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Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations


Authors: Daniele A. Di Pietro and Alexandre Ern
Journal: Math. Comp. 79 (2010), 1303-1330
MSC (2010): Primary 65N12, 65N30, 76D05, 35Q30, 76D07
DOI: https://doi.org/10.1090/S0025-5718-10-02333-1
Published electronically: March 3, 2010
MathSciNet review: 2629994
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Abstract: Two discrete functional analysis tools are established for spaces of piecewise polynomial functions on general meshes: (i) a discrete counterpart of the continuous Sobolev embeddings, in both Hilbertian and non-Hilbertian settings; (ii) a compactness result for bounded sequences in a suitable Discontinuous Galerkin norm, together with a weak convergence property for some discrete gradients. The proofs rely on techniques inspired by the Finite Volume literature, which differ from those commonly used in Finite Element analysis. The discrete functional analysis tools are used to prove the convergence of Discontinuous Galerkin approximations of the steady incompressible Navier-Stokes equations. Two discrete convective trilinear forms are proposed, a nonconservative one relying on Temam's device to control the kinetic energy balance and a conservative one based on a nonstandard modification of the pressure.


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Additional Information

Daniele A. Di Pietro
Affiliation: Institut Français du Pétrole, 1 & 4, avenue du Bois-Préau, 92852 Rueil-Malmaison Cedex, France
Email: daniele-antonio.di-pietro@ifp.fr

Alexandre Ern
Affiliation: Université Paris-Est, CERMICS, Ecole des Ponts, Champs sur Marne, 77455 Marne la Vallée Cedex 2, France
Email: ern@cermics.enpc.fr

DOI: https://doi.org/10.1090/S0025-5718-10-02333-1
Received by editor(s): May 12, 2008
Received by editor(s) in revised form: March 25, 2009
Published electronically: March 3, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.