A regularized multi-level technique for solving potential problems by the method of fundamental solutions

The method of fundamental solutions is investigated in the case when the source points are located along the boundary of the domain of the original problem and coincide with the collocation points. The appearing singularities are eliminated by several techniques: by using approximate but continuous fundamental solutions (regularization) and via auxiliary subproblems to avoid the stronger singularities that appear in the normal derivatives of the fundamental solution (desingularization). Both monopole and dipole formulations are investigated. A special iterative solution algorithm is presented, which converts the original (mixed) problem to a sequence of pure Dirichlet and pure Neumann subproblems. The pure subproblems can be handled efficiently by using conjugate gradients. The efficiency is significantly increased by embedding the resulting method in a natural multi-level context. At the same time, the problem of the use of highly ill-conditioned matrices is also avoided.