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{\bf HW \#4, 269C} Due during the week of May 27 - May 31st (note May 27 is a 
holiday). 

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{\bf [1]} Recall the ``coercive'' inhomogeneous Neumann problem: 
$$(D) \ \ \Big\{-\triangle u +u=f \mbox{ in }\Omega, \ \ \frac{\partial u}{\partial \vec{n}}=g \mbox{ on }\partial \Omega\Big\}.$$

We know that the corresponding weak variational problem (with $V=H^1(\Omega)$) is: find $u\in H^1(\Omega)$ such that
$$(V) \ \ \int_{\Omega}(\nabla u\cdot \nabla v +uv)dx =\int_{\Omega}fv dx +\int_{\partial \Omega}gv ds,$$
for all $v\in H^1(\Omega)$. We also know that $(D)\Rightarrow (V)$. 

Assume now that $u,f,g$ are sufficiently smooth. Show that, if $u$ is a smooth solution of $(V)$ (for example $u\in C^2(\overline{\Omega})$, then $u$ satisfies $(D)$; in other words, $(V)\Rightarrow (D)$. 


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{\bf [2]} Let $K$ be a tetrahedron with vertices $a^i$, $i=1,...,4$, and let 
$a^{ij}$ denote the midpoint on the straight line $a^ia^j$, $i<j$. Show that a
function $v\in P_2(K)$ is uniquely determined by the degrees of freedom: 
$v(a^i)$, $v(a^{ij})$, $i,j=1,...,4$, $i<j$. Show that the corresponding 
finite element space $V_h$ satisfies $V_h\subset C^0(\Omega)$. 

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{\bf [3]} Let $K$ be a triangle with vertices $a^i$, $i=1,2,3$. Suppose that
$v\in P_r(K)$ and that $v$ vanishes on the side $a^2a^3$. Prove that $v$ has
the form 
$$v(x)=\lambda_1(x)w_{r-1}(x),\ \ x\in K,$$
where $w_{r-1}\in P_{r-1}(K)$, and $\lambda_1(x)$ is the affine local basis function corresponding to the node $a^1$. 

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{\bf [4]} Let $K$ be a triangle with vertices $a^i$, $i=1,2,3$, and let $a^{ij}$, $i<j$, denote the midpoints of the sides of $K$. Let $a^{123}$ denote the 
center of gravity of $K$. Prove that $v\in P_4(K)$ is uniquely determined by 
the following degrees of freedom 
\begin{eqnarray*}
& &v(a^i),\\
& &\frac{\partial v}{\partial x_j}(a^i),\ i=1,2,3,\ j=1,2, \\
& &v(a^{ij}), \ i,j=1,2,3, \ i<j, \\ 
& &v(a^{123}), \ \frac{\partial v}{\partial x_j}(a^{123}), j=1,2,
\end{eqnarray*}
(typo in Exercise 3.8 in the textbook). 

Also show that the functions in the corresponding finite element space $V_h$ 
are continuous. 



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