On the application of the method of fundamental solutions to boundary value problems with jump discontinuities

- Applying a subtraction of singularity technique to circumvent MFS approximation problems (Gibbs phenomena) to discontinuous boundary data.
- Application of this technique to improve the MFS for other type of non regular data.
- Enrich the MFS approximation basis with crack singular functions, as an alternative technique.
- Application of both techniques to non trivial 2D geometries, for the Laplace equation.
- Excellent numerical results, with a practical advantage using the enrichment technique.

Two meshfree methods are proposed for the numerical solution of boundary value problems (BVPs) for the Laplace equation, coupled with boundary conditions with jump discontinuities. In the first case, the BVP is solved in two steps, using a subtraction of singularity approach. Here, the singular subproblem is solved analytically while the classical method of fundamental solutions (MFS) is applied for the solution of the regular subproblem. In the second case, the total BVP is solved using a variant of the MFS where its approximation basis is enriched with a set of harmonic functions with singular traces on the boundary of the domain. The same singularity-capturing functions, motivated by the boundary element method (BEM), are used for the singular part of the solution in the first method and for augmenting the MFS basis in the second method. Comparative numerical results are presented for 2D problems with discontinuous Dirichlet boundary conditions. In particular, the inappropriate oscillatory behavior of the classical MFS solution, due to the Gibbs phenomenon, is shown to vanish.