Fast multipole method for 3-D Laplace equation in layered media

Abstract

In this paper, a fast multipole method (FMM) is proposed for 3-D Laplace equations in layered media. The potential due to charges embedded in layered media is decomposed into a free space component and four types of reaction field components, and the latter can be associated with the potential of a polarization source defined for each type. New multipole expansions (MEs) and local expansions (LEs), as well as the multipole to local (M2L) translation operators are derived for the reaction components, based on which FMMs for reaction components are then developed. The resulting FMMs for charge interactions in layered media is a combination of the classic FMM for the free space component and the new FMMs for the reaction field components. With the help of a recurrence formula and contour deformation technique for the run-time computation of the Sommerfeld-type integrals required in M2L translation operators, pre-computations of a large number of tables are avoided. The new FMMs for the reaction components are found to be much faster than the classic FMM for the free space component due to the separation of equivalent polarization charges and target charges by a material interface. As a result, the FMM for potential in layered media costs almost the same as the classic FMM in the free space case. Numerical results validate the fast convergence of the MEs for the reaction components, and the O(N) complexity of the FMMs with a given truncation number p for charge interactions in 3-D layered media.