Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T19:44:38.855Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite volume schemes for the p-Laplacianon Cartesian meshes

Published online by Cambridge University Press:  15 December 2004

Boris Andreianov ,
Franck Boyer  and
Florence Hubert
Boris Andreianov
Affiliation:
Département de Mathématiques, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France.
Franck Boyer
Affiliation:
Laboratoire d'Analyse, Topologie et Probabilités, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. [email protected].
Florence Hubert
Affiliation:
Laboratoire d'Analyse, Topologie et Probabilités, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. [email protected].
Get access

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.

Keywords

Finite volume methodsp-Laplacianerror estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 6 , November 2004 , pp. 931 - 959
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Angot, P., Bruneau, C.-H. and Fabrie, P., A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81 (1999) 497–520. CrossRef
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).
Andreianov, B., Gutnic, M. and Wittbold, P., Convergence of finite volume approximations for a nonlinear elliptic-parabolic problem: A “continuous” approach. SIAM J. Numer. Anal. 42 (2004) 228–251. CrossRef
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.
Barrett, J.W. and Liu, W.B., Finite element approximation of the p-Laplacian. Math. Comp. 61 (1993) 523–537.
Chow, S., Finite element error estimates for non-linear elliptic equations of monotone type. Numer. Math. 54 (1989) 373–393. CrossRef
Coudière, Y., Vila, J.-P. and Villedieu, P., Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493–516. CrossRef
Diaz, J.I. and de Thelin, F., On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 25 (1994) 1085–1111. CrossRef
K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. (2004) (submitted).
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).
Eymard, R., Gallouët, T. and Herbin, R., Finite volume approximation of elliptic problems and convergence of an approximate gradient. Appl. Numer. Math. 37 (2001) 31–53. CrossRef
Eymard, R., Gallouët, T. and Herbin, R., A finite volume scheme for anisotropic diffusion problems. C.R. Acad. Sci. Paris 1 339 (2004) 299–302. CrossRef
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).
Glowinski, R. and Rappaz, J., Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology. ESAIM: M2AN 37 (2003) 175–186. CrossRef
Picasso, M., Rappaz, J., Reist, A., Funk, M. and Blatter, H., Numerical simulation of the motion of a two dimensional glacier. Int. J. Numer. Methods Eng. 60 (2004) 995–1009. CrossRef
Simon, J., Régularité de la solution d'un problème aux limites non linéaires. Ann. Fac. Sciences Toulouse 3 (1981) 247–274. CrossRef