The under-determined version of the MFS: Taking more sources than collocation points

In this study we investigate the approximation of the solutions of certain elliptic boundary value problems by the Method of Fundamental Solutions (MFS). In particular, we study the case in which the number of singularities (sources) exceeds the number of boundary (collocation) points. Two algorithms are proposed for the calculation of optimal solutions. An efficient numerical algorithm for the Dirichlet problem for Laplace's equation in the disk is described. Numerical experiments for a variety of geometries in two and three dimensions are presented.