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This work addresses the problem of a priori error analysis for the finite volume approxi- mation of solutions of the p-laplacian on cartesian grids. We first recall the way we constructed such schemes and the different error bounds we proved in our previous works. Then we concentrate par- ticularly on the case where the exact solution has only weak regularity properties (which are natural for this problem) of Besov kind with derivation index in between 1 and 2. In this framework, the usual techniques to obtain error estimates for finite volumes schemes are not straightfoward to apply. Hence, we propose to take advantage of the variational structure of the equation and the schemes in order to obtain the error bounds. In the case of uniform meshes, this strategy was succesfully applied in (2) to obtain optimal estimates. We propose in this work to extend this approach to a more general framework of non-uniform meshes called "smoothly refined" meshes. Resume. Ce travail concerne l'analyse d'erreur a priori pour l'approximation par une methode de volumes finis des solutions du p-laplacien sur un maillage cartesien. Nous commencons par rappeller la construction de tels schemas et les differents resultats obtenus dans nos travaux anterieurs. On s'interesse ensuitel'analyse d'erreur dans le cas ou la solution exacte possede seulement des proprietes de regularite assez faibles (naturelles pour le problemee) de type Besov d'indice de derivation strictement compris entre 1 et 2. Dans ce cadre, les methodes usuelles d'estimation d'erreur sont difficilement applicables. On propose donc de profiter de la structure variationnelle de l'´equation et des schemas pour obtenir les bornes souhaitees. Sous la restriction de l'uniformite des maillages, cette approche ae utilisee avec succes dans (2) pour obtenir des estimations optimales. Nous presentons, dans ce travail, l'extension de la methodeun cadre plus geral de maillages dits "regulierement raffines".

Discrete Besov framework for finite volume approximation of the p-laplacian on non-uniform cartesian grids