Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions (Communications in Computational Physics, 2018)
This paper describes an application of the recently developed sparse schemeof the method of fundamental solutions (MFS) for the simulation of three-dimensionalmodified Helmholtz problems. The solution to the given problems is approximatedby a two-step strategy which consists of evaluating the particular solution and the homogeneoussolution. The homogeneous solution is approximated by the traditionalMFS. The original dense system of the MFS formulation is condensed into a sparsesystem based on the exponential decay of the fundamental solutions. Hence, the homogeneoussolution can be efficiently obtained. The method of particular solutionswith polyharmonic spline radial basis functions and the localized method of approximateparticular solutions in combination with the Gaussian radial basis function areemployed to approximate the particular solution. Three numerical examples includinga near singular problem are presented to show the simplicity and effectiveness ofthis approach.