In this paper, we prove the exponential convergence of the multipole and local expansions, shifting and translation operators used in fast multipole methods (FMMs) for 3-dimensional Laplace equations in layered media. These theoretical results ensure the exponential convergence of the FMM which has been shown by the numerical results recently reported in [9]. As the free space components are calculated by the classic FMM, this paper will focus on the analysis for the reaction components of the Green's function for the Laplace equation in layered media. We first prove that the density functions in the integral representations of the reaction components are analytic and bounded in the right half complex plane. Then, using the Cagniard-de Hoop transform and contour deformations, estimate for the remainder terms of the truncated expansions is given, and, as a result, the exponential convergence for the expansions and translation operators is proven.
Taylor expansion based fast Multipole Methods for 3-D Helmholtz equations in Layered Media
Hunan Normal University（湖南师范大学）；Southern Methodist University（南方卫理公会大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；Indiana University（印第安纳大学）；University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校）
A Heterogeneous FMM for 2-D Layered Media Helmholtz Equation I: Two & Three Layers Cases
University of Massachusetts Lowell（马萨诸塞大学洛威尔分校）；University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；Beijing Computational Science Research Center（北京计算科学研究中心）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）
Flexibly imposing periodicity in kernel independent FMM: A Multipole-To-Local operator approach
Center for Computational Biology, Flatiron Institute（Flatiron研究所计算生物学中心）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
Accurate computation of Galerkin double surface integrals in the 3-D boundary element method
Department of Computer Science, University of Maryland, College Park（马里兰大学学院公园分校计算机科学系）；Fantalgo, LLC（Fantalgo有限责任公司）；Institute for Advanced Computer Studies, University of Maryland, College Park（马里兰大学学院公园分校高级计算机研究所）
