A Fast and Stable Multi-Level Solution Technique for the Method of Fundamental Solutions

Abstract

The classical form of the Method of Fundamental Solutions is applied. Instead of using a single set of subtly located external sources, a special strategy of defining several sets of external source points is introduced. The sets of sources are defined by the quadtree/octtree subdivision technique controlled by the boundary collocation points in a completely automatic way, resulting in a point set, the density of the spatial distribution of which decreases quickly far from the boundary. The ‘far’ sources are interpreted to form a ‘coarse grid’, while the densely distributed ‘near-boundary’ sources are considered a ‘fine grid’ (despite they need not to have any grid structure). Based on this classification, a multi-level technique is built up, where the smoothing procedure is defined by performing some familiar iterative technique e.g. the (conjugate) gradient method. The approximate solutions are calculated by enforcing the boundary conditions in the sense of least squares. The resulting multi-level method is robust and significantly reduces the computational cost. No weakly or strongly singular integrals have to be evaluated. Moreover, the problem of severely ill-conditioned matrices is completely avoided.