H2-matrices for translation-invariant kernel functions

IF 4.2 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY Engineering Analysis with Boundary Elements Pub Date : 2025-03-15 DOI:10.1016/j.enganabound.2025.106190

Steffen Börm, Janne Henningsen

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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

H^{2}

-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

O (log (n))

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

O (n)

units of storage.

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平移不变核函数的 H2 矩阵

椭圆偏微分方程的边界元方法通常会产生具有平移不变核函数的边界积分算子。如果使用一般的非结构网格和一般的基函数，利用这一特性并不简单，因为我们需要这些基函数的支持包含在具有平移对称性的子域层次中。在本文中，我们提出了在非结构准均匀网格上对 H2 矩阵进行修改的构造，该构造利用平移不变性显著降低了远场表示的存储要求。我们构建了轴平行盒的嵌套层次结构，从而保留了平移对称性，并在适度假设下证明了最优阶复杂度估计。特别是，我们只需要一个弱假设，就能证明整个远场只需要 O(log(n)) 个系数。值得一提的是，由于我们使用的是非结构网格和一般基函数，矩阵的近场仍然需要 O(n) 个单位的存储空间。

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

Engineering Analysis with Boundary Elements 工程技术-工程：综合

CiteScore

5.50

自引率

18.20%

发文量

368

审稿时长

56 days

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.