Math 6610 • Analysis of Numerical Methods I

{Finite-Element Methods}
Consider the problem:
\[
  -u''+u=f,\qquad \text{on }[0,1]
\]
with
\[
  u(0)=0\qquad \text{and}\qquad u(1)=1.
\]

Derive a variational formulations of this problem (be sure to define the space on
which your variational problem is to be solved). Formulate a Galerkin
finite-element method for the variational problem using piecewise linear functions.
Prove that the linear system which arises in the finite-element method has a
solution. In what sense is the solution optimal?