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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.


References [Enhancements On Off] (What's this?)

  • 1. J.H. Bramble.
    A proof of the inf-sup condition for the Stokes equations on Lipschitz domains.
    Mathematical Models and Methods in Applied Sciences, 13:361-371, 2003. MR 1977631 (2004d:35197)
  • 2. P. G. Ciarlet.
    Handbook of numerical analysis. Volume II : Finite elements methods - Basic error estimates for elliptic problems.
    In P. Ciarlet and J.L. Lions, editors, Handbook of Numerical Analysis, Volume II, pages 17-351. North Holland, 1991. MR 1115235 (92f:65001)
  • 3. M. Crouzeix and P.-A. Raviart.
    Conforming and nonconforming finite element methods for solving the stationary Stokes equations I.
    Revue Française d'Automatique, Informatique et Recherche Opérationnelle (R.A.I.R.O.), R-3:33-75, 1973. MR 0343661 (49:8401)
  • 4. A. Ern and J.-L. Guermond.
    Theory and practice of finite elements.
    Number 159 in Applied Mathematical Sciences. Springer, New York, 2004. MR 2050138 (2005d:65002)
  • 5. R. Eymard and R. Herbin.
    Entropy estimate for the approximation of the compressible barotropic Navier-Stokes equations using a collocated finite volume scheme.
    in preparation, 2008.
  • 6. T. Gallouët, L. Gastaldo, R. Herbin, and J.-C. Latché.
    An unconditionnally stable pressure correction scheme for compressible barotropic Navier-Stokes equations.
    Mathematical Modelling and Numerical Analysis, 42:303-331, 2008. MR 2405150 (2009b:76125)
  • 7. T. Gallouët, R. Herbin, and J.-C. Latché.
    A convergent finite element-finite volume scheme for the compressible Stokes problem. Part I - The isothermal case.
    to appear in Mathematics of Computation, 2009.
  • 8. L. Gastaldo, R. Herbin, and J.-C. Latché.
    A discretization of phase mass balance in fractional step algorithms for the drift-flux model.
    to appear in IMA Journal of Numerical Analysis, 2009.
  • 9. L. Gastaldo, R. Herbin, and J.-C. Latché.
    An entropy-preserving finite element-finite volume pressure correction scheme for the drift-flux model.
    submitted, 2009.
  • 10. M. Jobelin, B. Piar, P. Angot, and J.-C. Latché.
    Une méthode de pénalité-projection pour les écoulements dilatables.
    Revue Européene de mécanique numérique, 17:453-480, 2008.
  • 11. P.-L. Lions.
    Mathematical topics in fluid mechanics. Volume 2. Compressible models.
    Volume 10 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 1998. MR 1637634 (99m:76001)
  • 12. A. Novotný and I. Straškraba.
    Introduction to the mathematical theory of compressible flow.
    Volume 27 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, 2004. MR 2084891 (2005i:35220)

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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

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