A collocation and least squares p-singular boundary method without fictitious boundary

Abstract

This study proposes a new version, p-SBM, of the numerical singular boundary method (SBM) to solve general classes of elliptic PDEs such as: Laplace, Helmholtz and diffusion equations. In SBM, the fundamental solution (FS) of the problem must be given but unlike the method of fundamental solutions (MFS), a fictitious boundary is not required. Instead, the inverse interpolation technique (IIT) and least squares method for the calculation of the singular diagonal elements of the interpolation matrix allows us to avoid the singularity at origin. In this study, we enrich the traditional SBM by adding a constant parameter or a linear combination to the previous MFS approximation and use various types of internal, external and boundary nodes. The p-SBM is applied to some homogeneous Laplace, Helmholtz and Diffusion problems to show its ability and solution accuracy. The non-homogeneous problems can be handled by using the dual reciprocity method (DRM).