{"title":"A fast Fourier-Galerkin method for solving boundary integral equations on non-axisymmetric toroidal surfaces","authors":"Yiying Fang , Ying Jiang , Jiafeng Su","doi":"10.1016/j.apnum.2025.07.013","DOIUrl":null,"url":null,"abstract":"<div><div>We propose a fast Fourier–Galerkin method for solving boundary integral equations (BIEs) on smooth, non-axisymmetric toroidal surfaces. Our approach begins by analyzing the structure of the integral kernel, revealing an exponential decay pattern in the Fourier coefficients after a shear transformation. Leveraging this decay, we design a truncation strategy that compresses the dense representation matrix into a sparse form with only <span><math><mi>O</mi><mo>(</mo><mi>N</mi><msup><mrow><mi>ln</mi></mrow><mrow><mn>2</mn></mrow></msup><mo></mo><mi>N</mi><mo>)</mo></math></span> nonzero entries, where <em>N</em> denotes the degrees of freedom. We rigorously prove that the truncated system retains the stability of the original Fourier–Galerkin formulation and achieves a quasi-optimal convergence rate of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mi>p</mi><mo>/</mo><mn>2</mn></mrow></msup><mi>ln</mi><mo></mo><mi>N</mi><mo>)</mo></math></span>, with <em>p</em> denoting the regularity of the exact solution. Numerical experiments corroborate our theoretical results, demonstrating both high accuracy and computational efficiency. Furthermore, we extend the proposed strategy to BIEs defined on surfaces diffeomorphic to the sphere, confirming the sparsity structure remains exploitable under broader geometric settings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"218 ","pages":"Pages 73-90"},"PeriodicalIF":2.4000,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Numerical Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168927425001540","RegionNum":2,"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
We propose a fast Fourier–Galerkin method for solving boundary integral equations (BIEs) on smooth, non-axisymmetric toroidal surfaces. Our approach begins by analyzing the structure of the integral kernel, revealing an exponential decay pattern in the Fourier coefficients after a shear transformation. Leveraging this decay, we design a truncation strategy that compresses the dense representation matrix into a sparse form with only nonzero entries, where N denotes the degrees of freedom. We rigorously prove that the truncated system retains the stability of the original Fourier–Galerkin formulation and achieves a quasi-optimal convergence rate of , with p denoting the regularity of the exact solution. Numerical experiments corroborate our theoretical results, demonstrating both high accuracy and computational efficiency. Furthermore, we extend the proposed strategy to BIEs defined on surfaces diffeomorphic to the sphere, confirming the sparsity structure remains exploitable under broader geometric settings.
期刊介绍:
The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are:
(i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments.
(ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers.
(iii) Short notes, which present specific new results and techniques in a brief communication.