Localized method of fundamental solutions for three-dimensional inhomogeneous elliptic problems: theory and MATLAB code

Abstract

In this paper we investigate the application of the localized method of fundamental solutions (LMFS) for solving three-dimensional inhomogeneous elliptic boundary value problems. A direct Chebyshev collocation scheme (CCS) is employed for the approximation of the particular solutions of the given inhomogeneous problem. The Gauss–Lobatto collocation points are used in the CCS to ensure the pseudo-spectral convergence of the method. The resulting homogeneous equations are then calculated by using the LMFS. In the framework of the LMFS, the computational domain is divided into a set of overlapping local subdomains where the traditional MFS formulation and the moving least square method are applied. The proposed CCS-LMFS produces sparse and banded stiffness matrix which makes the method possible to perform large-scale simulations on a desktop computer. Numerical examples involving Poisson, Helmholtz as well as modified-Helmholtz equations (with up to 1,000,000 unknowns) are presented to illustrate the efficiency and accuracy of the proposed method.