Exponential Convergence Theory of the Multipole and Local Expansions for the 3-D Laplace Equation in Layered Media

Exponential Convergence Theory of the Multipole and Local Expansions for the 3-D Laplace Equation in Layered Media

Year:    2023

Author:    Wenzhong Zhang, Bo Wang, Wei Cai

Annals of Applied Mathematics, Vol. 39 (2023), Iss. 2 : pp. 99–148

Abstract

In this paper, we establish the exponential convergence theory for the multipole and local expansions, shifting and translation operators for the Green’s function of 3-dimensional Laplace equation in layered media. An immediate application of the theory is to ensure the exponential convergence of the FMM which has been shown by the numerical results reported in [27]. As the Green’s function in layered media consists of free space and reaction field components and the theory for the free space components is well known, this paper will focus on the analysis for the reaction components. We first prove that the density functions in the integral representations of the reaction components are analytic and bounded in the right half complex wave number plane. Then, by using the Cagniard-de Hoop transform and contour deformations, estimates for the remainder terms of the truncated expansions are given, and, as a result, the exponential convergence for the expansions and translation operators is proven.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/ 10.4208/aam.OA-2023-0005

Annals of Applied Mathematics, Vol. 39 (2023), Iss. 2 : pp. 99–148

Published online:    2023-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    50

Keywords:    Fast multipole method layered media multipole expansions local expansions 3-D Laplace equation Cagniard–de Hoop transform equivalent polarization sources.

Author Details

Wenzhong Zhang

Bo Wang

Wei Cai