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{\bf\large
 FUNDAMENTALS \\
of Numerical Analysis and Symbolic Computation\\[1ex]
	- Exercises on Discontinous Galerkin Methods -\\[2ex]
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WS 2012/2013
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NAME:\\[2ex]
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MATRIKELNUMMER:\\[1ex]
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{\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]
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{\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]
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{\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}
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{\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]
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%{\bf Exercise 5:} Show that (7) defines a norm on $V_k({\mathcal{T}}_h)$~!\\[4ex]
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\noindent
{\bf Remark:} All references (number) refer to formula markers from the lectures,
    see also lecture notes~!
	
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%BIBLIOGRAPHY                                                                                     %                   
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