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Fast Nyström Methods for Parabolic Boundary Integral Equations

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Fast Boundary Element Methods in Engineering and Industrial Applications

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 63))

Abstract

Time dependence in parabolic boundary integral operators appears in form of an integral over the previous time evolution of the problem. The kernels are singular only at the current time and get increasingly smooth for contributions that are further back in time. The thermal layer potentials can be regarded as generalized Abel operators where the kernel is a parameter dependent surface integral operator. This special form implies that discretization methods and fast evaluation methods must be significantly changed from the familiar elliptic case. After a brief review of recent developments in the area we discuss the different options to discretize Abel integral operators in time. These methods are combined with standard surface quadrature rules to obtain a Nyström method for parabolic integral equations. The method is explicit and we will show how a version of the fast multipole method in space and time can be used to evaluate the time stepping scheme efficiently.

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Authors and Affiliations

  1. Department of Mathematics, Southern Methodist University, Dallas, TX, 75275, USA

    Johannes Tausch

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  1. Johannes Tausch
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Correspondence to Johannes Tausch .

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Editors and Affiliations

  1. Inst. Numerische Mathematik, Universität Linz, Altenbergerstr. 69, Linz, 4040, Austria

    Ulrich Langer

  2. Inst. Allgemeine Mechanik, TU Graz, Technikerstr. 4, Graz, 8010, Austria

    Martin Schanz

  3. Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, Graz, 8010, Austria

    Olaf Steinbach

  4. Institut für Angewandte Analysis, und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, Stuttgart, 70569, Germany

    Wolfgang L. Wendland

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Tausch, J. (2012). Fast Nyström Methods for Parabolic Boundary Integral Equations. In: Langer, U., Schanz, M., Steinbach, O., Wendland, W. (eds) Fast Boundary Element Methods in Engineering and Industrial Applications. Lecture Notes in Applied and Computational Mechanics, vol 63. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25670-7_6

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  • DOI: https://doi.org/10.1007/978-3-642-25670-7_6

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  • Print ISBN: 978-3-642-25669-1

  • Online ISBN: 978-3-642-25670-7

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