Analysis of the method of fundamental solutions for the modified Helmholtz equation

The method of fundamental solutions (MFS) was first proposed by Kupradze in 1963. Since then, the MFS has been extensively applied for solving various kinds of problems in science and engineering. However, few theoretical works have been reported in the literature. In this paper, we devote our work to the error analysis and stability of the MFS for the case of the modified Helmholtz equation. For disk domains, a convergence analysis of the MFS was provided by Li (J. Comput. Appl. Math., 159:113125, 2004) for solving the modified Helmholtz equation. In this paper, the error bounds of the MFS for bounded and simply connected domains are derived for smooth solutions of the modified Helmholtz equation. The exponential convergence rates can be achieved for analytic solutions. The bounds of condition numbers of the MFS are derived for both disk domains and the bounded and simply connected domains, to give the exponential growth via the number of fundamental solutions used. Numerical experiments are carried out to support the theoretical analysis. Moreover, the analysis of this paper is applied to parabolic equations, and some reviews of proof techniques and the analytical characteristics of the MFS are addressed.