Convergence of a finite volume scheme for the convection-diffusion equation with 𝐿¹ data

Gallouët, T.; Larcher, A.; Latché, J.

doi:10.1090/S0025-5718-2011-02571-8

Convergence of a finite volume scheme for the convection-diffusion equation with $\mathrm {L}^1$ data
HTML articles powered by AMS MathViewer

by T. Gallouët, A. Larcher and J. C. Latché PDF

Math. Comp. 81 (2012), 1429-1454 Request permission

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

Philippe Bénilan, Lucio Boccardo, Thierry Gallouët, Ron Gariepy, Michel Pierre, and Juan Luis Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 2, 241–273. MR 1354907
D. Blanchard and F. Murat, Renormalised solutions of nonlinear parabolic problems with $L^1$ data: existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 6, 1137–1152. MR 1489429, DOI 10.1017/S0308210500026986
Lucio Boccardo, Andrea Dall’Aglio, Thierry Gallouët, and Luigi Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal. 147 (1997), no. 1, 237–258. MR 1453181, DOI 10.1006/jfan.1996.3040
Lucio Boccardo and Thierry Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), no. 1, 149–169. MR 1025884, DOI 10.1016/0022-1236(89)90005-0
Franck Boyer and Pierre Fabrie, Éléments d’analyse pour l’étude de quelques modèles d’écoulements de fluides visqueux incompressibles, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 52, Springer-Verlag, Berlin, 2006 (French). MR 2248409, DOI 10.1007/3-540-29819-3
Abdallah Bradji and Raphaèle Herbin, Discretization of coupled heat and electrical diffusion problems by finite-element and finite-volume methods, IMA J. Numer. Anal. 28 (2008), no. 3, 469–495. MR 2433209, DOI 10.1093/imanum/drm030
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 $L^1$, Numer. Math. 105 (2007), no. 3, 337–374. MR 2266830, DOI 10.1007/s00211-006-0033-2
S. Clain. Analyse mathématique et numérique d’un modèle de chauffage par induction. PhD thesis, EPFL, 1994.
Yves Coudière, Thierry Gallouët, and Raphaèle Herbin, Discrete Sobolev inequalities and $L^p$ error estimates for finite volume solutions of convection diffusion equations, M2AN Math. Model. Numer. Anal. 35 (2001), no. 4, 767–778. MR 1863279, DOI 10.1051/m2an:2001135
Jérôme Droniou, Thierry Gallouët, and Raphaèle Herbin, A finite volume scheme for a noncoercive elliptic equation with measure data, SIAM J. Numer. Anal. 41 (2003), no. 6, 1997–2031. MR 2034602, DOI 10.1137/S0036142902405205
Jérôme Droniou and Alain Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, NoDEA Nonlinear Differential Equations Appl. 14 (2007), no. 1-2, 181–205. MR 2346459, DOI 10.1007/s00030-007-5018-z
Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1086/phos.67.4.188705
R. Eymard, T. Gallouët, and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal. 30 (2010), no. 4, 1009–1043. MR 2727814, DOI 10.1093/imanum/drn084
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 J. Numer. Anal. 45 (2007), no. 1, 1–36. MR 2285842, DOI 10.1137/040613081
Thierry Gallouët, Laura Gastaldo, Raphaele Herbin, and Jean-Claude Latché, An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations, M2AN Math. Model. Numer. Anal. 42 (2008), no. 2, 303–331. MR 2405150, DOI 10.1051/m2an:2008005
Thierry Gallouët and Raphaèle Herbin, Finite volume approximation of elliptic problems with irregular data, Finite volumes for complex applications II, Hermes Sci. Publ., Paris, 1999, pp. 155–162. MR 2062134
Thierry Gallouët and Raphaèle Herbin, Convergence of linear finite elements for diffusion equations with measure data, C. R. Math. Acad. Sci. Paris 338 (2004), no. 1, 81–84 (English, with English and French summaries). MR 2038090, DOI 10.1016/j.crma.2003.11.024
Laura Gastaldo, Raphaèle Herbin, and Jean-Claude Latché, A discretization of the phase mass balance in fractional step algorithms for the drift-flux model, IMA J. Numer. Anal. 31 (2011), no. 1, 116–146. MR 2755939, DOI 10.1093/imanum/drp006
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
Alain Prignet, Existence and uniqueness of “entropy” solutions of parabolic problems with $L^1$ data, Nonlinear Anal. 28 (1997), no. 12, 1943–1954. MR 1436364, DOI 10.1016/S0362-546X(96)00030-2