The method of fundamental solutions for elliptic problems in circular domains with mixed boundary conditions

Abstract

We apply the method of fundamental solutions (MFS) for the solution of harmonic and biharmonic problems in circular domains subject to mixed boundary conditions. In contrast to the cases when boundary conditions of the same kind are prescribed on the whole boundary, for example, only Dirichlet conditions in the harmonic case, and Dirichlet and Neumann conditions in the biharmonic case, the resulting systems are neither circulant (harmonic case) nor block circulant (biharmonic case). However, by appropriately manipulating the matrices involved in the MFS discretization, the partial circulant/block circulant structure of these matrices can be exploited when certain iterative methods of solution are used for the solution of the resulting systems. This leads to efficient fast Fourier transform (FFT) algorithms which are tested on several numerical examples.