Finite volume schemes for the p-Laplacian on Cartesian meshes

Boris Andreianov; Franck Boyer; Florence Hubert

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 6, page 931-959
  • ISSN: 0764-583X

Abstract

top
This paper is concerned with the finite volume approximation of the p-Laplacian equation with homogeneous Dirichlet boundary conditions on rectangular meshes. A reconstruction of the norm of the gradient on the mesh's interfaces is needed in order to discretize the p-Laplacian operator. We give a detailed description of the possible nine points schemes ensuring that the solution of the resulting finite dimensional nonlinear system exists and is unique. These schemes, called admissible, are locally conservative and in addition derive from the minimization of a strictly convexe and coercive discrete functional. The convergence rate is analyzed when the solution lies in W2,p. Numerical results are given in order to compare different admissible and non-admissible schemes.

How to cite

top

Andreianov, Boris, Boyer, Franck, and Hubert, Florence. "Finite volume schemes for the p-Laplacian on Cartesian meshes." ESAIM: Mathematical Modelling and Numerical Analysis 38.6 (2010): 931-959. <http://eudml.org/doc/194250>.

@article{Andreianov2010,
abstract = { This paper is concerned with the finite volume approximation of the p-Laplacian equation with homogeneous Dirichlet boundary conditions on rectangular meshes. A reconstruction of the norm of the gradient on the mesh's interfaces is needed in order to discretize the p-Laplacian operator. We give a detailed description of the possible nine points schemes ensuring that the solution of the resulting finite dimensional nonlinear system exists and is unique. These schemes, called admissible, are locally conservative and in addition derive from the minimization of a strictly convexe and coercive discrete functional. The convergence rate is analyzed when the solution lies in W2,p. Numerical results are given in order to compare different admissible and non-admissible schemes. },
author = {Andreianov, Boris, Boyer, Franck, Hubert, Florence},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite volume methods; p-Laplacian; error estimates.; finite volume scheme; Cartesian meshes; numerical experiments},
language = {eng},
month = {3},
number = {6},
pages = {931-959},
publisher = {EDP Sciences},
title = {Finite volume schemes for the p-Laplacian on Cartesian meshes},
url = {http://eudml.org/doc/194250},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Andreianov, Boris
AU - Boyer, Franck
AU - Hubert, Florence
TI - Finite volume schemes for the p-Laplacian on Cartesian meshes
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 6
SP - 931
EP - 959
AB - This paper is concerned with the finite volume approximation of the p-Laplacian equation with homogeneous Dirichlet boundary conditions on rectangular meshes. A reconstruction of the norm of the gradient on the mesh's interfaces is needed in order to discretize the p-Laplacian operator. We give a detailed description of the possible nine points schemes ensuring that the solution of the resulting finite dimensional nonlinear system exists and is unique. These schemes, called admissible, are locally conservative and in addition derive from the minimization of a strictly convexe and coercive discrete functional. The convergence rate is analyzed when the solution lies in W2,p. Numerical results are given in order to compare different admissible and non-admissible schemes.
LA - eng
KW - Finite volume methods; p-Laplacian; error estimates.; finite volume scheme; Cartesian meshes; numerical experiments
UR - http://eudml.org/doc/194250
ER -

References

top
  1. P. Angot, C.-H. Bruneau and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math.81 (1999) 497–520.  
  2. B. Andreianov, F. Boyer and F. Hubert, Finite volume schemes for the p-Laplacian. Further error estimates. Preprint No. 03-29, LATP Université de Provence (2003).  
  3. B. Andreianov, M. Gutnic and P. Wittbold, Convergence of finite volume approximations for a nonlinear elliptic-parabolic problem: A “continuous” approach. SIAM J. Numer. Anal.42 (2004) 228–251.  
  4. J.W. Barrett and W.B. Liu, A remark on the regularity of the solutions of the p-Laplacian and its application to the finite element approximation, J. Math. Anal. Appl. 178 (1993) 470–487.  
  5. J.W. Barrett and W.B. Liu, Finite element approximation of the p-Laplacian. Math. Comp.61 (1993) 523–537.  
  6. S. Chow, Finite element error estimates for non-linear elliptic equations of monotone type. Numer. Math.54 (1989) 373–393.  
  7. Y. Coudière, J.-P. Vila and P. Villedieu, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN33 (1999) 493–516.  
  8. J.I. Diaz and F. de Thelin, On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal.25 (1994) 1085–1111.  
  9. K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. (2004) (submitted).  
  10. R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, Handbook Numer. Anal., P.G. Ciarlet and J.L. Lions Eds., North-Holland VII (2000).  
  11. R. Eymard, T. Gallouët and R. Herbin, Finite volume approximation of elliptic problems and convergence of an approximate gradient. Appl. Numer. Math.37 (2001) 31–53.  
  12. R. Eymard, T. Gallouët and R. Herbin, A finite volume scheme for anisotropic diffusion problems. C.R. Acad. Sci. Paris 1339 (2004) 299–302.  
  13. R. Glowinski and A. Marrocco, Sur l'approximation par éléments finis d'ordre un, et la résolution, par pénalisation-dualité, d'une classe de problèmes de Dirichlet non linéaires. RAIRO Sér. Rouge Anal. Numér.9 no R-2 (1975).  
  14. R. Glowinski and J. Rappaz, Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology. ESAIM: M2AN37 (2003) 175–186.  
  15. M. Picasso, J. Rappaz, A. Reist, M. Funk and H. Blatter, Numerical simulation of the motion of a two dimensional glacier. Int. J. Numer. Methods Eng.60 (2004) 995–1009.  
  16. J. Simon, Régularité de la solution d'un problème aux limites non linéaires. Ann. Fac. Sciences Toulouse 3 (1981) 247–274.  

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.