The Laplace equation in three dimensions by the method of fundamental solutions and the method of particular solutions

Abstract

For Laplace's equation in a bounded simply-connected domain Ω in 3D, the method of fundamental solutions (MFS) is studied in this paper. Although some numerical computations can be found in Chen et al. [10], the theoretical analysis is much behind (Li [23] only for unit sphere Ω). Our efforts are devoted to exploring a strict error analysis of the MFS. The error bounds are derived, and the optimal polynomial convergence rates can be achieved. Numerical experiments are carried out to support the analysis made, and several useful locations of source nodes are investigated numerically. The analysis in this paper may lay a theoretical basis of the MFS for 3D problems, as Bogomolny [8] and [24] for 2D problems. Besides, the method of particular solutions (MPS) in [26] is also studied by using the spherical harmonic functions (SHF). The optimal polynomial convergence rates and the exponential growth of condition number (Cond) are obtained. The source nodes are located based on the abscissas of quadrature rules on surfaces; they are “grid-like”. Since most of 3D problems, in reality, can not be simplified to 2D problems, and since the MFS has more advantages for 3D problems in algorithm simplicity and wide application, the study in this paper is essential and important to the MFS.