The Method of Particular Solutions for the Laplace Eigenvalue Problem

Timo Betcke

Manchester, UK

The Method of Particular Solutions (MPS) for the Laplace eigenvalue problem was popularized in a famous paper by Fox, Henrici and Moler in 1967. The idea is to approximate the eigenfunctions from a space of particular solutions that satisfy the eigenvalue equation but not necessarily the boundary conditions. However, for more complicated domains bases of particular solutions quickly tend to become ill-conditioned. In the first part of this talk we discuss tools from numerical linear algebra that allow to overcome these ill-conditioning problems and generalize these ideas to domain decomposition approaches that use particular solutions in each subdomain. In the second part of this talk we discuss the convergence of the MPS by using techniques from complex approximation theory. This allows us to give accurate exponential convergence estimates on certain domains. We present several examples including multiply connected domains, eigenvalue avoidance phenomena and localization of eigenfunctions.

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