Method of Fundamental Solutions: Singular Value Decomposition Analysis

The method of fundamental solutions (also known as the singularity or the source method) is a useful technique for solving linear partial di erential equations such as the Laplace or the Helmholtz equation. The procedure involves only boundary collocation or boundary tting and hence is a very fast procedure for the solution of these classes of problems. The resulting coe cient matrix, is however ill-conditioned and hence the solution accuracy is sensitive to the location of the source points. In this paper, an alternative solution procedure based on the singular value decomposition of the coe cient matrix is suggested and it is shown that the numerical results are extremely accurate (often within machine precision) and relatively independent of the location of the source points.