On the exponential convergence of the method of fundamental solutions

Abstract

It is well known that the method of fundamental solutions (MFS) is a numerical method of exponential convergence. In other words, the logarithmic error is proportional to the node number of spatial discretization. In this study, the exponential convergence of the MFS is demonstrated by solving the Laplace equation in domains of rectangles, ellipses, amoeba-like shapes, and rectangular cuboids. In the solution procedure, the sources of the MFS are located as far as possible and the instability resulted from the ill-conditioning of system matrix is avoided by using the multiple precision floating-point reliable (MPFR) library. The results converge faster for the cases of smoother boundary conditions and larger area/perimeter ratios. For problems with discontinuous boundary data, the exponential convergence is also accomplished using the enriched method of fundamental solutions (EMFS), which is constructed by the fundamental solutions and the local singular solutions. The computation is scalable in the sense that the required time increases only algebraically.