From cdeaa95d1981b1b59fec0c177911a680ba602389 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Mon, 23 Oct 2023 16:23:00 +0200 Subject: [PATCH] updated numerical methods --- .../Numerical_methods_for_PDEs.tex | 113 +++++++++++++++++- 1 file changed, 112 insertions(+), 1 deletion(-) diff --git a/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex b/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex index 52df153..5c8a5a8 100644 --- a/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex +++ b/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex @@ -122,5 +122,116 @@ \item $I_Kv=v$ $\forall v\in \mathcal{P}$, i.e.\ $I_K$ is a projection. \end{enumerate} \end{lemma} + \begin{definition} + A \emph{subdivision} of a bounded open set $\Omega\subset \RR^n$ is a collection $\mathcal{T}$ of open sets $K_i$ such that: + \begin{enumerate} + \item $K_i\cap K_j=\varnothing$ $\forall i\neq j$. + \item $\overline{\Omega}= \bigcup_{K\in\mathcal{T}}\overline{K}$. + \end{enumerate} + \end{definition} + \begin{definition} + Let $\mathcal{T}$ be a subdivision of $\Omega$ such that for each $K\in\mathcal{T}$ there exists a finite element $(K,\mathcal{P},\mathcal{N})$ with local interpolant $I_K$. Let $m$ be the order of the highest partial derivative appearing in any of the degrees of freedom of $\mathcal{N}$. We define the \emph{global interpolant} $I_\mathcal{T}v$ of $\mathcal{T}$, for $v\in \mathcal{C}^m(\overline{\Omega})$, as: + $$ + I_\mathcal{T}v|_K:=I_Kv\quad \forall K\in\mathcal{T} + $$ + \end{definition} + \begin{definition} + A \emph{triangulation} of a bounded open set $\Omega\subset \RR^2$ is a subdivision $\mathcal{T}$ of $\Omega$ such that: + \begin{enumerate} + \item Each $K\in\mathcal{T}$ is a triangle. + \item The intersection of two triangles is either empty or a common vertex or a common edge. + \end{enumerate} + \end{definition} + \begin{definition} + Let $(\widehat{K}, \widehat{\mathcal{P}}, \widehat{\mathcal{N}})$, $(K,\mathcal{P},\mathcal{N})$ be finite elements and $T:\RR^n\to\RR^n$ be an affine transformation. We say that these finite elements are \emph{affinely equivalent} by $T$ if: + \begin{enumerate} + \item $K=T(\widehat{K})$. + \item $\mathcal{P}=\{\widehat{p}\circ T^{-1}:\widehat{p}\in\widehat{\mathcal{P}}\}$. + \item $\mathcal{N}=\{N_i\}$, where $N_i(p)=\widehat{N}_i(p\circ T)$ $\forall p\in\mathcal{P}$. + \end{enumerate} + \end{definition} + \begin{lemma} + Let $(\widehat{K}, \widehat{\mathcal{P}}, \widehat{\mathcal{N}})$, $(K,\mathcal{P},\mathcal{N})$ be two affine equivalent finite elements by the affine transformation $\vf{T}_K$. Then: + $$ + I_{\widehat{K}}(v\circ \vf{T}_K)=I_Kv\circ T + $$ + \end{lemma} + \subsubsection{Polygonal interpolation in Sobolev spaces} + \begin{lemma}[Bramble-Hilbert lemma] + Let $F:W^{k,p}(\Omega)\to\RR$ be such that: + \begin{enumerate} + \item $\abs{F(v)}\leq c_1\abs{v}_{W^{k,p}(\Omega)}$ $\forall v\in W^{k,p}(\Omega)$, where $$ + \abs{v}_{W^{k,p}(\Omega)}:= + \begin{cases} + {\left({\sum_{\abs{\alpha}= k}\norm{\partial^\alpha v}_{L^p(\Omega)}}\right)}^{1/p} & \text{if }p<\infty \\ + \max_{\abs{\alpha}= k}\norm{\partial^\alpha v}_{L^\infty(\Omega)} & \text{if }p=\infty + \end{cases} + $$ + \item $\abs{F(u+v)}\leq c_2(\abs{F(u)}+\abs{F(v)})$ $\forall u,v\in W^{k,p}(\Omega)$. + \item $\abs{F(q)}=0$ $\forall q\in\mathcal{P}_{k-1}(\Omega)$, where $\mathcal{P}_{\ell}(\Omega)$ is the space of polynomials of degree less than $\ell$. + \end{enumerate} + Then, $\exists C>0$ such that $\forall v\in W^{k,p}(\Omega)$: + $$ + \abs{F(v)}\leq C\abs{v}_{W^{k,p}(\Omega)} + $$ + \end{lemma} + \begin{theorem} + Let $(K,\mathcal{P},\mathcal{N})$ be a finite element such that $\mathcal{P}_{k-1}\subseteq \mathcal{P}$ for some $k\in\NN$ and all $N\in\mathcal{N}$ be bounded in $W^{k,p}(K)$ for some $p\in[1,\infty]$. Then, $\exists C>0$ such that $\forall v\in W^{k,p}(K)$: + $$ + \abs{v-I_Kv}_{W^{\ell,p}(K)}\leq C\abs{v}_{W^{k,p}(K)} \quad\forall \ell\in\{0,\ldots,k\} + $$ + \end{theorem} + \begin{remark} + Let $(K,\mathcal{P},\mathcal{N})$ be a finite element and $(\widehat{K}, \widehat{\mathcal{P}}, \widehat{\mathcal{N}})$ be the reference element. From now on, if they are affine equivalent by $\vf{T}_K:\widehat{K}\to K$, we will assume that $\vf{T}_K\widehat{x}=\vf{A}_K \widehat{x}+\vf{b}_K$, with $\vf{A}_K$ invertible. + \end{remark} + \begin{lemma} + Let $k\in \NN$ and $p\in[1,\infty]$. Then, $\exists C>0$ such that $\forall K\subset\Omega$ and $\forall v\in W^{k,p}(\widehat{K})$: + \begin{align*} + \abs{v}_{W^{k,p}(\widehat{K})} & \leq C\norm{\vf{A}_K}^k\abs{\det \vf{A}_K}^{-1/p}\abs{v}_{W^{k,p}(K)} \\ + \abs{v}_{W^{k,p}(K)} & \leq C\norm{{\vf{A}_K}^{-1}}^k\abs{\det \vf{A}_K}^{1/p}\abs{v}_{W^{k,p}(\widehat{K})} + \end{align*} + \end{lemma} + \begin{definition} + Let $(K,\mathcal{P},\mathcal{N})$ be a finite element. We define the \emph{diameter} of $K$ as: + $$ + h_K:=\max_{x,y\in K}\norm{x-y} + $$ + We define the \emph{insphere diameter} of $K$ as: + $$ + \rho_K:=2\max\{\rho>0:B(x,\rho)\subset K\text{ for some }x\in K\} + $$ + We define the \emph{condition number} of $K$ as $\sigma_K:=\frac{h_K}{\rho_K}$. + \end{definition} + \begin{lemma} + Let $(K,\mathcal{P},\mathcal{N})$, $(\widehat{K}, \widehat{\mathcal{P}}, \widehat{\mathcal{N}})$ be affine equivalent finite elements by $\vf{T}_K:\widehat{K}\to K$. Then, $\abs{\det \vf{A}_K}=\frac{\vol(K)}{\vol(\widehat{K})}$, $\norm{\vf{A}_K}\leq \frac{h_K}{\rho_{\widehat{K}}}$ and $\norm{{\vf{A}_K}^{-1}}\leq \frac{h_{\widehat{K}}}{\rho_K}$. + \end{lemma} + \begin{theorem}[Local interpolation error] + Let $(\widehat{K}, \widehat{\mathcal{P}}, \widehat{\mathcal{N}})$ be a finite element with $\mathcal{P}_{k-1}\subseteq \mathcal{P}$ for some $k\in\NN$ and all $N\in\mathcal{N}$ be bounded in $W^{k,p}(\widehat{K})$ for some $p\in[1,\infty]$. Then, for all finite element $(K,\mathcal{P},\mathcal{N})$ affine equivalent to $(\widehat{K}, \widehat{\mathcal{P}}, \widehat{\mathcal{N}})$ by $\vf{T}_K:\widehat{K}\to K$, $\exists C>0$ (independent of $K$) such that $\forall v\in W^{k,p}(K)$: + $$ + \abs{v-I_Kv}_{W^{\ell,p}(K)}\leq C{h_K}^k{\sigma_K}^{\ell}\abs{v}_{W^{k,p}(K)} \quad\forall \ell\in\{0,\ldots,k\} + $$ + \end{theorem} + \begin{definition} + A subdivision $\mathcal{T}$ of $\Omega\in \RR^n$ is called \emph{regular} if $\exists C>0$ such that $\forall K\in\mathcal{T}$ we have $\sigma_K\leq C$. + \end{definition} + \begin{theorem}[Global interpolation error] + Let $\mathcal{T}$ be a regular subdivision of $\Omega\in \RR^n$ and $(\widehat{K}, \widehat{\mathcal{P}}, \widehat{\mathcal{N}})$ be a reference finite element with $\mathcal{P}_{k-1}\subseteq \mathcal{P}$ for some $k\in\NN$ and all $N\in\mathcal{N}$ be bounded in $W^{k,p}(\widehat{K})$ for some $p\in[1,\infty]$. Let $h:= \max_{K\in\mathcal{T}}h_K$. Then, $\exists C>0$ (independent of $h$) such that $\forall v\in W^{k,p}(\Omega)$: + \begin{multline*} + \abs{v-I_\mathcal{T}v}_{W^{\ell,p}(\Omega)}+\sum_{\ell=1}^k\left(h^\ell\sum_{K\in\mathcal{T}} \abs{v-I_Kv}_{W^{\ell,p}(K)}^p\right)^{1/p}\leq \\\leq C h^k\abs{v}_{W^{k,p}(\Omega)} + \end{multline*} + if $p<\infty$ and: + \begin{multline*} + \abs{v-I_\mathcal{T}v}_{W^{\ell,\infty}(\Omega)}+\sum_{\ell=1}^kh^\ell \max_{K\in\mathcal{T}}\abs{v-I_Kv}_{W^{\ell,\infty}(K)}\leq\\\leq C h^k\abs{v}_{W^{k,\infty}(\Omega)} + \end{multline*} + if $p=\infty$. + \end{theorem} + \subsubsection{Error estimates for finite element approximation} + \begin{theorem} + Let $\Omega\subset\RR^n$ be open and bounded, $u\in H^1(\Omega)$ be the solution of the boundary value problem and $\mathcal{T}$ be a regular triangulation of $\Omega$ with reference element $(\widehat{K}, \widehat{\mathcal{P}}, \widehat{\mathcal{N}})$ such that $\mathcal{P}_{k-1}\subseteq \mathcal{P}$ for some $k\in\NN$. Let $u_h\in V_h$ be the solution of the Galerkin method. Then, if $u\in H^m$, with $\frac{n}{2}< m0$ (independent of $h$ and $u$) such that: + $$ + \norm{u-u_h}_{H^1(\Omega)}\leq C h^{m-1}\norm{u}_{H^m(\Omega)} + $$ + \end{theorem} + \subsection{Spectral methods} \end{multicols} -\end{document} \ No newline at end of file +\end{document}