论文标题
多层和局部扩展的指数收敛及其在分层介质中的源翻译：三维拉普拉斯方程
Exponential convergence for multipole and local expansions and their translations for sources in layered media: three-dimensional Laplace equation
论文作者
论文摘要
在本文中，我们证明了在层次介质中用于三维拉普拉斯方程的多极和局部扩展，转换和翻译运算符的指数收敛。这些理论结果确保了FMM的指数收敛性，这是由[9]中最近报道的数值结果所表明的。由于自由空间组件是由经典FMM计算的，因此本文将重点介绍分层介质中绿色函数的反应组件的分析。我们首先证明反应组件的积分表示中的密度功能是分析性的，并在右半复杂平面中界定。然后，使用cagniard-de箍变换和轮廓变形，给出了截止扩展的其余条款的估计，结果证明了扩展和翻译运算符的指数收敛性。
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.
