{"title":"Error bound of the multilevel fast multipole method for 3‐D scattering problems","authors":"Wenhui Meng","doi":"10.1002/num.23148","DOIUrl":null,"url":null,"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.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"21 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerical Methods for Partial Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/num.23148","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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.
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.
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