Finite volume schemes for the $p$-laplacian on cartesian meshes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 6, p. 931-959

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 ${W}^{2,p}$. Numerical results are given in order to compare different admissible and non-admissible schemes.

DOI : https://doi.org/10.1051/m2an:2004045
Classification:  35J65,  65N15,  74S10
Keywords: finite volume methods, p-laplacian, error estimates
@article{M2AN_2004__38_6_931_0,
author = {Andreianov, Boris and Boyer, Franck and Hubert, Florence},
title = {Finite volume schemes for the $p$-laplacian on cartesian meshes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {38},
number = {6},
year = {2004},
pages = {931-959},
doi = {10.1051/m2an:2004045},
zbl = {1081.65105},
mrnumber = {2108939},
language = {en},
url = {http://www.numdam.org/item/M2AN_2004__38_6_931_0}
}

Andreianov, Boris; Boyer, Franck; Hubert, Florence. Finite volume schemes for the $p$-laplacian on cartesian meshes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 6, pp. 931-959. doi : 10.1051/m2an:2004045. http://www.numdam.org/item/M2AN_2004__38_6_931_0/

[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. | Zbl 0921.76168

[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. | Zbl 1080.65081

[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. | Zbl 0799.35085

[5] J.W. Barrett and W.B. Liu, Finite element approximation of the $p$-Laplacian. Math. Comp. 61 (1993) 523-537. | Zbl 0791.65084

[6] S. Chow, Finite element error estimates for non-linear elliptic equations of monotone type. Numer. Math. 54 (1989) 373-393. | Zbl 0643.65058

[7] 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. | Numdam | Zbl 0937.65116

[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. | Zbl 0808.35066

[9] K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. (2004) (submitted). | Numdam | MR 2195910 | Zbl 1086.65108

[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). | MR 1804748 | Zbl 0981.65095

[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. | Zbl 0982.65122

[12] R. Eymard, T. Gallouët and R. Herbin, A finite volume scheme for anisotropic diffusion problems. C.R. Acad. Sci. Paris 1 339 (2004) 299-302. | Zbl 1055.65124

[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). | Numdam | Zbl 0368.65053

[14] R. Glowinski and J. Rappaz, Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology. ESAIM: M2AN 37 (2003) 175-186. | Numdam | Zbl 1046.76002

[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. | Zbl 1060.76577

[16] J. Simon, Régularité de la solution d'un problème aux limites non linéaires. Ann. Fac. Sciences Toulouse 3 (1981) 247-274. | Numdam | Zbl 0487.35015