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 . Numerical results are given in order to compare different admissible and non-admissible schemes.
Mots-clés : 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: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {931--959},
publisher = {EDP-Sciences},
volume = {38},
number = {6},
year = {2004},
doi = {10.1051/m2an:2004045},
mrnumber = {2108939},
zbl = {1081.65105},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2004045/}
}
TY - JOUR AU - Andreianov, Boris AU - Boyer, Franck AU - Hubert, Florence TI - Finite volume schemes for the $p$-laplacian on cartesian meshes JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2004 SP - 931 EP - 959 VL - 38 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2004045/ DO - 10.1051/m2an:2004045 LA - en ID - M2AN_2004__38_6_931_0 ER -
%0 Journal Article %A Andreianov, Boris %A Boyer, Franck %A Hubert, Florence %T Finite volume schemes for the $p$-laplacian on cartesian meshes %J ESAIM: Modélisation mathématique et analyse numérique %D 2004 %P 931-959 %V 38 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2004045/ %R 10.1051/m2an:2004045 %G en %F M2AN_2004__38_6_931_0
Andreianov, Boris; Boyer, Franck; Hubert, Florence. Finite volume schemes for the $p$-laplacian on cartesian meshes. ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 6, pp. 931-959. doi : 10.1051/m2an:2004045. http://www.numdam.org/articles/10.1051/m2an:2004045/
[1] , and, A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81 (1999) 497-520. | Zbl
[2] , and, Finite volume schemes for the p-Laplacian. Further error estimates. Preprint No. 03-29, LATP Université de Provence (2003).
[3] , and, Convergence of finite volume approximations for a nonlinear elliptic-parabolic problem: A “continuous” approach. SIAM J. Numer. Anal. 42 (2004) 228-251. | Zbl
[4] and, A remark on the regularity of the solutions of the -Laplacian and its application to the finite element approximation, J. Math. Anal. Appl. 178 (1993) 470-487. | Zbl
[5] and, Finite element approximation of the -Laplacian. Math. Comp. 61 (1993) 523-537. | Zbl
[6] , Finite element error estimates for non-linear elliptic equations of monotone type. Numer. Math. 54 (1989) 373-393. | Zbl
[7] , and, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem. ESAIM: M2AN 33 (1999) 493-516. | Numdam | Zbl
[8] and, On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 25 (1994) 1085-1111. | Zbl
[9] and, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. (2004) (submitted). | Numdam | MR | Zbl
[10] , and, Finite Volume Methods, Handbook Numer. Anal., P.G. Ciarlet and J.L. Lions Eds., North-Holland VII (2000). | MR | Zbl
[11] , and, Finite volume approximation of elliptic problems and convergence of an approximate gradient. Appl. Numer. Math. 37 (2001) 31-53. | Zbl
[12] , and, A finite volume scheme for anisotropic diffusion problems. C.R. Acad. Sci. Paris 1 339 (2004) 299-302. | Zbl
[13] and, 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
[14] and, Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology. ESAIM: M2AN 37 (2003) 175-186. | Numdam | Zbl
[15] ,,, and, Numerical simulation of the motion of a two dimensional glacier. Int. J. Numer. Methods Eng. 60 (2004) 995-1009. | Zbl
[16] , Régularité de la solution d'un problème aux limites non linéaires. Ann. Fac. Sciences Toulouse 3 (1981) 247-274. | Numdam | Zbl
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