Approximation theory methods for solving elliptic eigenvalue problems

Eigenvalue problems with elliptic operators L on a domain G⊂R² are considered. We discuss approximation properties of some special classes of solutions of Lu=0 on G. These solutions are used as trial functions in a method for solving the eigenvalue problem which is based on a-posteriori error bounds. Singular trial functions are applied to smooth the problem at corner points of G. In special situations, this method can produce approximations of eigenvalues and eigenfunctions with extremely high accuracy by only using a low number of trial functions. Some illustrative numerical examples for the eigenvalue problem with the Laplacian are presented. We discuss two problems from plasma physics ('relaxed plasma', 'MHD-equation') to which the approach can be applied.