Discrete sobolev-poincaré inequalities for voronoi finite volume approximations

SIAM Journal on Numerical Analysis

Volumn 48, Issue 1, 2010, Pages 372-391

  Glitzky, Annegret ^a   Griepentrog, Jens A ^a  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

Discrete Sobolev inequality; Sobolev integral representation; Voronoi finite volume mesh

Indexed keywords

FINITE VOLUME; INTEGRAL REPRESENTATION; SOBOLEV; SOBOLEV INEQUALITIES; VORONOI;

EID: 77951855159     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/09076502X     Document Type: Article

References (16)

1. R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
2. A. Buffa and C. Ortner, Compact embeddings of broken Sobolev spaces and applications, IMA J. Numer. Anal., 29 (2009), pp. 827-855.
3. Y. Coudìere, T. Galloüet, and R. Herbin, Discrete Sobolev Inequalities and Lp error estimates for finite volume solutions of convection diffusion equations, M2AN Math. Model. Numer. Anal., 35 (2001), pp. 767-778.
4. D. Di Pietro and A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with applications to the incompressible Navier-Stokes equations, Math. Comp., to appear. Available online as HAL report 00278925, http://hal.archives-ouvertes.fr/hal- 00278925/en (2008).
5. J. Droniou, T. Galloüet, and R. Herbin, A finite volume scheme for a noncoercive elliptic equation with measure data, SIAM J. Numer. Anal., 41 (2003), pp. 1997-2031. (Pubitemid 40044140)
6. R. Eymard, T. Galloüet, and R. Herbin, The finite volume method, in Handbook of Numerical Analysis VII, Ph. Ciarlet and J. L. Lions, eds., North-Holland, Amsterdam, 2000, pp. 723-1020.
7. R. Eymard, T. Galloüet, and R. Herbin, Discretization schemes for heterogeneous and anisotropic diffusion problems on general nonconforming meshes, IMA J. Numer. Anal., to appear. Available online as HAL report 00203269, http://hal.archives-ouvertes.fr/hal- 00203269/en (2008).
8. R. Eymard, T. Galloüet, and R. Herbin, Cell centered discretisation of non linear elliptic problems on general multidimensional polyhedral grids, J. Numer. Math., 17 (2009), pp. 173-193.
9. T. Galloüet, R. Herbin, and M. H. Vignal, Error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions, SIAM J. Numer. Anal., 37 (2000), pp. 1935-1972.
10. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, Heidelberg, New York, 1977.
11. A. Glitzky, Exponential decay of the free energy for discretized electro-reaction-diffusion systems, Nonlinearity, 21 (2008), pp. 1989-2009.
12. A. Glitzky, Uniform Exponential Decay of the Free Energy for Voronoi Finite Volume Discretized Reaction-Diffusion Systems, Preprint 1443, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, 2009.
13. A. Glitzky and K. G̈artner, Energy estimates for continuous and discretized electro-reactiondiffusion systems, Nonlinear Anal., 70 (2009), pp. 788-805.
14. A. Kufner, O. John, and S. Fucik, Function Spaces, Academia, Prague, 1977.
15. W. I. Smirnow, Lehrgang der ḧoheren Mathematik V, Deutscher Verlag der Wissenschaften, Berlin, 1962.
16. S. L. Sobolev, Einige Anwendungen der Funktionalanalysis auf Gleichungen der mathematischen Physik, Akademie-Verlag, Berlin, 1964.