Fast Solution for Solving the Modified Helmholtz Equation with the Method of Fundamental Solutions (Communications in Computational Physics, 2018)
The method of fundamental solutions (MFS) is known as an effective boundarymeshless method. However, the formulation of the MFS results in a dense andextremely ill-conditioned matrix. In this paper we investigate the MFS for solvinglarge-scale problems for the nonhomogeneous modified Helmholtz equation. The keyidea is to exploit the exponential decay of the fundamental solution of the modifiedHelmholtz equation, and consider a sparse or diagonal matrix instead of the originaldense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately.A standard two-step solution process which consists of evaluating the particularsolution and the homogeneous solution is applied. Polyharmonic spline radial basisfunctions are employed to evaluate the particular solution. Five numerical examples inirregular domains and a large number of boundary collocation points are presented toshow the simplicity and effectiveness of our approach for solving large-scale problems.