Analysis of Boundary Element Methods for Laplacian Eigenvalue Problems

The aim of the book is to provide an analysis of the boundary element method for the numerical solution of Laplacian eigenvalue problems. The representation of Laplacian eigenvalue problems in the form of boundary integral equations leads to nonlinear eigenvalue problems for related boundary integral operators. The solution of boundary element discretizations of such eigenvalue problems requires appropriate methods for algebraic nonlinear eigenvalue problems. Although the numerical solution of eigenvalue problems for partial differential operators using boundary element methods has a long tradition, a rigorous numerical analysis has not been established so far. One of the main goals of this work is to develop a convergence and error analysis of the Galerkin boundary element approximation of Laplacian eigenvalue problems. To this end, the concept of eigenvalue problems for so–called holomorphic Fredholm operator functions is used. This concept is a generalization of the theory for eigenvalue problems of bounded linear operators. The analysis of the approximation of eigenvalue problems for holomorphic Fredholm operator functions is usually done in the framework of the concept of regular approximation schemes. In this work convergence results and error estimates are derived for Galerkin discretizations of such eigenvalue problems. These results are then applied to the discretizations of Laplacian boundary integral operator eigenvalue problems. Furthermore, numerical methods for the solution of algebraic nonlinear eigenvalue problems are reviewed. The little–known Kummer’s method is presented and its convergence behavior for algebraic holomorphic eigenvalue problems is analyzed by using the concept of holomorphic operator functions. Finally, a numerical example is considered and results of a boundary element and a finite element approximation of the eigenvalues are presented which confirm the theoretical results.