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\noindent{\bf 269C\ \ \ HW\#2, due on Monday, May 4} 

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\noindent{\bf [1]} Find the linear basis functions for the triangle $K$ with vertices at 
$(0,0)$, $(h,0)$ and $(0,h)$. Show that the corresponding element stiffness
matrix is given by 
$$
\left[
\begin{array}{ccc}
1 & -\frac{1}{2} & -\frac{1}{2} \\
-\frac{1}{2} & \frac{1}{2} & 0\\
-\frac{1}{2} & 0 & \frac{1}{2}
\end{array}
\right]
$$
Using this result show that the linear system (1.25) of Example 1.1. has the
stated form (pages 30-31) (there is a typo in the textbook regarding the linear system). 

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\noindent{\bf [2]} (a) Give a weak variational formulation of the problem 
$$\frac{d^4 u}{dx^4}=f\ \mbox{ for }\ 0<x<1,$$
$$u(0)=u''(0)=u'(1)=u'''(1)=0,$$
and show that the assumptions of the Lax-Milgram Lemma are satisfied. Which boundary conditions
are essential and which are natural ? 

(b) Solve the same problem with the following alternative boundary conditions: 
$$u(0)=-u''(0)+\gamma u'(0)=0,\ \ u(1)=u''(1)+\gamma u'(1)=0,$$
where $\gamma$ is a positive constant. 

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\noindent{\bf [3]}  Give a weak variational formulation of the Robin's problem 
$$-\triangle u=f\ \mbox{ in }\Omega,\ \ \gamma u+\frac{\partial u}{\partial n}=g
\mbox{ on }\Gamma,$$
where $\gamma$ is a constant. When are the assumptions of the Lax-Milgram Lemma satisfied ? 

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\noindent{\bf [4]}  Consider the Neumann problem 
\begin{eqnarray*}
-\triangle u =f \mbox{ in }\Omega,\\
\frac{\partial u}{\partial n}=g\mbox{ on }\Gamma=\partial\Omega,\\
\int_{\Omega}u(x)dx=0.
\end{eqnarray*}
where $f:\Omega\rightarrow R$ and $g:\partial\Omega\rightarrow R$ satisfy the {\it compatibility condition} 
$$\int_{\Omega}f(x)dx+\int_{\partial\Omega}g(x)d\sigma(x)=0.$$

(a) Why condition ``$\int_{\Omega}u(x)dx=0$'' was added here ? Why do we need the compatibility condition ? 

(b) Give a weak variational formulation of the problem,  
and prove that the conditions of the Lax-Milgram Lemma are satisfied, under the necessary assumptions on $f$ and $g$ that you would specify. 


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