{"title":"H2-matrices for translation-invariant kernel functions","authors":"Steffen Börm, Janne Henningsen","doi":"10.1016/j.enganabound.2025.106190","DOIUrl":null,"url":null,"abstract":"<div><div>Boundary element methods for elliptic partial differential equations typically lead to boundary integral operators with translation-invariant kernel functions. Taking advantage of this property is not straightforward if general unstructured meshes and general basis functions are used, since we need the supports of these basis functions to be contained in a hierarchy of subdomains with translational symmetry.</div><div>In this article, we present a modified construction for <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-matrices on unstructured quasi-uniform meshes that uses translation-invariance to significantly reduce the storage requirements for the farfield representation.</div><div>We construct a nested hierarchy of axis-parallel boxes so that translational symmetry is preserved and prove optimal-order complexity estimates under moderate assumptions. In particular, we need only one weak assumption for proving that the entire farfield requires only <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mo>log</mo><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> coefficients.</div><div>It should be mentioned that, since we are working with an unstructured mesh and general basis functions, the nearfield of the matrix still requires <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> units of storage.</div></div>","PeriodicalId":51039,"journal":{"name":"Engineering Analysis with Boundary Elements","volume":"175 ","pages":"Article 106190"},"PeriodicalIF":4.2000,"publicationDate":"2025-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Engineering Analysis with Boundary Elements","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0955799725000785","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
Boundary element methods for elliptic partial differential equations typically lead to boundary integral operators with translation-invariant kernel functions. Taking advantage of this property is not straightforward if general unstructured meshes and general basis functions are used, since we need the supports of these basis functions to be contained in a hierarchy of subdomains with translational symmetry.
In this article, we present a modified construction for -matrices on unstructured quasi-uniform meshes that uses translation-invariance to significantly reduce the storage requirements for the farfield representation.
We construct a nested hierarchy of axis-parallel boxes so that translational symmetry is preserved and prove optimal-order complexity estimates under moderate assumptions. In particular, we need only one weak assumption for proving that the entire farfield requires only coefficients.
It should be mentioned that, since we are working with an unstructured mesh and general basis functions, the nearfield of the matrix still requires units of storage.
期刊介绍:
This journal is specifically dedicated to the dissemination of the latest developments of new engineering analysis techniques using boundary elements and other mesh reduction methods.
Boundary element (BEM) and mesh reduction methods (MRM) are very active areas of research with the techniques being applied to solve increasingly complex problems. The journal stresses the importance of these applications as well as their computational aspects, reliability and robustness.
The main criteria for publication will be the originality of the work being reported, its potential usefulness and applications of the methods to new fields.
In addition to regular issues, the journal publishes a series of special issues dealing with specific areas of current research.
The journal has, for many years, provided a channel of communication between academics and industrial researchers working in mesh reduction methods
Fields Covered:
• Boundary Element Methods (BEM)
• Mesh Reduction Methods (MRM)
• Meshless Methods
• Integral Equations
• Applications of BEM/MRM in Engineering
• Numerical Methods related to BEM/MRM
• Computational Techniques
• Combination of Different Methods
• Advanced Formulations.