Fast multipole method for 3-D Laplace equation in layered media
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In this paper, a fast multipole method (FMM) is proposed for 3-D Laplace equation in layered media. The potential due to source charges embedded in layered media is decomposed into a free space and reaction component(s). The free space component will be handled by the classic FMM while new multipole expansions (MEs), as well as the multipole to local (M2L) translation operators, will be developed for the reaction components, for which ME-based FMMs can be then developed. Based on the convergence analysis of the MEs for the reaction fields, equivalent polarization charges can be introduced for the reaction components, and are combined with the original source charges for the implementation of the FMM algorithm. It is found that the FMMs for the reaction field components are much faster than the FMM for the free space component due to the fact that the polarization charges are separated from the original source charges by a material interface. As a result, the FMM for charges in layered media costs the same as the classic FMM in the free space case. Numerical results validate the fast convergence of the MEs for the reaction field components, and the O(N) complexity of the FMM for charges in 3-D layered media.
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