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Preconditioning a class of fourth order problems by operator splitting

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Abstract

We develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space. We consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two (discrete) linear second order elliptic operators, and exploit this property in designing the preconditioners. The underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available. We propose symmetric and non-symmetric preconditioners, along with theory and numerical experiments. They both document crucial properties of the preconditioners as well as their practical performance. It is important to note that we neither need H s-regularity, s > 1, of the continuous problem nor quasi-uniform grids.

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Author information

Authors and Affiliations

  1. Angewandte Mathematik III, Friedrich-Alexander-Universität Erlangen-Nürnberg, Haberstr. 2, 91058, Erlangen, Germany

    Eberhard Bänsch

  2. Instituto de Matemática Aplicada del Litoral, CONICET—Universidad Nacional del Litoral, Güemes 3450, S3000GLN, Santa Fe, Argentina

    Pedro Morin

  3. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

Authors
  1. Eberhard Bänsch
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  2. Pedro Morin
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  3. Ricardo H. Nochetto
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Corresponding author

Correspondence to Pedro Morin.

Additional information

P. Morin partially supported by CONICET through Grant PIP 5811, ANPCyT through Grant PICT 2008-0622 and Universidad Nacional del Litoral through Grant CAI+D PI-62-312. R. H. Nochetto partially supported by NSF grants DMS-0505454, DMS-0807811 and INT-0126272.

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Bänsch, E., Morin, P. & Nochetto, R.H. Preconditioning a class of fourth order problems by operator splitting. Numer. Math. 118, 197–228 (2011). https://doi.org/10.1007/s00211-010-0333-4

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  • DOI: https://doi.org/10.1007/s00211-010-0333-4

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