Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Convergence of a finite volume scheme for the convection-diffusion equation with $ \mathrm{L}^1$ data

Authors: T. Gallouët, A. Larcher and J. C. Latché
Journal: Math. Comp. 81 (2012), 1429-1454
MSC (2010): Primary 35K10, 65M12, 65M08
Published electronically: December 21, 2011
MathSciNet review: 2904585
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove the convergence of a finite-volume scheme for the time-dependent convection-diffusion equation with an $ \mathrm {L}^1$ right-hand side. To this purpose, we first prove estimates for the discrete solution and for its discrete time and space derivatives. Then we show the convergence of a sequence of discrete solutions obtained with more and more refined discretizations, possibly up to the extraction of a subsequence, to a function which meets the regularity requirements of the weak formulation of the problem; to this purpose, we prove a compactness result, which may be seen as a discrete analogue to the Aubin-Simon lemma. Finally, such a limit is shown to be indeed a weak solution.

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

  • 1. P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, and J.-L. Vazquez.
    An $ L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations.
    Annali della Scuola Normale Superiora di Pisa, Classe de Scienze, 22:240-273, 1995. MR 1354907 (96k:35052)
  • 2. D. Blanchard and F. Murat.
    Renormalized solutions of nonlinear parabolic problems with $ L^1$ data: existence and uniqueness.
    Proceedings of the Royal Society of Edinburgh Section A, 127:1137-1152, 1997. MR 1489429 (98i:35096)
  • 3. L. Boccardo, A. Dall'Aglio, T. Gallouët, and L. Orsina.
    Nonlinear parabolic equations with measure data.
    Journal of Functional Analysis, 147:237-258, 1997. MR 1453181 (98c:35093)
  • 4. L. Boccardo and T. Gallouët.
    Non-linear elliptic and parabolic equations involving measure data.
    Journal of Functional Analysis, 87:149-169, 1989. MR 1025884 (92d:35286)
  • 5. F. Boyer and P. Fabrie.
    Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles.
    In Mathématiques et Applications. Springer, 2006. MR 2248409 (2009f:76040)
  • 6. A. Bradji and R. Herbin.
    Discretization of the coupled heat and electrical diffusion problems by the finite element and the finite volume methods.
    IMA Journal of Numerical Analysis, 28:469-495, 2008. MR 2433209 (2009e:65167)
  • 7. J. Casado-Díaz, T. Chacón Rebollo, V. Girault, M. Gómez Mármol, and F. Murat.
    Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in $ L^1$.
    Numerische Mathematik, 105:337-374, 2007. MR 2266830 (2007k:65175)
  • 8. S. Clain.
    Analyse mathématique et numérique d'un modèle de chauffage par induction.
    PhD thesis, EPFL, 1994.
  • 9. Y. Coudière, T. Gallouët, and R. Herbin.
    Discrete Sobolev Inequalities and $ L^p$ Error Estimates for Approximate Finite Volume Solutions of Convection Diffusion Equations.
    Mathematical Modelling and Numerical Analysis, 35:767-778, 1998. MR 1863279 (2002h:65167)
  • 10. J. Droniou, T. Gallouët, and R. Herbin.
    A finite volume scheme for a noncoercive elliptic equation with measure data.
    SIAM Journal on Numerical Analysis, 41:1997-2031, 2003. MR 2034602 (2005b:65117)
  • 11. J. Droniou and A. Prignet.
    Equivalence between entropy and renormalized solutions for parabolic equations with soft measure data.
    Nonlinear Differential Equations and Applications, 14:181-205, 2007. MR 2346459 (2009b:35177)
  • 12. R. Eymard, T. Gallouët, and R. Herbin.
    Finite volume methods.
    In P. Ciarlet and J.L. Lions, editors, Handbook of Numerical Analysis, Volume VII, pages 713-1020. North Holland, 2000. MR 1804748 (2002e:65138)
  • 13. R. Eymard, T. Gallouët, and R. Herbin.
    Discretisation of heterogeneous and anisotropic diffusion problems on general nonconforming meshes - SUSHI: a scheme using stabilization and hybrid interfaces.
    IMA Journal of Numerical Analysis, 30:1009-1043, 2010. MR 2727814
  • 14. R. Eymard, R. Herbin, and J.-C. Latché.
    Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2D or 3D meshes.
    SIAM Journal on Numerical Analysis, 45:1-36, 2007. MR 2285842 (2008f:65182)
  • 15. T. Gallouët, L. Gastaldo, R. Herbin, and J.-C. Latché.
    An unconditionally stable pressure correction scheme for compressible barotropic Navier-Stokes equations.
    Mathematical Modelling and Numerical Analysis, 42:303-331, 2008. MR 2405150 (2009b:76125)
  • 16. T. Gallouët and R. Herbin.
    Finite volume approximation of elliptic problems with irregular data.
    In Finite Volume for Complex Applications II. Hermes, 1999. MR 2062134
  • 17. T. Gallouët and R. Herbin.
    Convergence of linear finite elements for diffusion equations with measure data.
    Comptes Rendus de l'Académie des Sciences de Paris, Série I, Mathématiques, 338:81-84, 2004. MR 2038090
  • 18. L. Gastaldo, R. Herbin, and J.-C. Latché.
    A discretization of phase mass balance in fractional step algorithms for the drift-flux model.
    IMA Journal of Numerical Analysis, 31:116-146, 2011. MR 2755939
  • 19. J.-L. Lions.
    Quelques méthodes de résolution des problèmes aux limites non linéaires.
    Dunod, Paris, 1969. MR 0259693 (41:4326)
  • 20. A. Prignet.
    Existence and uniqueness of ``entropy'' solutions of parabolic problems with $ L^1$ data.
    Nonlinear Analysis, 28:1943-1954, 1997. MR 1436364 (98b:35085)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 35K10, 65M12, 65M08

Retrieve articles in all journals with MSC (2010): 35K10, 65M12, 65M08

Additional Information

T. Gallouët
Affiliation: Université de Provence, France

A. Larcher
Affiliation: Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France

J. C. Latché
Affiliation: Institut de Radioprotection et de Sûreté Nucléaire (IRSN), France

Keywords: Convection-diffusion equation, finite volumes, irregular data
Received by editor(s): July 23, 2010
Received by editor(s) in revised form: April 18, 2011
Published electronically: December 21, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society