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Local projection stabilization of equal order interpolation applied to the Stokes problem


Authors: Sashikumaar Ganesan, Gunar Matthies and Lutz Tobiska
Journal: Math. Comp. 77 (2008), 2039-2060
MSC (2000): Primary 65N12; Secondary 65N30
DOI: https://doi.org/10.1090/S0025-5718-08-02130-3
Published electronically: May 9, 2008
MathSciNet review: 2429873
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Abstract: The local projection stabilization allows us to circumvent the Babuška-Brezzi condition and to use equal order interpolation for discretizing the Stokes problem. The projection is usually done in a two-level approach by projecting the pressure gradient onto a discontinuous finite element space living on a patch of elements. We propose a new local projection stabilization method based on (possibly) enriched finite element spaces and discontinuous projection spaces defined on the same mesh. Optimal order of convergence is shown for pairs of approximation and projection spaces satisfying a certain inf-sup condition. Examples are enriched simplicial finite elements and standard quadrilateral/hexahedral elements. The new approach overcomes the problem of an increasing discretization stencil and, thus, is simple to implement in existing computer codes. Numerical tests confirm the theoretical convergence results which are robust with respect to the user-chosen stabilization parameter.


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Additional Information

Sashikumaar Ganesan
Affiliation: Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany
Email: ga.sashikumaar@mathematik.uni-magdeburg.de

Gunar Matthies
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780 Bochum, Germany
Email: gunar.matthies@ruhr-uni-bochum.de

Lutz Tobiska
Affiliation: Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany
Email: tobiska@mathematik.uni-magdeburg.de

DOI: https://doi.org/10.1090/S0025-5718-08-02130-3
Received by editor(s): April 17, 2007
Received by editor(s) in revised form: September 21, 2007
Published electronically: May 9, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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