# Boundary element methods for Laplacian eigenvalue problems

## Dr. Gerhard Unger

**June 30, 2011, 2:30 p.m. HF 136**

For the numerical solution of Laplacian eigenvalue problems we propose a boundary element method. While the standard approach for the solution of Laplacian eigenvalue problems is the finite element method, the use of boundary element methods seems to be a competitive alternative in particular when considering problems in unbounded domains, and for rather complicated geometries. The formulation of Laplacian eigenvalue problems in terms of boundary integral equations results in boundary integral operator eigenvalue problems which are nonlinear in the eigenvalue. The concept of eigenvalue problems for holomorphic Fredholm operator functions is used to establish a convergence and error analysis for a Galerkin boundary element discretization.

The discretization of the boundary integral operator eigenvalue problems leads to algebraic nonlinear eigenvalue problems. We use the recently proposed contour integral method which reduces the algebraic nonlinear eigenvalue problems to linear ones. This method is based on a contour integral representation of the resolvent operator and it is suitable for the extraction of all eigenvalues in a predefined domain which is enclosed by the contour. Numerical examples demonstrate the reliability of this method for the solution of discretized boundary integral eigenvalue problems.