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JSchoeberl · Mar 11, 2024 · 90129d6 · 90129d6
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diff --git a/_images/picproject.png b/_images/picproject.png
diff --git a/_images/picture.png b/_images/picture.png
diff --git a/_sources/abstracttheory/BasicProperties.ipynb b/_sources/abstracttheory/BasicProperties.ipynb
@@ -186,6 +186,7 @@
    "metadata": {},
    "source": [
     "**Lemma:** Let $S$ be a subspace (not necessarily closed) of $V$. Then\n",
+    "\n",
     "$$\n",
     "S^\\bot := \\{ v \\in V : (v,w) = 0 \\; \\forall \\, w \\in S \\}\n",
     "$$\n",
@@ -243,7 +244,7 @@
     "and let $W$ be a complete space. Let $T : S \\rightarrow W$ be a bounded linear operator \n",
     "with respect to the norm $\\| T \\|_{V \\rightarrow W}$. Then, the operator can be uniquely extended onto $V$.\n",
     "\n",
-    "*Proof:* Let $u \\in V$, and let $v_n$ be a sequence such that $v_n \\rightarrow u$. Thus, $v_n$ is Cauchy. $T v_n$ is a well defined sequence in $W$. Since $T$ is continuous, $T v_n$ is also Cauchy. Since $W$ is complete, there exists a limit $w$ such that $T v_n \\rightarrow w$. The limit is independent of the sequence, and thus $T u$ can be defined as the limit $w$."
+    "*Proof:* Let $u \\in V$, and let $v_n$ be a sequence in $S$ such that $v_n \\rightarrow u$. Thus, $v_n$ is Cauchy. $T v_n$ is a well defined sequence in $W$. Since $T$ is bounded, $T v_n$ is also Cauchy. Since $W$ is complete, there exists a limit $w$ such that $T v_n \\rightarrow w$. The limit is independent of the sequence, and thus $T u$ can be defined as the limit $w$."
    ]
   },
   {

diff --git a/_sources/abstracttheory/exercises.ipynb b/_sources/abstracttheory/exercises.ipynb
@@ -133,7 +133,7 @@
    "metadata": {},
    "source": [
     "## One sup is enough\n",
-    "Let $A(.,.)$ be a continuous and symmetric bilinear-form on the Hilbert-space $V$. Proof that\n",
+    "Let $A(.,.)$ be a continuous and symmetric bilinear-form on the Hilbert-space $V$. Prove that\n",
     "\n",
     "$$\n",
     "\\sup_{x \\in V} \\sup_{y \\in V} \\frac{A(x,y)}{\\| x \\| \\, \\| y \\| } =\n",

diff --git a/_sources/abstracttheory/subspaceprojection.ipynb b/_sources/abstracttheory/subspaceprojection.ipynb
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bcbfab77-338a-47d8-aa0f-20054cdc5a0c",
+   "id": "09f3026f-87bd-47e7-956c-2ee6dc7c791a",
    "metadata": {},
    "source": [
     "> **Theorem:** \n",
@@ -53,15 +53,29 @@
     "$\\varphi(\\cdot)$ is a convex function, it takes its unique minimum $d$ at $t=0$. Thus\n",
     "\n",
     "$$\n",
-    "0 = \\frac{d \\varphi(t)}{dt}|_{t=0} = \\{ 2 (u-u_0, w) - 2 t (w,w) \\} |_{t=0} = \n",
-    "   2 (u-u_0,w)\n",
+    "0 = \\frac{d \\varphi(t)}{dt}|_{t=0} = \\{ -2 (u-u_0, w) + 2 t (w,w) \\} |_{t=0} = \n",
+    "   -2 (u-u_0,w)\n",
     "$$ \n",
     "We obtained $u-u_0 \\bot S$. If there were two minimizers $u_0 \\neq u_1$, \n",
     "then $u_0-u_1 = (u_0-u) - (u_1-u) \\bot S$ and $u_0-u_1 \\in S$, which implies\n",
     "$u_0-u_1 = 0$, a contradiction.\n",
-    " $\\Box$\n",
-    "\n",
-    "\n",
+    " $\\Box$"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "id": "bfdcb7c7-8a09-4f9b-9227-98b965c79f71",
+   "metadata": {},
+   "source": [
+    "![picproject.png](picproject.png)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6471497e-8bba-4765-af99-78486481d171",
+   "metadata": {},
+   "source": [
     "The theorem says that given an $u \\in V$, we can uniquely\n",
     "decompose it as\n",
     "\n",
@@ -102,7 +116,7 @@
   {
    "cell_type": "code",
    "execution_count": null,
-   "id": "f9bc9f14-fc3f-4de1-bf43-7f57d96d015d",
+   "id": "0f5f6865-6942-46f0-b84f-80517d248252",
    "metadata": {},
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    "source": []

diff --git a/abstracttheory/BasicProperties.html b/abstracttheory/BasicProperties.html
@@ -561,11 +561,12 @@ <h1><span class="section-number">7. </span>Basic properties<a class="headerlink"
 Now, let <span class="math notranslate nohighlight">\((u_n) \in \operatorname{ker} T^{\mathbb N}\)</span> converge to <span class="math notranslate nohighlight">\(u \in V\)</span>. Since
 <span class="math notranslate nohighlight">\(T\)</span> is continuous, <span class="math notranslate nohighlight">\(T u_n \rightarrow T u\)</span>, and thus <span class="math notranslate nohighlight">\(T u = 0\)</span> and <span class="math notranslate nohighlight">\(u \in \operatorname{ker} T\)</span>.
 <span class="math notranslate nohighlight">\(\Box\)</span></p>
-<p><strong>Lemma:</strong> Let <span class="math notranslate nohighlight">\(S\)</span> be a subspace (not necessarily closed) of <span class="math notranslate nohighlight">\(V\)</span>. Then
-$<span class="math notranslate nohighlight">\(
+<p><strong>Lemma:</strong> Let <span class="math notranslate nohighlight">\(S\)</span> be a subspace (not necessarily closed) of <span class="math notranslate nohighlight">\(V\)</span>. Then</p>
+<div class="math notranslate nohighlight">
+\[
 S^\bot := \{ v \in V : (v,w) = 0 \; \forall \, w \in S \}
-\)</span>$
-is a closed subspace.</p>
+\]</div>
+<p>is a closed subspace.</p>
 <p>The proof is similar to the Lemma on the kernel.</p>
 <p><strong>Definition:</strong> Let <span class="math notranslate nohighlight">\(V\)</span> and <span class="math notranslate nohighlight">\(W\)</span> be vector spaces. A <strong>linear operator</strong> <span class="math notranslate nohighlight">\(T : V \rightarrow W\)</span> is a linear mapping from <span class="math notranslate nohighlight">\(V\)</span> to <span class="math notranslate nohighlight">\(W\)</span>. The operator is called <strong>bounded</strong> if its operator-norm</p>
 <div class="math notranslate nohighlight">
@@ -587,7 +588,7 @@ <h1><span class="section-number">7. </span>Basic properties<a class="headerlink"
 <p><strong>Extension principle:</strong> Let <span class="math notranslate nohighlight">\(S\)</span> be a dense subspace of the normed space <span class="math notranslate nohighlight">\(V\)</span>,
 and let <span class="math notranslate nohighlight">\(W\)</span> be a complete space. Let <span class="math notranslate nohighlight">\(T : S \rightarrow W\)</span> be a bounded linear operator
 with respect to the norm <span class="math notranslate nohighlight">\(\| T \|_{V \rightarrow W}\)</span>. Then, the operator can be uniquely extended onto <span class="math notranslate nohighlight">\(V\)</span>.</p>
-<p><em>Proof:</em> Let <span class="math notranslate nohighlight">\(u \in V\)</span>, and let <span class="math notranslate nohighlight">\(v_n\)</span> be a sequence such that <span class="math notranslate nohighlight">\(v_n \rightarrow u\)</span>. Thus, <span class="math notranslate nohighlight">\(v_n\)</span> is Cauchy. <span class="math notranslate nohighlight">\(T v_n\)</span> is a well defined sequence in <span class="math notranslate nohighlight">\(W\)</span>. Since <span class="math notranslate nohighlight">\(T\)</span> is continuous, <span class="math notranslate nohighlight">\(T v_n\)</span> is also Cauchy. Since <span class="math notranslate nohighlight">\(W\)</span> is complete, there exists a limit <span class="math notranslate nohighlight">\(w\)</span> such that <span class="math notranslate nohighlight">\(T v_n \rightarrow w\)</span>. The limit is independent of the sequence, and thus <span class="math notranslate nohighlight">\(T u\)</span> can be defined as the limit <span class="math notranslate nohighlight">\(w\)</span>.</p>
+<p><em>Proof:</em> Let <span class="math notranslate nohighlight">\(u \in V\)</span>, and let <span class="math notranslate nohighlight">\(v_n\)</span> be a sequence in <span class="math notranslate nohighlight">\(S\)</span> such that <span class="math notranslate nohighlight">\(v_n \rightarrow u\)</span>. Thus, <span class="math notranslate nohighlight">\(v_n\)</span> is Cauchy. <span class="math notranslate nohighlight">\(T v_n\)</span> is a well defined sequence in <span class="math notranslate nohighlight">\(W\)</span>. Since <span class="math notranslate nohighlight">\(T\)</span> is bounded, <span class="math notranslate nohighlight">\(T v_n\)</span> is also Cauchy. Since <span class="math notranslate nohighlight">\(W\)</span> is complete, there exists a limit <span class="math notranslate nohighlight">\(w\)</span> such that <span class="math notranslate nohighlight">\(T v_n \rightarrow w\)</span>. The limit is independent of the sequence, and thus <span class="math notranslate nohighlight">\(T u\)</span> can be defined as the limit <span class="math notranslate nohighlight">\(w\)</span>.</p>
 <p><strong>Definition:</strong> A bounded linear operator <span class="math notranslate nohighlight">\(T : V \rightarrow W\)</span> is called <strong>compact</strong> if for every bounded sequence <span class="math notranslate nohighlight">\((u_n) \in V^{\mathbb N}\)</span>, the sequence <span class="math notranslate nohighlight">\((T u_n)\)</span> contains a convergent sub-sequence.</p>
 <p><strong>Lemma:</strong> Let <span class="math notranslate nohighlight">\(V, W\)</span> be Hilbert spaces. An operator is
 compact if and only if there exists a complete orthogonal system

diff --git a/abstracttheory/Coercive.html b/abstracttheory/Coercive.html
@@ -32,7 +32,7 @@
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+    <link rel="stylesheet" type="text/css" href="../_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css" />
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diff --git a/abstracttheory/exercises.html b/abstracttheory/exercises.html
@@ -32,7 +32,7 @@
     <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" />
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+    <link rel="stylesheet" type="text/css" href="../_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css" />
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@@ -577,7 +577,7 @@ <h3><span class="section-number">12.3.3. </span>complex-valued problem as real s
 </section>
 <section id="one-sup-is-enough">
 <h2><span class="section-number">12.4. </span>One sup is enough<a class="headerlink" href="#one-sup-is-enough" title="Link to this heading">#</a></h2>
-<p>Let <span class="math notranslate nohighlight">\(A(.,.)\)</span> be a continuous and symmetric bilinear-form on the Hilbert-space <span class="math notranslate nohighlight">\(V\)</span>. Proof that</p>
+<p>Let <span class="math notranslate nohighlight">\(A(.,.)\)</span> be a continuous and symmetric bilinear-form on the Hilbert-space <span class="math notranslate nohighlight">\(V\)</span>. Prove that</p>
 <div class="math notranslate nohighlight">
 \[
 \sup_{x \in V} \sup_{y \in V} \frac{A(x,y)}{\| x \| \, \| y \| } =

diff --git a/abstracttheory/infsup.html b/abstracttheory/infsup.html
@@ -32,7 +32,7 @@
     <link rel="stylesheet" type="text/css" href="../_static/styles/sphinx-book-theme.css?v=384b581d" />
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-    <link rel="stylesheet" type="text/css" href="../_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css?v=be8a1c11" />
+    <link rel="stylesheet" type="text/css" href="../_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css" />
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diff --git a/abstracttheory/subspaceprojection.html b/abstracttheory/subspaceprojection.html
@@ -508,13 +508,14 @@ <h1><span class="section-number">8. </span>Projection onto subspaces<a class="he
 <span class="math notranslate nohighlight">\(\varphi(\cdot)\)</span> is a convex function, it takes its unique minimum <span class="math notranslate nohighlight">\(d\)</span> at <span class="math notranslate nohighlight">\(t=0\)</span>. Thus</p>
 <div class="math notranslate nohighlight">
 \[
-0 = \frac{d \varphi(t)}{dt}|_{t=0} = \{ 2 (u-u_0, w) - 2 t (w,w) \} |_{t=0} = 
-   2 (u-u_0,w)
+0 = \frac{d \varphi(t)}{dt}|_{t=0} = \{ -2 (u-u_0, w) + 2 t (w,w) \} |_{t=0} = 
+   -2 (u-u_0,w)
 \]</div>
 <p>We obtained <span class="math notranslate nohighlight">\(u-u_0 \bot S\)</span>. If there were two minimizers <span class="math notranslate nohighlight">\(u_0 \neq u_1\)</span>,
 then <span class="math notranslate nohighlight">\(u_0-u_1 = (u_0-u) - (u_1-u) \bot S\)</span> and <span class="math notranslate nohighlight">\(u_0-u_1 \in S\)</span>, which implies
 <span class="math notranslate nohighlight">\(u_0-u_1 = 0\)</span>, a contradiction.
 <span class="math notranslate nohighlight">\(\Box\)</span></p>
+<p><img alt="picproject.png" src="../_images/picproject.png" /></p>
 <p>The theorem says that given an <span class="math notranslate nohighlight">\(u \in V\)</span>, we can uniquely
 decompose it as</p>
 <div class="math notranslate nohighlight">

diff --git a/primal/elasticity3D.html b/primal/elasticity3D.html
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diff --git a/reports/DG/splitting.err.log b/reports/DG/splitting.err.log
diff --git a/reports/MPIparallel/PETSc_interface.err.log b/reports/MPIparallel/PETSc_interface.err.log