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Wavelet-Based Numerical Homogenization with Applications

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Multiscale and Multiresolution Methods

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 20))

Abstract

Classical homogenization is an analytic technique for approximating multiscale differential equations. The numbers of scales are reduced and the resulting equations are easier to analyze or numerically approximate. The class of problems that classical homogenization applies to is quite restricted. We shall describe a numerical procedure for homogenization, which starts from a discretization of the multiscale differential equation. In this procedure the discrete operator is represented in a wavelet space and projected onto a coarser subspace. The wavelet homogenization applies to a wider class of problems than classical homogenization. The projection procedure is general and we give a presentation of a framework in Hilbert space, which also applies to the differential equation directly. The wavelet based homogenization technique is applied to discretizations of the Helmholtz equation. In one problem from electromagnetic compatibility a subgrid scale geometrical detail is represented on a coarser grid. In another a wave-guide filter is efficiently approximated in a lower dimension. The technique is also applied to the derivation of effective equations for a nonlinear problem and to the derivation of coarse grid operators in multigrid. These multigrid methods work very well for equations with highly oscillatory or discontinuous coefficients.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, 90095-1555, USA

    Björn Engquist

  2. Royal Institute of Technology, 100 44, Stockholm

    Björn Engquist

  3. Department of Mathematics, Princeton University, PACM, Princeton, NJ, 08544-1000, USA

    Olof Runborg

Authors
  1. Björn Engquist
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  2. Olof Runborg
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Editor information

Editors and Affiliations

  1. NAS Division, NASA Ames Research Center, Moffett Field, CA, 94035, USA

    Timothy J. Barth

  2. Department of Mathematics, University of California, Los Angeles, CA, 90095-1555, USA

    Tony Chan

  3. Department of Aeronautics and Astronautics, MIT - 37-467, Cambridge, MA, 02139

    Robert Haimes

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© 2002 Springer-Verlag Berlin Heidelberg

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Engquist, B., Runborg, O. (2002). Wavelet-Based Numerical Homogenization with Applications. In: Barth, T.J., Chan, T., Haimes, R. (eds) Multiscale and Multiresolution Methods. Lecture Notes in Computational Science and Engineering, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56205-1_2

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  • DOI: https://doi.org/10.1007/978-3-642-56205-1_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42420-8

  • Online ISBN: 978-3-642-56205-1

  • eBook Packages: Springer Book Archive

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