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Inf-Sup stability of the discrete duality finite volume method for the 2D Stokes problem


Authors: Franck Boyer, Stella Krell and Flore Nabet
Journal: Math. Comp. 84 (2015), 2705-2742
MSC (2010): Primary 65N08, 65N12, 76D07, 76M12
Published electronically: April 29, 2015
MathSciNet review: 3378845
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Abstract: ``Discrete Duality Finite Volume'' schemes (DDFV for short) on general 2D meshes, in particular, non-conforming ones, are studied for the Stokes problem with Dirichlet boundary conditions. The DDFV method belongs to the class of staggered schemes since the components of the velocity and the pressure are approximated on different meshes. In this paper, we investigate from a numerical and theoretical point of view, whether or not the stability condition holds in this framework for various kinds of mesh families. We obtain that different behaviors may occur depending on the geometry of the meshes.

For instance, for conforming acute triangle meshes, we prove the unconditional Inf-Sup stability of the scheme, whereas for some conforming or non-conforming Cartesian meshes we prove that Inf-Sup stability holds up to a single unstable pressure mode. In any case, the DDFV method appears to be very robust.


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Additional Information

Franck Boyer
Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France
Email: franck.boyer@univ-amu.fr

Stella Krell
Affiliation: Université de Nice Sophia-Antipolis, CNRS, LJAD UMR 7351, Nice, France
Email: krell@unice.fr

Flore Nabet
Affiliation: Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France
Email: flore.nabet@univ-amu.fr

DOI: https://doi.org/10.1090/mcom/2956
Keywords: Finite-volume methods, Stokes problem, DDFV methods, Inf-Sup stability
Received by editor(s): February 27, 2013
Received by editor(s) in revised form: December 20, 2013, and March 12, 2014
Published electronically: April 29, 2015
Article copyright: © Copyright 2015 American Mathematical Society