Approximation theory methods for solving elliptic eigenvalue problems

Eigenvalue problems with elliptic operators L on a domain G C R2 are considered. By applying results from complex approximation theory we obtain results on the approximation properties of 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).