Fast-multipole accelerated singular boundary method for large-scale three-dimensional potential problems

Abstract

The singular boundary method (SBM) is a relatively new meshless boundary collocation method for the numerical solution of certain elliptic boundary value problems. The method involves a coupling between the boundary element method (BEM) and the method of fundamental solutions (MFS). The main idea is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration-free attributes of the later. This makes it particularly attractive for problems in complex geometries and three dimensions. However, similar to the traditional BEM, the SBM produces dense and unsymmetrical coefficient matrices, which requires O(N2) memory and another O(N3) operations to solve the system with direct solvers. This paper documents the first attempt to apply the fast multipole method (FMM) to accelerate the solutions of the SBM for the solution of large-scale problems. The FMM formulations for the SBM are presented for three-dimensional (3D) potential problems. Numerical examples with up to 900,000 unknowns are solved successfully on a desktop computer using the developed FMM-SBM code. These results clearly demonstrate the efficiency, accuracy and potentials of the FMM-SBM for solving large-scale problems.