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On the efficient realization of sparse matrix techniques for integral equations with focus on panel clustering, cubature and software design aspects


Hackbusch, W; Lage, C; Sauter, S (1997). On the efficient realization of sparse matrix techniques for integral equations with focus on panel clustering, cubature and software design aspects. In: Wendland, W. Boundary element topics (Stuttgart, 1995). Berlin: Springer, 51-75.

Abstract

The paper is organized as follows. In the next section, we go briefly through the setting of Petrov-Galerkin discretization of boundary integral equations and introduce the notations necessary for the sequel. Then, in Section 3, we investigate the computation of the surface integrals arising when computing the matrix entries of the linear system. First, we have to provide some analytical properties of the kernel functions that arise, which then are used in the design of efficient cubature strategies. (Note that in more than one dimension the term quadrature is replaced by cubature). In Section 4, we explain the panel clustering algorithm. After having developed the basic principle, we formulate the algorithm and present an error analysis. Estimates of the asymptotic complexity of the algorithm follow and remarks on their relevance for practical problem sizes are given. Section 6 is devoted to the design of modern BEM software in C++. It is explained which data structures are well suited to making the different algorithms efficient and flexible with respect to various kinds of integral equations, discretization schemes such as, e.g., collocation or Galerkin BEMs, and various geometries and cubature techniques.

The paper is organized as follows. In the next section, we go briefly through the setting of Petrov-Galerkin discretization of boundary integral equations and introduce the notations necessary for the sequel. Then, in Section 3, we investigate the computation of the surface integrals arising when computing the matrix entries of the linear system. First, we have to provide some analytical properties of the kernel functions that arise, which then are used in the design of efficient cubature strategies. (Note that in more than one dimension the term quadrature is replaced by cubature). In Section 4, we explain the panel clustering algorithm. After having developed the basic principle, we formulate the algorithm and present an error analysis. Estimates of the asymptotic complexity of the algorithm follow and remarks on their relevance for practical problem sizes are given. Section 6 is devoted to the design of modern BEM software in C++. It is explained which data structures are well suited to making the different algorithms efficient and flexible with respect to various kinds of integral equations, discretization schemes such as, e.g., collocation or Galerkin BEMs, and various geometries and cubature techniques.

Citations

Additional indexing

Other titles:Proceedings of the final conference of the Priority Research Programme "Boundary Element Methods 1989–1995'' of the German Research Foundation held in Stuttgart, October 2–4, 1995
Item Type:Book Section, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:1997
Deposited On:29 Nov 2010 16:28
Last Modified:05 Apr 2016 13:26
Publisher:Springer
ISBN:3-540-62850-9
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0884.65114

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