Finite volume schemes for the p-laplacian on cartesian meshes

Boris Andreianov (1) , Franck Boyer (2) , Florence Hubert (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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.

Mots clés: finite volume methods; p-laplacian; error estimates

Domaines: Equations aux dérivées partielles [math.AP]

Boris Andreianov, Franck Boyer, Florence Hubert. Finite volume schemes for the p-laplacian on cartesian meshes. ESAIM: Mathematical Modelling and Numerical Analysis, 2004, 38 no 6, pp 931-959. ⟨10.1051/m2an:2004045⟩. ⟨hal-00004088⟩