The problem of numerically computing eigenvalues and eigenfunctions of the Laplacian, with Dirichlet (zero) boundary conditions, on a plane domain, is computationally intensive and there is a lot of theory behind finding efficient algorithms. Proving convergence rates is likewise an interesting theoretical problem. Recently, Barnett and Barnett-Hassell have shown that the method of particular solutions (MPS), a standard method, is more accurate by an order of E1/2, where E is the eigenvalue, then previously shown.
Analyzing the scaling method, which is a more efficient method for finding large blocks of eigenvalues simultaneously, is planned for 2009. There are good projects possible here for those who like to combine theory and computation.