be continuous. This we obtain by recovering the flux in an $H(\\operatorname{div})$ finite element space: "
    ]
   },
   {

_sources/aposteriori/residualEE.ipynb

@@ -109,20 +109,22 @@
     "*Proof:* First, choose a reference patch $\\widehat \\omega_T$ of \n",
     "dimension \n",
     "$\\simeq 1$. The quasi-interpolation operator is bounded on $H^1(\\omega_T)$:\n",
-    "\\begin{equation}\n",
+    "\n",
+    "$$\n",
     "% \\label{equ_clement_bh}\n",
     "\\| v - \\Pi_h v \\|_{L_2(\\widehat T)} + \\| \\nabla (v - \\Pi_h v) \\|_{L_2(\\widehat T)} \\preceq \\| v \\|_{H^1(\\widehat \\omega_T)}\n",
-    "\\end{equation}\n",
+    "$$\n",
     "If $v$ is constant on $\\omega_T$, then the mean values in the vertices\n",
     "take the same values, and also $(\\Pi_h v)_{|T}$ is the same constant.\n",
     "The constant function (on $\\omega_T$) is in the kernel of \n",
     "$\\| v - \\Pi_h v \\|_{H^1(T)}$. Due to\n",
     "the Bramble-Hilbert lemma, we can replace the norm on the right hand side\n",
-    "of (\\ref{equ_clement_bh}) by the semi-norm:\n",
-    "\\begin{equation}\n",
+    " by the semi-norm:\n",
+    "\n",
+    "$$ \n",
     "% \\label{equ_clement_bh2}\n",
     "\\| v - \\Pi_h v \\|_{L_2(\\widehat T)} + \\| \\nabla (v - \\Pi_h v) \\|_{L_2(\\widehat T)} \\preceq \\| \\nabla v \\|_{L_2(\\widehat \\omega_T)}\n",
-    "\\end{equation}\n",
+    "$$\n",
     "\n",
     "The rest follows from scaling. Let $F : x \\rightarrow h x$ scale the reference patch $\\widehat \\omega_T$ to the actual patch $\\omega_T$. Then\n",
     "\n",

aposteriori/residualEE.html

@@ -583,22 +583,22 @@ <h2><span class="section-number">27.1. </span>The Clément- operator<a class="he
 <p><em>Proof:</em> First, choose a reference patch <span class="math notranslate nohighlight">\(\widehat \omega_T\)</span> of
 dimension
 <span class="math notranslate nohighlight">\(\simeq 1\)</span>. The quasi-interpolation operator is bounded on <span class="math notranslate nohighlight">\(H^1(\omega_T)\)</span>:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-f3376a05-20fc-435c-9d5e-6f1a9ad0510c">
-<span class="eqno">(27.1)<a class="headerlink" href="#equation-f3376a05-20fc-435c-9d5e-6f1a9ad0510c" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="math notranslate nohighlight">
+\[
 % \label{equ_clement_bh}
 \| v - \Pi_h v \|_{L_2(\widehat T)} + \| \nabla (v - \Pi_h v) \|_{L_2(\widehat T)} \preceq \| v \|_{H^1(\widehat \omega_T)}
-\end{equation}\]</div>
+\]</div>
 <p>If <span class="math notranslate nohighlight">\(v\)</span> is constant on <span class="math notranslate nohighlight">\(\omega_T\)</span>, then the mean values in the vertices
 take the same values, and also <span class="math notranslate nohighlight">\((\Pi_h v)_{|T}\)</span> is the same constant.
 The constant function (on <span class="math notranslate nohighlight">\(\omega_T\)</span>) is in the kernel of
 <span class="math notranslate nohighlight">\(\| v - \Pi_h v \|_{H^1(T)}\)</span>. Due to
 the Bramble-Hilbert lemma, we can replace the norm on the right hand side
-of (\ref{equ_clement_bh}) by the semi-norm:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-8e827d51-a913-471d-adcb-d29804d4624e">
-<span class="eqno">(27.2)<a class="headerlink" href="#equation-8e827d51-a913-471d-adcb-d29804d4624e" title="Permalink to this equation">#</a></span>\[\begin{equation}
+by the semi-norm:</p>
+<div class="math notranslate nohighlight">
+\[ 
 % \label{equ_clement_bh2}
 \| v - \Pi_h v \|_{L_2(\widehat T)} + \| \nabla (v - \Pi_h v) \|_{L_2(\widehat T)} \preceq \| \nabla v \|_{L_2(\widehat \omega_T)}
-\end{equation}\]</div>
+\]</div>
 <p>The rest follows from scaling. Let <span class="math notranslate nohighlight">\(F : x \rightarrow h x\)</span> scale the reference patch <span class="math notranslate nohighlight">\(\widehat \omega_T\)</span> to the actual patch <span class="math notranslate nohighlight">\(\omega_T\)</span>. Then</p>
 <div class="math notranslate nohighlight">
 \[