Localized method of fundamental solutions for interior Helmholtz problems with high wave number

Abstract

This paper introduces a localized version of the method of fundamental solutions (MFS) named as the localized MFS (LMFS) for two-dimensional (2D) interior Helmholtz problems with high wave number. Due to its full interpolation matrix, the traditional MFS is low-efficiency to solve the above-mentioned problem that requires a large number of boundary nodes for obtaining availably numerical results. For the LMFS, the computational domain is first divided into some overlap subdomains based on the distributed nodes. In each subdomain, physical variables are then represented as linear combinations of the fundamental solution of the governing equation as same as in the traditional MFS. A sparse and banded system matrix is finally formed for the LMFS by satisfying Helmholtz equation and boundary conditions, and thus the developed method is inherently efficient for large-scale problems. Three numerical examples are provided to verify the accuracy and the stability of the LMFS.