An implicit Method of Fundamental Solutions using twofold domains

A novel meshless method is presented. It is based on completing the original problem with an additional one which is defined in the same domain and connected with the original problem through the boundary only. The boundary conditions are enforced by some internal sources appearing in the supplementary problem only. The fundamental solutions are not used explicitly. The two problems are solved simultaneously using computationally quite efficient multi-level tools. No dense and ill-conditioned linear systems have to be solved. Moreover, no external or boundary sources are to be located. The approach can easily be extended to more general elliptic problems in a natural way, since the corresponding fundamental solution is not needed to be known.