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A wavelet Galerkin method for the stokes equations

Eine Wavelet-Galerkin Methode für die Stokes-Gleichungen

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Abstract

The purpose of this paper is to investigate Galerkin schemes for the Stokes equations based on a suitably adapted multiresolution analysis. In particular, it will be shown that techniques developed in connection with shift-invariant refinable spaces give rise to trial spaces of any desired degree of accuracy satisfying the Ladyšenskaja-Babuška-Brezzi condition for any spatial dimension. Moreover, in the time dependent case efficient preconditioners for the Schur complements of the discrete systems of equations can be based on corresponding stable multiscale decompositions. The results are illustrated by some concrete examples of adapted wavelets and corresponding numerical experiments.

Zusammenfassung

In dieser Arbeit werden Galerkin-Verfahren für das Stokes-Problem untersucht, die auf speziell angepaßten Multiresolution-Ansätzen beruhen. Insbesondere wird gezeigt, daß gewisse Konstruktionsprinzipien für Wavelets auf gleichförmigen Gittern für jede Raumdimension und beliebige gewünschte Exaktheitsordnung auf Parre von Ansatzräumen führen, die die Ladyšenskaja-Babuška-Brezzi-Bedingung erfüllen. Darüber hinaus ergeben sich auch im instationären Fall aus den entsprechenden stabilen Multiskalenzerlegungen effiziente Vorkonditionierer für die Schurkomplemente entsprechenden systemmatrizen. Die Ergebnisse werden anhand einiger konkreter Realisierungen und numerischer Tests illustriert.

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Authors and Affiliations

  1. Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-52056, Aachen, Germany

    W. Dahmen

  2. Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-52056, Aachen, Germany

    K. Urban

  3. Weierstrass-Institut für Angewandte, Analysis und Stochastik (WIAS), Mohrenstrasse 39, D-10117, Berlin, Germany

    A. Kunoth

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  1. W. Dahmen
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The work of this author is supported by the Deutsche Forschungsgemeinschaft.

The work of this author is supported by the Graduiertenkolleg “Analyse und Konstruktion in der Mathematik” funded by the Deutsche Forschungsgemeinschaft at the RWTH Aachen.

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Dahmen, W., Kunoth, A. & Urban, K. A wavelet Galerkin method for the stokes equations. Computing 56, 259–301 (1996). https://doi.org/10.1007/BF02238515

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