# A boundary element method for Laplacian eigenvalue problems

## Dr. Gerhard Unger

**Oct. 12, 2010, 1:30 p.m. P 004**

For the solution of Laplacian eigenvalue problems we propose a boundary element method which is used to solve equivalent nonlinear eigenvalue problems for related boundary integral operators. 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 interval which is enclosed by the contour. The dimension of the resulting linear eigenvalue problem corresponds to the number of eigenvalues which lie inside the contour. The main numerical effort consists in the evaluation of the resolvent operator for the contour integral which requires the solution of several linear systems. Compared with other methods for nonlinear eigenvalue problems no initial approximations of the eigenvalues and eigenvectors are needed. Numerical examples demonstrate the reliability of the method.