Math 6420 • PDE

\begin{enumerate}
\item[(a)] What is the weak formulation for the boundary value problem \(u''=f(x)\) subject to
boundary conditions \(u(0)=u(1)=0\)?
\item[(b)] Let \(\phi(x)=N_2(x+1)\) where \(N_2(x)\) is the piecewise linear function given in
Problem A.2. Divide the unit interval \([0,1]\) interval \(N\) uniform subintervals with
endpoints at \(x_k=\frac{k}{N}\), \(k=0,\ldots,N\). Use \(\phi(x)\) to define a finite element
basis for continuous piecewise linear functions on the unit interval which interpolates
function values at \(x_k\). Suppose \(u_k=u(x_k)\) is known. Express \(u(x)\) in terms of
\(\phi(x)\).
\item[(c)] Use the Galerkin method to find equations for \(u_k=u(x_k)\) that approximately
solve the boundary value problem in part (a), using the piecewise linear finite element
basis.
\end{enumerate}