The method of fundamental solutions applied to boundary eigenvalue problems

We develop methods based on fundamental solutions to compute the Steklov, Wentzell and Laplace–Beltrami eigenvalues in the context of shape optimization. In the class of smooth simply connected two dimensional domains the numerical method is accurate and fast. A theoretical error bound is given along with comparisons with mesh-based methods. We illustrate the use of this method in the study of a wide class of shape optimization problems in two dimensions. We extend the method to the computation of the Laplace–Beltrami eigenvalues on surfaces and we investigate some spectral optimal partitioning problems.