Skip to content

Commit

Permalink
Update documentation
Browse files Browse the repository at this point in the history
  • Loading branch information
@JSchoeberl
JSchoeberl committed Mar 18, 2024
1 parent 83d7801 commit 30e8314
Show file tree
Hide file tree
Showing 3 changed files with 58 additions and 17 deletions.
30 changes: 25 additions & 5 deletions _sources/sobolevspaces/exercises.ipynb
View file
Original file line number Diff line number Diff line change
Expand Up @@ -43,16 +43,35 @@
"source": [
"## Poincaré inequality\n",
"\n",
"### Eqivalent versions of the Poincaré inequality\n",
"\n",
"Prove that the following inequalities are equivalent\n",
"\n",
"\n",
"1. There exists a constant $c_p$ such that\n",
"\n",
" $$ \\| u - \\overline u \\|_{L_2}^2 \\leq c_P^2 \\, \\| \\nabla u \\|^2_{L_2} \\; \\forall \\, u \\in H^1(\\Omega)$$\n",
"\n",
"Here, $\\overline u$ is the mean value of $u$, which is $\\frac{1}{|\\Omega|} \\int_\\Omega u(x) dx$, understood as a constant function.\n",
"\n",
"2. There exists a constant $c_p$ such that\n",
"\n",
" $$ \\| u \\|_{L_2}^2 \\leq c_P^2 \\, \\| \\nabla u \\|_{L_2}^2 + \\frac{1}{|\\Omega|} \\left( \\int_\\Omega u \\right)^2 $$\n",
"\n",
"Hint: Pythagoras. First, observe that $\\overline u$ and $u - \\overline u$ are orthogonal with respect to the $L_2$ inner product.\n",
"\n",
"### Poincaré inequality in 1D\n",
"\n",
"Prove the Poincaré inequality on the interval $I = (a,b)$ with elementary tools:\n",
"\n",
"$$\n",
"\\| u \\|_{L_2(I)}^2 \\leq c_P \\left( \\| u^\\prime \\|_{L_2(I)}^2 + \\big( \\int_I u dx \\big)^2 \\right)\n",
"\\| u - \\overline u \\|_{L_2(I)}^2 \\leq c_P^2 \\| u^\\prime \\|_{L_2(I)}^2\n",
"\\qquad \\forall \\, u \\in C^1(\\overline I)\n",
"$$\n",
"\n",
"Hint: ..."
"How does the Poincaré constant depend on $a$ and $b$ ? \n",
"\n",
"Hint: Bring the left hand side to something similar to $\\int \\left( \\int u(y) - u(x) dx\\right) ^2 dy$. Then, use the fundamental theorem of calculus: $u(y)-u(x) = \\int_x^y u^\\prime(s) \\, ds$. finally: Cauchy-Schwarz"
]
},
{
Expand All @@ -71,18 +90,19 @@
"$$ \n",
"where $\\overline u := | \\Omega |^{-1} \\int_\\Omega u \\, dx$ is the mean value.\n",
"\n",
"Hint: apply the Bramble-Hilbert theorem\n",
"Use the Bramble-Hilbert theorem\n",
"\n",
"### Scaled domain\n",
"\n",
"Let $\\Omega = B_r(p)$ a ball with center $p$ and radius $r$. Prove that there exists a\n",
"constant $c$, independent of $r$ and $p$, such that\n",
"\n",
"$$\n",
"\\| u - \\overline u \\|_{L_2(\\Omega)} \\leq c r \\, \\| \\nabla u \\|_{L_2(\\Omega)} \\qquad \\forall \\, u \\in H^1(\\Omega)\n",
"\\| u - \\overline u \\|_{L_2} \\leq c r \\, \\| \\nabla u \\|_{L_2} \\qquad \\forall \\, u \\in H^1(B_r(p))\n",
"$$ \n",
"\n",
"Hint: Prove the estimate for the unit ball $B_1(0)$. Define a function $\\Phi$ mapping the unit-ball to the arbitrary ball $B_r(p)$. For $u \\in H^1(B_r(p))$, define the pull-back $u \\circ \\Phi \\in H^1(B_1(0))$\n"
"Hint: Prove the estimate for the unit ball $B_1(0)$. Define a function $\\Phi$ mapping the unit-ball to the arbitrary ball $B_r(p)$. For $u \\in H^1(B_r(p))$, define the pull-back $u \\circ \\Phi \\in H^1(B_1(0))$.\n",
"Does mean-value and pull-back commute, i.e. does there hold $\\overline u \\circ \\Phi = \\overline{ u \\circ \\Phi}$ ? \n"
]
},
{
Expand Down
2 changes: 1 addition & 1 deletion searchindex.js
View file

Large diffs are not rendered by default.

43 changes: 32 additions & 11 deletions sobolevspaces/exercises.html
View file
Original file line number Diff line number Diff line change
Expand Up @@ -63,7 +63,7 @@
<link rel="index" title="Index" href="../genindex.html" />
<link rel="search" title="Search" href="../search.html" />
<link rel="next" title="20. Finite Element Method" href="../FEM/finiteelements.html" />
<link rel="prev" title="18. Experiments with norms (WIP)" href="experiments.html" />
<link rel="prev" title="18. Experiments with norms" href="experiments.html" />
<meta name="viewport" content="width=device-width, initial-scale=1"/>
<meta name="docsearch:language" content="en"/>
</head>
Expand Down Expand Up @@ -203,7 +203,7 @@
<li class="toctree-l1"><a class="reference internal" href="Traces.html">15. Trace theorems and their applications</a></li>
<li class="toctree-l1"><a class="reference internal" href="equivalentnorms.html">16. Equivalent norms on <span class="math notranslate nohighlight">\(H^1\)</span> and on sub-spaces</a></li>
<li class="toctree-l1"><a class="reference internal" href="preciseweak.html">17. The weak formulation of the Poisson equation</a></li>
<li class="toctree-l1"><a class="reference internal" href="experiments.html">18. Experiments with norms (WIP)</a></li>
<li class="toctree-l1"><a class="reference internal" href="experiments.html">18. Experiments with norms</a></li>
<li class="toctree-l1 current active"><a class="current reference internal" href="#">19. Exercises</a></li>
</ul>
<p aria-level="2" class="caption" role="heading"><span class="caption-text">Finite Element Method</span></p>
Expand Down Expand Up @@ -476,7 +476,8 @@ <h2> Contents </h2>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#poincare-inequality">19.2. Poincaré inequality</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#poincare-inequality-in-1d">19.2.1. Poincaré inequality in 1D</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#eqivalent-versions-of-the-poincare-inequality">19.2.1. Eqivalent versions of the Poincaré inequality</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#poincare-inequality-in-1d">19.2.2. Poincaré inequality in 1D</a></li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bramble-hilbert-lemma">19.3. Bramble-Hilbert Lemma</a><ul class="nav section-nav flex-column">
Expand Down Expand Up @@ -525,15 +526,33 @@ <h3><span class="section-number">19.1.2. </span>Friedrichs’ inequality on the
</section>
<section id="poincare-inequality">
<h2><span class="section-number">19.2. </span>Poincaré inequality<a class="headerlink" href="#poincare-inequality" title="Link to this heading">#</a></h2>
<section id="eqivalent-versions-of-the-poincare-inequality">
<h3><span class="section-number">19.2.1. </span>Eqivalent versions of the Poincaré inequality<a class="headerlink" href="#eqivalent-versions-of-the-poincare-inequality" title="Link to this heading">#</a></h3>
<p>Prove that the following inequalities are equivalent</p>
<ol class="arabic simple">
<li><p>There exists a constant <span class="math notranslate nohighlight">\(c_p\)</span> such that</p></li>
</ol>
<div class="math notranslate nohighlight">
\[ \| u - \overline u \|_{L_2}^2 \leq c_P^2 \, \| \nabla u \|^2_{L_2} \; \forall \, u \in H^1(\Omega)\]</div>
<p>Here, <span class="math notranslate nohighlight">\(\overline u\)</span> is the mean value of <span class="math notranslate nohighlight">\(u\)</span>, which is <span class="math notranslate nohighlight">\(\frac{1}{|\Omega|} \int_\Omega u(x) dx\)</span>, understood as a constant function.</p>
<ol class="arabic" start="2">
<li><p>There exists a constant <span class="math notranslate nohighlight">\(c_p\)</span> such that</p>
<div class="math notranslate nohighlight">
\[ \| u \|_{L_2}^2 \leq c_P^2 \, \| \nabla u \|_{L_2}^2 + \frac{1}{|\Omega|} \left( \int_\Omega u \right)^2 \]</div>
</li>
</ol>
<p>Hint: Pythagoras. First, observe that <span class="math notranslate nohighlight">\(\overline u\)</span> and <span class="math notranslate nohighlight">\(u - \overline u\)</span> are orthogonal with respect to the <span class="math notranslate nohighlight">\(L_2\)</span> inner product.</p>
</section>
<section id="poincare-inequality-in-1d">
<h3><span class="section-number">19.2.1. </span>Poincaré inequality in 1D<a class="headerlink" href="#poincare-inequality-in-1d" title="Link to this heading">#</a></h3>
<h3><span class="section-number">19.2.2. </span>Poincaré inequality in 1D<a class="headerlink" href="#poincare-inequality-in-1d" title="Link to this heading">#</a></h3>
<p>Prove the Poincaré inequality on the interval <span class="math notranslate nohighlight">\(I = (a,b)\)</span> with elementary tools:</p>
<div class="math notranslate nohighlight">
\[
\| u \|_{L_2(I)}^2 \leq c_P \left( \| u^\prime \|_{L_2(I)}^2 + \big( \int_I u dx \big)^2 \right)
\| u - \overline u \|_{L_2(I)}^2 \leq c_P^2 \| u^\prime \|_{L_2(I)}^2
\qquad \forall \, u \in C^1(\overline I)
\]</div>
<p>Hint: …</p>
<p>How does the Poincaré constant depend on <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> ?</p>
<p>Hint: Bring the left hand side to something similar to <span class="math notranslate nohighlight">\(\int \left( \int u(y) - u(x) dx\right) ^2 dy\)</span>. Then, use the fundamental theorem of calculus: <span class="math notranslate nohighlight">\(u(y)-u(x) = \int_x^y u^\prime(s) \, ds\)</span>. finally: Cauchy-Schwarz</p>
</section>
</section>
<section id="bramble-hilbert-lemma">
Expand All @@ -546,17 +565,18 @@ <h3><span class="section-number">19.3.1. </span>Mean-value interpolation<a class
\| u - \overline u \|_{L_2(\Omega)} \leq c \, \| \nabla u \|_{L_2(\Omega)} \qquad \forall \, u \in H^1(\Omega),
\]</div>
<p>where <span class="math notranslate nohighlight">\(\overline u := | \Omega |^{-1} \int_\Omega u \, dx\)</span> is the mean value.</p>
<p>Hint: apply the Bramble-Hilbert theorem</p>
<p>Use the Bramble-Hilbert theorem</p>
</section>
<section id="scaled-domain">
<h3><span class="section-number">19.3.2. </span>Scaled domain<a class="headerlink" href="#scaled-domain" title="Link to this heading">#</a></h3>
<p>Let <span class="math notranslate nohighlight">\(\Omega = B_r(p)\)</span> a ball with center <span class="math notranslate nohighlight">\(p\)</span> and radius <span class="math notranslate nohighlight">\(r\)</span>. Prove that there exists a
constant <span class="math notranslate nohighlight">\(c\)</span>, independent of <span class="math notranslate nohighlight">\(r\)</span> and <span class="math notranslate nohighlight">\(p\)</span>, such that</p>
<div class="math notranslate nohighlight">
\[
\| u - \overline u \|_{L_2(\Omega)} \leq c r \, \| \nabla u \|_{L_2(\Omega)} \qquad \forall \, u \in H^1(\Omega)
\| u - \overline u \|_{L_2} \leq c r \, \| \nabla u \|_{L_2} \qquad \forall \, u \in H^1(B_r(p))
\]</div>
<p>Hint: Prove the estimate for the unit ball <span class="math notranslate nohighlight">\(B_1(0)\)</span>. Define a function <span class="math notranslate nohighlight">\(\Phi\)</span> mapping the unit-ball to the arbitrary ball <span class="math notranslate nohighlight">\(B_r(p)\)</span>. For <span class="math notranslate nohighlight">\(u \in H^1(B_r(p))\)</span>, define the pull-back <span class="math notranslate nohighlight">\(u \circ \Phi \in H^1(B_1(0))\)</span></p>
<p>Hint: Prove the estimate for the unit ball <span class="math notranslate nohighlight">\(B_1(0)\)</span>. Define a function <span class="math notranslate nohighlight">\(\Phi\)</span> mapping the unit-ball to the arbitrary ball <span class="math notranslate nohighlight">\(B_r(p)\)</span>. For <span class="math notranslate nohighlight">\(u \in H^1(B_r(p))\)</span>, define the pull-back <span class="math notranslate nohighlight">\(u \circ \Phi \in H^1(B_1(0))\)</span>.
Does mean-value and pull-back commute, i.e. does there hold <span class="math notranslate nohighlight">\(\overline u \circ \Phi = \overline{ u \circ \Phi}\)</span> ?</p>
</section>
</section>
<section id="fractional-sobolev-spaces">
Expand Down Expand Up @@ -627,7 +647,7 @@ <h3><span class="section-number">19.4.2. </span>Point evaluation functional<a cl
<i class="fa-solid fa-angle-left"></i>
<div class="prev-next-info">
<p class="prev-next-subtitle">previous</p>
<p class="prev-next-title"><span class="section-number">18. </span>Experiments with norms (WIP)</p>
<p class="prev-next-title"><span class="section-number">18. </span>Experiments with norms</p>
</div>
</a>
<a class="right-next"
Expand Down Expand Up @@ -661,7 +681,8 @@ <h3><span class="section-number">19.4.2. </span>Point evaluation functional<a cl
</ul>
</li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#poincare-inequality">19.2. Poincaré inequality</a><ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#poincare-inequality-in-1d">19.2.1. Poincaré inequality in 1D</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#eqivalent-versions-of-the-poincare-inequality">19.2.1. Eqivalent versions of the Poincaré inequality</a></li>
<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#poincare-inequality-in-1d">19.2.2. Poincaré inequality in 1D</a></li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bramble-hilbert-lemma">19.3. Bramble-Hilbert Lemma</a><ul class="nav section-nav flex-column">
Expand Down

0 comments on commit 30e8314

Please sign in to comment.