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\documentstyle[12pt,a4wide]{article} % %\pagestyle{empty} %\input{command.inp} % \begin{document} % \noindent \begin{center} {\bf\large FUNDAMENTALS \\ of Numerical Analysis and Symbolic Computation\\[1ex] - Exercises on Discontinous Galerkin Methods -\\[2ex] } WS 2012/2013 \end{center} \bigskip \noindent \hrule %\bigskip %\noindent% % % \hrule \vspace*{5mm} \bigskip \noindent % \def\solution{0} % without solution %\def\solution{1} % with solution % % \noindent NAME:\\[2ex] % MATRIKELNUMMER:\\[1ex] \hrule \vspace*{2ex} % %WS 2011/2012 % % \noindent {\bf Exercise 1:} Show that the gradient $\nabla u$ of the weak solution $u \in V_0 := H^1_0(\Omega)$ of the variational problem (1) belongs not only to $\left[L_2(\Omega)\right]^d$ but also to $H(\mbox{div}):= \{v \in \left[L_2(\Omega)\right]^d: \mbox{div}(v) \in L_2(\Omega)\}$, and, moreover, $ \mbox{div}(\nabla u) = -f$ ! \\[2ex] % % Solution 1 % \ifnum\solution=1 \input{solution1.inp} \fi % % \noindent {\bf Exercise 2:} Show the DG-identities (3), i.e. \[ \sum_{\delta \in {\mathcal{T}}_h} \left(\nabla u \cdot n,v\right)_{\partial \delta} = \sum_{\delta \in {\mathcal{T}}_h} \left(\{\nabla u\},v\cdot n\right)_{\partial \delta} = \sum_{e \in {\overline{\mathcal{E}}}_h} \left(\{\nabla u\},[v]\right)_e %~! \] for a weak solution $u \in H^1_0(\Omega) \cap H^s({\mathcal{T}}_h)$ of the variational problem (1) and for all $v \in H^s({\mathcal{T}}_h)$ with some $s > 3/2$~! \\[2ex] % % % Solution 2 % \ifnum\solution=1 \input{solution1.inp} \fi % % \noindent {\bf Exercise 3:} Prove the consistency theorem: Let $s > 3/2$. Then the following statements are valid: \begin{enumerate} \item Assume that the weak solution $u$ of (1), i.e. the solution of $ (1)_{\mbox{\tiny VF}} $ ($\exists$ ! due to Lax \& Milgram), belongs to $ H^s({\mathcal{T}}_h) $. Then $u$ satisfies the DG variational formulation (5). \item Conversely, if $u \in H^1_0(\Omega) \cap H^s({\mathcal{T}}_h)$ satisfies the DG variational formulation (5), then $u$ is also the solution of our variational problem $ (1)_{\mbox{\tiny VF}} $. \end{enumerate} %\\[2ex] % % Solution 3 % \ifnum\solution=1 \input{solution1.inp} \fi % % % \noindent {\bf Exercise 4:} Show that the Dirichlet boundary condition $u=0$ on $\Gamma$ is incorporated in (5) resp. (6)~! Derive the DG variational formulation (5) resp. the DG scheme (6) for the case of inhomogeneous Dirichlet boundary conditions $u=g$ on $\Gamma$ and piecewise constant coefficients $a|_\delta = a_\delta = const > 0$ for all $\delta \in {\mathcal{T}}_h$~!.\\[2ex] % % Solution 4 % \ifnum\solution=1 \input{solution1.inp} \fi % % % %\noindent %{\bf Exercise 5:} Show that (7) defines a norm on $V_k({\mathcal{T}}_h)$~!\\[4ex] % % % Solution 5 % % \ifnum\solution=1 \input{solution1.inp} \fi % % % %------------------------------------------------------------------------------------------- \noindent {\bf Remark:} All references (number) refer to formula markers from the lectures, see also lecture notes~! %------------------------------------------------------------------------------------------- %BIBLIOGRAPHY % %\bibliographystyle{abbrv} %\bibliographystyle{plain} %\bibliography{fundamentals-langer} % %------------------------------------------------------------------------------------------- \end{document}