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269C

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{\bf First computational project:} {\bf due on Monday, May 9}

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{\bf (i)} Show that the problem 
$$
\left\{
\begin{array}{l}
-u''(x)+u(x)=f(x), \ \ 0<x<1,\\
u(0)=0,\ u'(1)=0
\end{array}
\right.
$$
can be given a weak variational formulation.

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{\bf (ii)} Formulate a FEM for this problem using 
piecewise-linear functions. Determine the corresponding linear system of 
equations in the case of a uniform partition and study in particular how the
boundary condition $u'(1)=0$ is approximated by the method. 

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{\bf (iii)} Write a computer program for the piecewise-linear FEM applied to
this problem. 
Present calculations with $h=0.1$ and $h=0.2$, for $f(x)=2+2x-x^2$. 
Approximate $\int f(x)\varphi_j(x)dx$ by a quadrature formula of your choice. The exact solution of the problem is $u(x)=x(2-x)$. 

(see section 1.8: Remarks on programming) 

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