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Multiscale, Nonlinear and Adaptive Approximation pp 249–294Cite as

Rapid solution of boundary integral equations by wavelet Galerkin schemes

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Abstract

The present paper aims at reviewing the research on the wavelet-based rapid solution of boundary integral equations. When discretizing boundary integral equations by appropriate wavelet bases the system matrices are quasi-sparse. Discarding the non-relevant matrix entries is called wavelet matrix compression. The compressed system matrix can be assembled within linear complexity if an exponentially convergent hp-quadrature algorithm is used. Therefore, in combination with wavelet preconditioning, one arrives at an algorithm that solves a given boundary integral equation within discretization error accuracy, offered by the underlying Galerkin method, at a computational expense that stays proportional to the number of unknowns. By numerical results we illustrate and quantify the theoretical findings.

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Authors and Affiliations

  1. Institute for Numerical Simulation, Bonn University, Wegelerstr. 6, 53115, Bonn, Germany

    Helmut Harbrecht

  2. Institute of Mathematics, Technical University of Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Reinhold Schneider

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  1. Helmut Harbrecht
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  1. Dept. Mathematics, Texas A & M University, Blocker Hall 608K, College Station, 77843, U.S.A.

    Ronald DeVore

  2. Institut für Mathematik, Universität Paderborn, Warburger Str. 100, Paderborn, 33098, Germany

    Angela Kunoth

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Harbrecht, H., Schneider, R. (2009). Rapid solution of boundary integral equations by wavelet Galerkin schemes. In: DeVore, R., Kunoth, A. (eds) Multiscale, Nonlinear and Adaptive Approximation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03413-8_8

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