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{\bf Math 269C. L. Vese. HW\#5.} 


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%\noindent{\bf [7]} Let $\Omega$ be a bounded domain in the plane and let the boundary $\Gamma$
%of $\Omega$ be devided into two parts $\Gamma_1$ and $\Gamma_2$. Give a 
%variational formulation of the following problem: 
%\begin{eqnarray*}
%-\triangle u+u=f & \mbox{ in } \Omega,\\
%\frac{\partial u}{\partial \vec{n}}=g & \mbox{ on }\Gamma_1,\\
%u=u_0 & \mbox{ on }\Gamma_2,
%\end{eqnarray*}
%where $f$, $u_0$ and $g$ are given functions. Then formulate a FEM for this
%problem. 

%\noindent{\bf [8]} Consider the inhomogeneous Dirichlet problem 
%$$-\triangle u +b(x)u=f \mbox{ in }\Omega,\ \ \ \ u=u_0\mbox{ on } \Gamma,$$
%where $b\in L^{\infty}(\Omega)$, $b(x)\geq 0$ a.e. in $\Omega$, $f\in L^2(\Omega)$, and $u_0|_{\Gamma}$ is the trace of some $u_0\in H^1(\Omega)$ (with $u_0|_{\Gamma}$ being %at least $L^2(\Gamma)$). 
%To solve this problem by Lax-Milgram, modify it into a homogeneous 
%Dirichlet problem, and analyze the resulting weak variational problem.  

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\noindent{\bf [1]} Let $I=[0,h]$ and let $\pi v\in P_1(I)$ be the linear interpolant 
that agrees with $v\in C^2(I)$ at the end points of $I$. Using the technique 
of the proof of Thm. 4.1, prove estimates for $\|v-\pi v\|_{L^{\infty}(I)}$ and $\|v'-(\pi v)'\|_{L^{\infty}(I)}$, cf. (1.12) and (1.13). 

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\noindent{\bf [2]} Consider the PDE (in the distributional sense)
$$-\triangle u+k^2u=f\ \ \mbox{ in } R^n,$$ 
where $k$ is a constant. 
Let $s\in R$. Show that, for all $f\in H^{s}(R^n)$, there exists a unique 
$u\in H^{s+2}(R^n)$, solution of the PDE, with $k\in R$, $k\neq0$. 

Hint: use the Fourier transform (see handout of useful results for notations of Sobolev spaces with arbitrary exponent $s$). 

%{\bf [3]} Using the general results from Chapter 4, estimate the error $\|u-u_h\|_{H^2}$ for Problem 1.5 and Example 2.4. 

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%{\bf [4]} Using polar coordinates $(r,\theta)$, let $\Omega=\{(r,\theta): \ 0<r<1, \ 0<\theta<\omega\}$ be a pie-shaped domain of angle $\omega$. 

%(i) Prove that the function $u(r,\theta)=r^{\gamma}\sin(\gamma \theta)$, $\gamma=\frac{\pi}{\omega}$, satisfies: $\triangle u=0$ in $\Omega$, $u=0$ on the straight parts of the boundary of $\Omega$. 

%(ii) Consider the cases $\omega>\pi$ and $\omega<\pi$ and study the condition $u\in H^2(\Omega)$ (see the discusssion in Section 4.5). 

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\noindent{\bf Optional Problems} 

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{\bf [3]} The following elliptic problem is approximated by the finite 
element method,
\begin{eqnarray*}
-\mbox{div}\Big( a(x)\nabla u(x)\Big)&=&f(x),\ x\in \Omega\subset R^2,\\
u(x)&=&0,\ x\in \partial\Omega_1,\\
\frac{\partial u(x)}{\partial x_1}+u(x)&=&0, \ x\in \partial\Omega_2,\\
\frac{\partial u(x)}{\partial x_2}&=&0,\ x\in\partial\Omega_3,
\end{eqnarray*}
where 
\begin{eqnarray*}
\Omega&=&\{(x_1,x_2): \ 0<x_1<1,\ 0<x_2<1\},\\
\Gamma_1=\partial\Omega_1&=&\{(x_1,x_2): \ x_1=0,\ 0\leq x_2\leq1\},\\
\Gamma_2=\partial\Omega_2&=&\{(x_1,x_2): \ x_1=1,\ 0\leq x_2\leq1\},\\
\Gamma_3=\partial\Omega_3&=&\{(x_1,x_2): \ 0<x_1<1,\ x_2=0,\ 1\},\\
\end{eqnarray*}
and
$$0<A\leq a(x)\leq B.$$

(a) Determine an appropriate weak formulation of the problem. 

(b) Prove conditions on the corresponding linear and bilinear forms which are 
needed for existence and uniqueness and for the convergence of a finite 
element method (assume $f\in L^2(\Omega)$, $a\in L^{\infty}(\Omega)$). 

(c) Describe briefly a finite element mesh, a FEM using $P_1$ elements, and a set of basis functions such that the linear system from the finite element approximation is sparse and of band structure. 

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\noindent{\bf [4]} (a) Develop and describe the piecewise linear Galerkin finite element 
approximation of, 
\begin{eqnarray*}
-\nabla\cdot (a(x)\nabla u) +b(x)u=f(x),&\ x=(x_1,x_2)\in\Omega,\\
u=2,&\ x\in \partial\Omega_1,\\
\frac{\partial u}{\partial x_1}+\frac{\partial u}{\partial x_2}+u=2,&\ x\in
\partial\Omega_2,
\end{eqnarray*}
where $f\in L^2(\Omega)$, 

$\Omega=\{x| \ x_1>0, \ x_2>0, \ x_1+x_2<1\}$,

$\partial\Omega_1=\{x| \ x_1=0,\ 0\leq x_2\leq 1\}\cup\{x| \ x_2=0,\ 0\leq x_1\leq 1\} $,

$\partial\Omega_2=\{x| \ x_1>0,\ x_2>0,\ x_1+x_2=1\}$,

$0<a\leq a(x)\leq A$, $0<b\leq b(x)\leq B$. 

(b) Justify the approximation by analyzing the appropriate bilinear and linear
forms. Give a convergence estimate and quote the appropriate thoerems for
convergence. 


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