On convergence and inherent oscillations within computational methods employing fictitious sources

Recent studies of the method of auxiliary sources (MAS) and the method of fundamental solutions (MFS) have shown that it is possible to have convergence of the field, which is the finally desired quantity, together with divergence of the fictitious sources, which are intermediate quantities. The aforementioned studies concern a simple scattering problem, where the scattered field satisfies the Helmholtz equation and the fictitious sources are placed symmetrically. In the present paper, we extend the previous results in two directions, one having to do with the geometry and the other with frequency. More precisely, we first show that the above convergence/divergence phenomena can also occur when the fictitious sources are placed asymmetrically and then study the nature of this divergence in detail. Second, we show that all findings carry over to the case of Laplace’s equation. We, additionally, point out certain differences of the discussed phenomena with the well-known ones of internal resonances.