TY - JOUR
T1 - Exponential Convergence Theory of the Multipole and Local Expansions for the 3-D Laplace Equation in Layered Media
AU - Zhang , Wenzhong
AU - Wang , Bo
AU - Cai , Wei
JO - Annals of Applied Mathematics
VL - 2
SP - 99
EP - 148
PY - 2023
DA - 2023/06
SN - 39
DO - http://doi.org/ 10.4208/aam.OA-2023-0005
UR - https://global-sci.org/intro/article_detail/aam/21831.html
KW - Fast multipole method, layered media, multipole expansions, local expansions, 3-D Laplace equation, Cagniard–de Hoop transform, equivalent polarization sources.
AB - <p style="text-align: justify;">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.</p>