The method of fundamental solutions: A weighted least-squares approach

Abstract

We investigate the Method of Fundamental Solutions (MFS) for the solution of certain elliptic boundary value problems. In particular, we study the case in which the number of collocation points exceeds the number of singularities, which leads to an over-determined linear system. In such a case, the resulting linear system is over-determined and the proposed algorithm chooses the approximate solution for which the error, when restricted to the boundary, minimizes a suitably defined discrete Sobolev norm. This is equivalent to a weighted least-squares treatment of the resulting over-determined system. We prove convergence of the method in the case of the Laplace's equation with Dirichlet boundary data in the disk. We develop an alternative way of implementing the numerical algorithm, which avoids the inherent ill-conditioning of the MFS matrices. Finally, we present numerical experiments suggesting that introduction of Sobolev weights improves the approximation.