Discrete Sobolev inequalities and L p error estimates for finite volume solutions of convection diffusion equations

Yves Coudière; Thierry Gallouët; Raphaèle Herbin

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 4, page 767-778
  • ISSN: 0764-583X

Abstract

top
The topic of this work is to obtain discrete Sobolev inequalities for piecewise constant functions, and to deduce L p error estimates on the approximate solutions of convection diffusion equations by finite volume schemes.

How to cite

top

Coudière, Yves, Gallouët, Thierry, and Herbin, Raphaèle. "Discrete Sobolev inequalities and $L^p$ error estimates for finite volume solutions of convection diffusion equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.4 (2001): 767-778. <http://eudml.org/doc/194072>.

@article{Coudière2001,
abstract = {The topic of this work is to obtain discrete Sobolev inequalities for piecewise constant functions, and to deduce $L^p$ error estimates on the approximate solutions of convection diffusion equations by finite volume schemes.},
author = {Coudière, Yves, Gallouët, Thierry, Herbin, Raphaèle},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite volume methods; $\{L^p\}$ error estimates; unstructured meshes; convection-diffusion equations; error estimates; convergence; discrete Sobolev inequalities},
language = {eng},
number = {4},
pages = {767-778},
publisher = {EDP-Sciences},
title = {Discrete Sobolev inequalities and $L^p$ error estimates for finite volume solutions of convection diffusion equations},
url = {http://eudml.org/doc/194072},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Coudière, Yves
AU - Gallouët, Thierry
AU - Herbin, Raphaèle
TI - Discrete Sobolev inequalities and $L^p$ error estimates for finite volume solutions of convection diffusion equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 4
SP - 767
EP - 778
AB - The topic of this work is to obtain discrete Sobolev inequalities for piecewise constant functions, and to deduce $L^p$ error estimates on the approximate solutions of convection diffusion equations by finite volume schemes.
LA - eng
KW - finite volume methods; ${L^p}$ error estimates; unstructured meshes; convection-diffusion equations; error estimates; convergence; discrete Sobolev inequalities
UR - http://eudml.org/doc/194072
ER -

References

top
  1. [1] R.E. Bank and D.J. Rose, Some error estimates for the box method. SIAM J. Numer. Anal. 24 (1987) 777–787. Zbl0634.65105
  2. [2] R. Belmouhoub, Modélisation tridimensionnelle de la genèse des bassins sédimentaires. Thesis, École Nationale Supérieure des Mines de Paris, France (1996). 
  3. [3] Z. Cai, On the finite volume element method. Numer. Math. 58 (1991) 713–735. Zbl0731.65093
  4. [4] Z. Cai, Mandel J. and S. Mc Cormick, The finite volume element method for diffusion equations on general triangulations. SIAM J. Numer. Anal. 28 (1991) 392–402. Zbl0729.65086
  5. [5] W.J. Coirier and K.G. Powell, A Cartesian, cell-based approach for adaptative-refined solutions of the Euler and Navier-Stokes equations. AIAA J. 0566 (1995). 
  6. [6] M. Dauge, Elliptic boundary value problems in corner domains. Lecture Notes in Math. 1341 Springer-Verlag, Berlin (1988). Zbl0668.35001MR961439
  7. [7] Y. Coudière and P. Villedieu, A finite volume scheme for the linear convection-diffusion equation on locally refined meshes, in7-th international colloquium on numerical analysis, Plovdiv, Bulgaria (1998). 
  8. [8] Y. Coudière, J.P. Vila and P. Villedieu, Convergence of a finite volume scheme for a diffusion problem, in Finite volumes for complex applications, problems and perspectives, F. Benkhaldoun and R. Vilsmeier Eds., Hermès, Paris (1996) 161–168. 
  9. [9] Y. Coudière, J.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection diffusion problem. ESAIM: M2AN 33 (1999) 493–516. Zbl0937.65116
  10. [10] Y. Coudière and P. Villedieu, Convergence of a finite volume scheme for a two dimensional diffusion convection equation on locally refined meshes. ESAIM: M2AN 34 (2000) 1109–1295. Zbl0972.65081
  11. [11] R. Eymard, T. Gallouët and R. Herbin, Convergence of finite volume schemes for semilinear convection diffusion equations. Numerische Mathematik. 82 (1999) 91–116. Zbl0930.65118
  12. [12] R. Eymard and T. Gallouët, Convergence d’un schéma de type éléments finis-volumes finis pour un système couplé elliptique-hyperbolique. RAIRO Modél. Math. Anal. Numér. 27 (1993) 843–861. Zbl0792.65073
  13. [13] R. Eymard, T. Gallouët and R. Herbin, The finite volume method, in Handbook of numerical analysis, P.G. Ciarlet and J.L. Lions, Eds., Elsevier Science BV, Amsterdam (2000) 715–1022. Zbl0981.65095
  14. [14] I. Faille, A control volume method to solve an elliptic equation on a 2D irregular meshing. Comput. Methods Appl. Mech. Engrg. 100 (1992) 275–290. Zbl0761.76068
  15. [15] P.A. Forsyth, A control volume finite element approach to NAPL groundwater contamination. SIAM J. Sci. Stat. Comput. 12 (1991) 1029–1057. Zbl0725.76087
  16. [16] P.A. Forsyth and P.H. Sammon, Quadratic convergence for cell-centered grids. Appl. Numer. Math. 4 (1988) 377–394. Zbl0651.65086
  17. [17] T. Gallouët, R. Herbin and M.H. Vignal, Error estimate for the approximate finite volume solutions of convection diffusion equations with Dirichlet, Neumann or Fourier boundary conditions. SIAM J. Numer. Anal. 37 (2000) 1035–1072. Zbl0986.65099
  18. [18] B. Heinrich, Finite difference methods on irregular networks. A generalized approach to second order elliptic problems. Internat. Ser. Numer. Math. 82, Birkhäuser-Verlag, Stuttgart (1987). Zbl0623.65096MR875416
  19. [19] R. Herbin, An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh. Numer. Methods. Partial Differ. Equations 11 (1995) 165–173. Zbl0822.65085
  20. [20] R. Herbin, Finite volume methods for diffusion convection equations on general meshes, in Finite volumes for complex applications, problems and perspectives, F. Benkhaldoun and R. Vilsmeier, Eds., Hermès, Paris (1996) 153–160. 
  21. [21] F. Jacon and D. Knight, A Navier-Stokes algorithm for turbulent flows using an unstructured grid and flux difference splitting. AIAA J. 2292 (1994). 
  22. [22] R.D. Lazarov and I.D. Mishev, Finite volume methods for reaction diffusion problems, in Finite volumes for complex applications, problems and perspectives, F. Benkhaldoun and R. Vilsmeier, Eds., Hermès, Paris (1996) 233–240 . 
  23. [23] R.D. Lazarov, I.D. Mishev and P.S. Vassilevski, Finite volume methods for convection-diffusion problems. SIAM J. Numer. Anal. 33 (1996) 31–55. Zbl0847.65075
  24. [24] T.A. Manteufel and A.B. White, The numerical solution of second order boundary value problem on non uniform meshes. Math. Comput. 47 (1986) 511–536. Zbl0635.65092
  25. [25] K.W. Morton and E. Süli, Finite volume methods and their analysis. IMA J. Numer. Anal. 11 (1991) 241–260. Zbl0729.65087
  26. [26] D. Trujillo, Couplage espace-temps de schémas numériques en simulation de réservoir. Ph.D. Thesis, University of Pau, France (1994). 
  27. [27] P.S. Vassileski, S.I. Petrova and R.D. Lazarov, Finite difference schemes on triangular cell-centered grids with local refinement. SIAM J. Sci. Statist. Comput. 13 (1992) 1287–1313. Zbl0813.65115

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.