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
\[
N_2(x)=
\begin{cases}
x, & 0<x<1,\\
2-x, & 1\le x<2,\\
0, & \text{elsewhere}.
\end{cases}
\]
Divide the unit interval \([0,1]\) into \(N\) 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}