Extending the Range of Application of the Method of Fundamental Solutions

The method of fundamental solutions (MFS) is a meshfree boundary collocation method, originally developed for the numerical solution of homogeneous PDEs, coupled with sufficiently regular boundary conditions. In this talk we show how to extend the range of application of the classical MFS to two more general situations.

In the first case, the numerical solution of a non-homogeneous PDE is addressed. Here, fundamental solutions with different source points and test frequencies are used as shape functions in order to approximate the solution of the Cauchy-Navier equations of elastodynamics.

In the second situation, a potential problem with singular (discontinuous) boundary conditions is solved. In order to reduce the effects of the Gibbs phenomenon in the neighborhood of the singularities, the MFS approximation basis is augmented with a set of harmonic functions which possess a discontinuous inner trace at the boundary of the domain.

In both methods presented here the meshfree characteristics of the original MFS are preserved and the total approximate solution of the BVP is calculated by solving a single collocation linear system. Several 2D numerical examples will be presented in order to illustrate the performance of the methods.