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A convergent finite element-finite volume scheme for the compressible Stokes problem. Part II: the isentropic case


Authors: R. Eymard, T. Gallouët, R. Herbin and J. C. Latché
Journal: Math. Comp. 79 (2010), 649-675
MSC (2000): Primary 35Q30, 65N12, 65N30, 76M25
DOI: https://doi.org/10.1090/S0025-5718-09-02310-2
Published electronically: December 8, 2009
MathSciNet review: 2600538
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Abstract: In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with an equation of state of the form $ p=\rho^\gamma$ (where $ p$ stands for the pressure and $ \rho$ for the density). This scheme is based on Crouzeix-Raviart approximation spaces. The discretization of the momentum balance is obtained by the usual finite element technique. The discrete mass balance is obtained by a finite volume scheme, with an upwinding of the density, and two additional stabilization terms. We prove a priori estimates for the discrete solution, which yield its existence. Then the convergence of the scheme to a solution of the continuous problem is established. The passage to the limit in the equation of state requires the a.e. convergence of the density. It is obtained by adapting at the discrete level the ``effective viscous pressure lemma'' of the theory of compressible Navier-Stokes equations.


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Additional Information

R. Eymard
Affiliation: Université de Marne-la-Vallée, France
Email: eymard@univ-mlv.fr

T. Gallouët
Affiliation: Université de Provence, France
Email: gallouet@latp.univ-mrs.fr

R. Herbin
Affiliation: Université de Provence, France
Email: herbin@latp.univ-mrs.fr

J. C. Latché
Affiliation: Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France
Email: jean-claude.latche@irsn.fr

DOI: https://doi.org/10.1090/S0025-5718-09-02310-2
Keywords: Compressible Stokes equations, finite element methods, finite volume methods
Received by editor(s): October 2, 2008
Received by editor(s) in revised form: April 6, 2009
Published electronically: December 8, 2009
Article copyright: © Copyright 2009 American Mathematical Society