Periodic conduction problems: the fast multipole method and convergence of integral equations and lattice sums

Abstract

This paper presents a version of the fast multipole method (FMM) for integral equations describing conduction through three–dimensional periodic heterogeneous media. The proposed method is based on the standard rather than periodic fundamental solution, and therefore it is very close to the original FMM. In deriving the method, particular attention is paid to convergence of arising integral equations and lattice sums. It is shown that convergence can be achieved without introducing artificial compensatory sources or boundary conditions.