The Poincare-Friedrichs inequality on polygon and its applications.tex

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\title{\emph{VEM Theory Analysis}\\\textbf{The Poincar\'{e}-Friedrichs inequality on polygon and its applications}}
% Title
\author{\Large{\emph{Submitted by:}}\\\\\Large{\textbf{JGH Seminars}}\\\textbf{School of Mathematical Science}\\\textbf{Shanghai Jiao Tong University in Shanghai}\\}
%\author{JGH Seminars\\{\small(SJTU, School of Mathematical Science, Shanghai, 200240)}}
% Author name
\date{}
\maketitle

\begin{center}
{\Large 多角形上的Poincar\'{e}-Friedrichs不等式及其应用}
\end{center}

%\noindent{\bf Key Words}: \qquad Polygon Division,Centroid Coordinate,Matlab code.
\vspace{1pc}
%\noindent{\bf Abstract}: This article mainly help students to understand the homeworks better.
%\vspace{1pc}

%%%\begin{figure}[b]
%%%\rule[-2.5truemm]{5cm}{0.1truemm}\\[2mm]{
%%%\quad\quad {\bf收稿日期:}\,2016-09-27\, \\
%%%{\bf 作者简介:}\,黄建国, 男, 教授, 博士生导师,\\
%%%  \mbox{}\qquad\qquad~\,林森, 男, 博士研究生, 研究方向以有限元方法为主. \\
%%%  \mbox{}\qquad\qquad~\,电话: 021-54743151-2105; \\
%%%  %\mbox{}\qquad\qquad~\,E-mail:jghuang@sjtu.edu.cn \& linsenlyj@163.com }
%%%  \mbox{}\qquad\qquad~\,E-mail:jghuang@sjtu.edu.cn \& sjtu\_Einsteinlin7@sjtu.edu.cn }
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%%\rule[-2.5truemm]{5cm}{0.1truemm}\\[2mm]{
%%\quad {\bf Submitted date:}\,2016-10-08\, \\
%%{\bf Author:}\,Sen Lin, Male,PhD.\\
%%  \mbox{}\qquad\quad~\,Research on the Finite Element Method(FEM). \\
%%  \mbox{}\qquad\quad~\,Tel:13262292127; \\
%%  \mbox{}\qquad\quad~\,E-mail:linsenlyj@163.com.}
%%  %\mbox{}\qquad\qquad~\,E-mail:jghuang@sjtu.edu.cn \& sjtu\_Einsteinlin7@sjtu.edu.cn }
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%%% Section 1
\section{多角形的Poincar\'{e}-Friedrichs不等式}
由等价模定理可以推得
\begin{itemize}
  \item Poincar\'{e}不等式:
       \[
         \|v\|_{0,K}\leq{C}\big(|v|_{1,K} + |\int_{K}v(x)\rm{d}x|\big)
         \quad \forall v \in H^1(K);
       \]
%%%       \begin{align*}
%%%           \|v\|_{0,K}\leq{C}\big(|v|_{1,K} + |\int_{K}v(x)\rm{d}x|\big)
%%%           \quad \forall v \in H^1(K);
%%%       \end{align*}
  \item Friedrichs不等式:
       \[
         \|v\|_{0,K}\leq{C}\big(|v|_{1,K} + |\int_{\partial{K}}v(x)\rm{d}s(x)|\big)
         \quad \forall v \in H^1(K).
       \]
\end{itemize}
上述两式是$p=2$的结果,进一步,我们可以得到更一般的结果.
\begin{equation*}
\label{Eq:PFIeq}%%%Poincar\'{e}-Friedrichs inequality for W^{1,p}.
\left\{
\begin{aligned}
   \|v\|_{L^p(K)} & \leq{C}\big(|v|_{W^{1,p}(K)}+|\int_{K}v(x)\rm{d}x|\big)
         & \forall v \in W^{1,p}(K);       \\
   \|v\|_{L^p(K)} & \leq{C}\big(|v|_{W^{1,p}(K)}+|\int_{\partial{K}}v(x)\rm{d}s(x)|\big)
         & \forall v \in W^{1,p}(K).
\end{aligned}
\right.
\end{equation*}
其中,$K\subset\mathbb{R}^2$是一个有界且连通的多角形区域和$1\leq{p}\leq\infty$.
\begin{itemize}
  \item 1.由紧性论证技巧可证明:$C=C(K)$,即正常数$C$ 仅取决于$K$.
  \item 2.若$K$为三角形,可由\textbf{scaling-argument} 来获得一致估计.
\end{itemize}

\begin{question}
\label{Q1}
当$K$为一般的多角形时,如何处理?即它需要满足什么要求时,可以得到一致估计?
\end{question}

下面,我们假设多角形$K$满足如下\textbf{性质P}:
\begin{property}[\large\textbf{P}] \label{Pro1}
有以下三点: \\
(1)存在$K$的正规三角剖分$\mathcal{T}_{h_K}(K)$,这里正规(shape-regularity),
   意指,存在常数$C_r>0$使
   \[
     h_{\tau}/\rho_{\tau} \leq C_r < \infty, \quad \forall \tau \in \mathcal{T}_{h_K}(K),
   \]
   且$K$的每条边必为$\mathcal{T}_{h_K}(K)$中某一单元$\tau$之边; \\
(2)$\mathcal{T}_{h_K}(K)$中的单元个数是一致有界的,即$\#\mathcal{T}_{h_K}(K)\leq{L}$; \\
(3)在$K$中包含以半径为$r$的一个圆,且$h_K\lesssim{r}$, 其中生成的正常数$C$有界.
\end{property}

进一步,我们给出星形区域的定义,并给出两个例子来说明.
\begin{definition}[Star-shaped区域]\label{D1} \mbox{}\par
   我们称区域$\Omega\subset\mathbb{R}^2$关于某个圆$B$(3 维为球)是星形的,
如果对于$\forall{x}\in\Omega$,$\{x\}\cup{B}$ 的闭凸包还是$\Omega$的子集.
\end{definition}

\begin{figure}[!h]
  \centering
    \subfigure[Domain star-shaped with respect to $B_1$ but not $B_2$]{
    \label{Fig1:subfig:1}      %% label for first subfigure
    \includegraphics[width=2.1in,height=2.1in]{StarDomain01.eps}} %{StarDomain1.eps}
    \hspace{0.05in}
    \subfigure[Domain which is not star-shaped with respect to any circle $B$]{
    \label{Fig1:subfig:2}      %% label for second subfigure
    \includegraphics[width=2.1in,height=2.1in]{StarDomain02.eps}}
    \caption{星形区域的两个例子}
    \label{Fig1} %% label for entire figure
\end{figure}
从图\ref{Fig1}可以看出图\ref{Fig1:subfig:1} 中的区域$\Omega$是星形区域,而
图\ref{Fig1:subfig:2}中的区域$\Omega$却不是星形区域.

\begin{proposition}\label{P1}
若多角形$K$为星形区域(star-shaped domain)且相应圆的半径$r$满足$h_K\lesssim{r}$,
且$\mathcal{T}_{h_K}(K)$中的每一单元$\tau$满足$h_{\tau}\approx{h_{K}}$,则
显然$K$满足\textbf{性质\ref{Pro1}}.
\end{proposition}

\begin{remark} \label{R1} %\mbox{}\par
在此,我们想说明一点:一方面,$h_{\tau}\lesssim{h_{K}}$ 显然成立;另一方面,由
\[
  L*\pi(\frac{h_\tau}{2})\geq |K| \geq \pi{r}^2  \gtrsim\pi h_K^2
\]
可得 $h_{K}\lesssim{h_{\tau}}$. \\
正如下面的示意图所示,我们发现当多角形不满足性质\ref{Pro1}中(3)的条件
$h_K\lesssim{r}$时,由上述推导可知,我们无法得到$h_{\tau}\approx{h_{K}}$,进而无法
得到$|u-u_h|_{1,K}$的一致估计.这进一步说明了性质\ref{Pro1}(3)的重要性!
\begin{figure}[!ht]\label{Fig2}
   \centering
     %\includegraphics[width=3.2in,height=1.2in]{Diamond1.eps}
     \includegraphics[width=3.6in,height=1.2in]{Diamond2.eps}
   \caption{不满足性质\ref{Pro1}中的第3条的多角形示意图}
\end{figure}

\end{remark}

\begin{lemma}\label{L1}
给定一个三角形单元$\tau$,其满足$h_\tau/\rho_{\tau}\leq{C_r}<\infty$,设$e$为
$K$的任一边,则成立
\begin{equation}
\label{Eq:FIeq}
  \|v\|_{0,p,K} \lesssim {h}_{\tau}^{-1}|v|_{1,p,K} + h_{\tau}^{2/p-1}|\int_ev\rm{d}s|
  \quad \forall v\in W^{1,p}(K).
\end{equation}
式中$1\leq p<\infty$.
\end{lemma}

由\textbf{scaling argument}和参考元上的Friedrichs不等式立知结果.


\begin{theorem}[带几何尺度的Friedrichs不等式]
\label{T1}\mbox{}\par
假设多角形$K$满足\textbf{性质\ref{Pro1}},则对任何$v\in{W}^{1,p}(K)$
使$\int_{\partial{K}}v\rm{d}s=0$,成立估计
\begin{equation}
\label{Eq:GFIeq} %%%  Friedrichs Inequality with Geometric scale
  \|v\|_{L^p(K)} \lesssim {h_K}|v|_{1,p,K}
\end{equation}
式中生成常数与$h_K$无关.
\end{theorem}

\begin{proof}
\label{Proof1}
我们分以下几步给出证明. \\
\textbf{step 1.}设$h_K=diam(K)=1$,故由$h_{\tau}\approx{h_K}$知
$h_{\tau}\approx1$.设$\partial{K}$由$N$条边$e_1,e_2,\cdots,e_N$ 组成,
对任一边$e$,定义
\[
  A_ev = \frac{1}{|e|}\int_ev\rm{d}s.
\]
则由条件$\int_{\partial{K}}v\rm{d}s=0$知
\[
  \sum_{i=1}^N |e_i|A_{e_i}v = 0 \Rightarrow
  \sum_{i=1}^N \frac{|e_i|}{|\partial{K}|} A_{e_i}v = 0.
\]
进一步,有
\begin{equation}
\label{Eq:GFIeq_LpEst1}
  \begin{aligned}
   \|v\|_{L^p(K)} & =     \|v - \sum_{i=1}^N \frac{|e_i|}{|\partial{K}|} A_{e_i}v\|_{L^p(K)}    \\
                  & =     \|\sum_{i=1}^N \frac{|e_i|}{|\partial{K}|} (v - A_{e_i}v)\|_{L^p(K)}  \\
                  & \leq  \sum_{i=1}^N \frac{|e_i|}{|\partial{K}|} \|(v - A_{e_i}v)\|_{L^p(K)}  \\
                  & \leq  \max_{1\leq{i}\leq{N}} \|(v - A_{e_i}v)\|_{L^p(K)}.
  \end{aligned}
\end{equation}
\textbf{step 2.} 记$w_i = v - A_{e_i}v$,$1\leq{i}\leq{N}$.由定义知
\begin{equation}
\label{Eq:GFIeq_LpEst2}
   \|w_i\|_{L^p(K)} \leq \sum_{\tau\in\mathcal{T}_{h_K}(K)} \|w_i\|_{L^p(\tau)}
        \leq {L} \|w_i\|_{L^p(\tau^{*})}
\end{equation}
式中
\[
  \|w_i\|_{L^p(\tau^{*})} = \max_{\tau\in\mathcal{T}_{h_K}(K)} \|w_i\|_{L^p(\tau)}
\]
对于$\tau^{*}$可以找到一列三角形单元$\tau_1,\tau_2,\cdots,\tau_l$使得
$\tau_1=\tau^{*}$,$\tau_l=$以$e_i$为边的三角形(可参考下面示意图),其中$\tau_j$和
$\tau_{j+1}$的公共边为$f_j$,$1\leq{j}\leq{l-1}$,则由$h_{\tau_j}\approx 1$,引理\ref{L1}
和迹定理知
\begin{equation}
\label{Eq:GFIeq_LpEst3}
\begin{aligned}
   \|w_i\|_{L^p(\tau^{*})}
        & =         \|w_i\|_{L^p(\tau_1)}  \\
        & \lesssim  |w_i|_{1,p,\tau_1} + |\int_{f_1}w_i\rm{d}s|
                    &  \text{(引理\ref{L1})}                     \\
        & \lesssim  |w_i|_{1,p,\tau_1} + \|w_i\|_{L^p(f_1)}      \\
        & \lesssim  |w_i|_{1,p,\tau_1} + \|w_i\|_{1,p,\tau_2}
                    &  \text{(迹定理)}                           \\
        & \lesssim  |w_i|_{1,p,\tau_1} + |w_i|_{1,p,\tau_2} + |\int_{f_2}w_i\rm{d}s|
                    &  \text{($\tau_2$和$\tau_3$的公共边$f_2$)}  \\
        & \lesssim  \sum_{j=1}^{l-1} |w_i|_{1,p,\tau_j} + \|w_i\|_{1,p,\tau_l}
                    &  \text{(依此类推)}                         \\
        & \lesssim  \sum_{j=1}^{l} |w_i|_{1,p,\tau_j}
                    &  \text{(引理\ref{L1} 和 $A_ev$ 定义)}      \\
        & \lesssim  \big(\sum_{j=1}^{l} |w_i|_{1,p,\tau_j}^{p}\big)^{1/p}
                    \big(\sum_{j=1}^l 1\big)^{\frac{p-1}{p}}
                    &  \text{(C-S不等式)}                        \\
        & \lesssim  l^{\frac{p-1}{p}} |w_i|_{1,p,K}
\end{aligned}
\end{equation}
由\eqref{Eq:GFIeq_LpEst1}-\eqref{Eq:GFIeq_LpEst3} 知
\begin{equation}
\label{Eq:GFIeq_1} %%%  Friedrichs Inequality with Geometric scale one for h_K=1
  \|v\|_{L^p(K)} \lesssim |v|_{1,p,K}
\end{equation}
\begin{figure}[!ht]\label{Fig3}
   \centering
%   \includegraphics[width=3.2in]{PolygonPartition1.eps}
    \includegraphics[width=3.2in,height=3.2in]{PolygonPartition1.eps}
   \caption{多角形$K$作正规剖分后得到一系列子三角形}
\end{figure}
%\begin{figure}[!h]\label{Fig2}
%   \centering
%   \includegraphics[width=3.2in]{PolygonPartition.eps}
%   \caption{多角形$K$作正规剖分后得到一系列子三角形}
%\end{figure}
\textbf{step 3.} 对于一般情形,作变量代换$x=h_K\hat{x}$,$K\rightarrow\hat{K}$,
$diam(\hat{K})=1$,则由\eqref{Eq:GFIeq_1}立知结果\eqref{Eq:GFIeq}.
\end{proof}

使用与定理\ref{T1}中类似的推导可得
\begin{theorem}[带几何尺度的Poincar\'{e}不等式]
\label{T2}\mbox{}\par
假设多角形$K$满足\textbf{性质\ref{Pro1}},则对任何$v\in{W}^{1,p}(K)$ 使
$\int_{K}v\rm{d}s=0$,成立估计
\begin{equation}
\label{Eq:GPIeq} %%%  Poincar\'{e} Inequality with Geometric scale
  \|v\|_{L^p(K)} \lesssim {h}_K^{-1}|v|_{1,p,K}
\end{equation}
式中$1\leq{p}<\infty$,生成常数$C$与$h_K$无关.
\end{theorem}



%%% Section 2
\section{函数逼近的误差估计}
本节,我们主要给出关于函数逼近的误差估计的两个定理.
\begin{theorem}
\label{T3} %%%\mbox{}\par
假设多角形$K$满足\textbf{性质\ref{Pro1}},则对$\forall{v}\in{W}^{1,p}(K)$,
存在$v_{\pi}\in{P_k(K)}$使
\begin{equation}
\label{Eq:PAEst} %%% Polynomial Approximation Estimate
   |v - v_{\pi}|_{l,p,K} \lesssim {h}_K^{k+1-l} |v|_{k+1,p,K},
   \quad 0\leq{l}\leq{k}.
\end{equation}
\begin{proof}
\label{Proof2}
不妨设$h_K=1$证明结果即可(对于一般情形,作变量代换$x=h_K\hat{x}$). \\
找$v_{\pi}\in{P_k(K)}$使
\[
  \int_K\partial^{\alpha}(v - v_{\pi}) \rm{d}x = 0, \quad |\alpha|\leq{k}.
\]
则由定理\ref{T2}知
\begin{align*}
  \|v-v_{\pi}\|_{L^p(K)}
            & \lesssim |v-v_{\pi}|_{1,p,K}   \\
            & \lesssim |v-v_{\pi}|_{2,p,K}   \\
            & \lesssim \cdots                \\
            & \lesssim |v-v_{\pi}|_{k+1,p,K} \\
            & \lesssim |v|_{k+1,p,K}.
\end{align*}
\end{proof}

对于$\mathcal{T}_{h_K}(K)$,记
\[
  \overline{V}_{h_K}(K) := \{ v\in{C}(\overline{K}): v|_e\in{P_k(e)},
                              \forall{e}\in\mathcal{T}_{h_K}(K) \}.
\]
$E_K$:$\overline{V}_{h_K}(K)$相应的$k$次Lagrange插值算子. \\
对$\forall{v}\in{H}^{k+1}(K)$,记$w=I_Kv$,由以下条件确定:
\begin{equation}
\label{Eq:IPDK} %%% Interpolation Projector definition on K
   \left\{
   \begin{aligned}
      \Delta{w} & = \Delta{v_\pi} \in {P_{k-2}(K)}
                & \text{in} \quad K; \\
              w & = E_Kv\in{B}_k(\partial{K})
                & \text{on} \quad \partial{K}.
   \end{aligned}
   \right.
\end{equation}
显然$w=I_Kv\in{V_k(K)}$. \eqref{Eq:IPDK}即
\begin{equation}
\label{Eq:IPDK1} %%% Interpolation Projector definition 1 on K
   \left\{
   \begin{aligned}
      \Delta(w-v_\pi) & = 0
                      & \text{in} \quad K; \\
              w-v_\pi & = E_Kv - v_\pi
                & \text{on} \quad \partial{K}.
   \end{aligned}
   \right.
\end{equation}
由能量最小原理知
\[
  |w-v_\pi|_{1,K} \leq |E_Kv - v_\pi|_{1,K}
  \leq |E_Kv - w|_{1,K} + |w - v_\pi|_{1,K}
\]
进一步,有
\begin{equation}
\label{Eq:IPEsti_K} %%% Interpolation Projector Estimate on K
   \begin{aligned}
     |v-I_Kv|_{1,K} & = |v- w|_{1,K}                   \\
                    & \leq |v-v_\pi|_{1,K} + |w - v_\pi|_{1,K}
                       & \text{($\pm{v_\pi}$)}         \\
                    & \leq |v-v_\pi|_{1,K} + |E_Kv - v_\pi|_{1,K} \\
                    & \leq 2|v-v_\pi|_{1,K} + |v - E_Kv|_{1,K}
                       & \text{($\pm{v}$)}              \\
                    & \leq 2[|v-v_\pi|_{1,K} + |v - E_Kv|_{1,K}]  \\
                    & \lesssim {h}_K^k |v|_{k+1,K}.
                       & \text{(\eqref{Eq:PAEst} 和$E_Kv$ 定义)}
   \end{aligned}
\end{equation}
由定理\ref{T1}知
\begin{align*}
  \|I_Kv - E_Kv\|_{0,K} & = \|w - E_Kv\|_{0,K}  \\
                        & \lesssim {h}_K |w - E_Kv|_{1,K} \\
                        & \lesssim {h}_K^k |v|_{k+1,K}
\end{align*}
\begin{align*}
  \|v - I_Kv\|_{0,K} & \leq \|v - E_Kv\|_{0,K} + \|I_Kv - E_Kv\|_{0,K}
                       & \text{($\pm{E_Kv}$)}               \\
                     & \lesssim {h}_K^{k+1} |v|_{k+1,K}
\end{align*}
对$\forall{v}\in{H}^{k+1}(\Omega)$,定义:
\[
  (I_hv)(x) := (I_Kv)(x) \quad x\in{K}.
\]

\end{theorem}

\begin{theorem}
\label{T4} %%%\mbox{}\par
假设多角形$K$满足\textbf{性质\ref{Pro1}},则对$\forall{v}\in{H}^{k+1}(K)$,
有$I_hv\in{V_h}$,且成立
\begin{equation}
\label{Eq:IPAEst} %%% Interpolation Projector Approximation Estimate
  \left\{
  \begin{aligned}
  |v-I_hv|_{1,K}   & \lesssim {h}_K^{k}|v|_{k+1,K}; \\
  \|v-I_hv\|_{0,K} & \lesssim {h}_K^{k+1}|v|_{k+1,K}.
  \end{aligned}
  \right.
\end{equation}
\end{theorem}
由\eqref{Eq:IPDK}-\eqref{Eq:IPEsti_K},立即可得\eqref{Eq:IPAEst}.



%%% Section 3
\section{在VEM分析中的应用}
FEM: 找$u\in{V}:=H_0^1(\Omega)$使
\[
   a(u,v) = f(v) \quad \forall {v}\in V.
\]

VEM: 找$u_h\in{V}_h$使
\[
   a_h(u_h,v) = f_h(v) \quad \forall {v}\in V_h.
\]
式中$V_h\subset{V}$.  \\
由第一Strang引理知
\begin{align*}
    |u - u_h|_{1,\Omega}
      & \lesssim  |u - I_hu|_{1,\Omega} + \sup_{v\in{V_h}}|a(I_hu,v) - a_h(I_hu,v) |/|v|_{1,\Omega} 
                  + \|f - f_h\|_{-1,\Omega}  \\
                & \quad \text{(由$a(u_\pi,v)=a_h(u_\pi,v)$,其中分片$u_\pi\in\mathbb{P}_k$)}  \\
%      & \lesssim  |u - I_hu|_{1,\Omega} + \sup_{v\in{V_h}}|a(I_hu-u_\pi,v) - a_h(I_hu-u_\pi,v)|/|v|_{1,\Omega}
%                  + \|f - f_h\|_{-1,\Omega}  \\
      & \lesssim  |u - I_hu|_{1,\Omega} + |I_hu - u_\pi |_{1,\Omega} + \|f - f_h\|_{-1,\Omega}  \\
                & \quad \text{(由$|a(I_hu-u_\pi,v) - a_h(I_hu-u_\pi,v)|\leq{C}|I_hu - u_\pi|_{1,\Omega}|v|_{1,\Omega}$)}  \\
      & \lesssim  |u - I_hu|_{1,\Omega} + |u - u_\pi|_{1,\Omega} + \|f - f_h\|_{-1,\Omega}  \\
                & \quad  \text{(由$I_hu - u_\pi = (I_hu - u) + (u-u_\pi)$及三角不等式)}                                    \\
      & \lesssim h^k |u|_{k+1,\Omega} + h^k |u|_{k+1,\Omega} + h^k |f|_{k-1,\Omega}         \\
                & \quad  \text{(由定理\ref{T4},定理\ref{T3}和$k\geq2$时)} \\
      & \lesssim \mathcal{O}(h^k).
\end{align*}



\end{CJK*}
\end{document}