A Novel Technique for Estimating the Numerical Error in Solving the Helmholtz Equation

In this study, we applied a defined auxiliary problem in a novel error estimation technique to estimate the numerical error in the method of fundamental solutions (MFS) for solving the Helmholtz equation. The defined auxiliary problem is substituted for the real problem, and its analytical solution is generated using the complementary solution set of the governing equation. By solving the auxiliary problem and comparing the solution with the quasianalytical solution, an error curve of the MFS versus the source location parameters can be obtained. Thus, the optimal location parameter can be identified. The convergent numerical solution can be obtained and applied to the case of an unavailable analytical solution condition in the real problem. Consequently, we developed a systematic error estimation scheme to identify an optimal parameter. Through numerical experiments, the optimal location parameter of the source points and the optimal number of source points in the MFS were studied and obtained using the error estimation technique.