Error bound of the multilevel fast multipole method for 3‐D scattering problems

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Numerical Methods for Partial Differential Equations Pub Date : 2024-09-04 DOI:10.1002/num.23148

Wenhui Meng

引用次数: 0

Abstract

The multilevel fast multipole method (MLFMM) is widely used to accelerate the solutions of acoustic and electromagnetic scattering problems. In the expansions and translation operators of the MLFMM for 3‐D scattering problems, some special functions are used, including spherical Bessel functions, spherical harmonics and Wigner symbol. This makes it difficult to analyze the truncation errors. In this paper, we first give sharp bounds for the truncation errors of the expansions used in the MLFMM, then derive the overall error formula of the MLFMM and estimate its upper bound, the result is finally applied to the cube octree structure. Some numerical examples are performed to validate the proposed results. The method in this paper can also be used to the MLFMM for other 3‐D problems, such as potential problems, elastostatic problems, Stokes flow problems and so on.

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三维散射问题多级快速多极子方法的误差约束

多级快速多极法（MLFMM）被广泛用于加速声学和电磁散射问题的求解。在 MLFMM 用于三维散射问题的展开和平移算子中，使用了一些特殊函数，包括球面贝塞尔函数、球面谐波和 Wigner 符号。这给截断误差分析带来了困难。本文首先给出了 MLFMM 中所用展开式的截断误差的尖锐边界，然后推导出 MLFMM 的总体误差公式并估计其上限，最后将结果应用于立方体八叉树结构。本文通过一些数值示例验证了所提出的结果。本文的方法还可用于其他三维问题的 MLFMM，如势能问题、弹性问题、斯托克斯流问题等。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Numerical Methods for Partial Differential Equations 数学-应用数学

CiteScore

7.20

自引率

2.60%

发文量

审稿时长

9 months

期刊介绍： An international journal that aims to cover research into the development and analysis of new methods for the numerical solution of partial differential equations, it is intended that it be readily readable by and directed to a broad spectrum of researchers into numerical methods for partial differential equations throughout science and engineering. The numerical methods and techniques themselves are emphasized rather than the specific applications. The Journal seeks to be interdisciplinary, while retaining the common thread of applied numerical analysis.