On the finite-volume approximation of regular solutions of the p-Laplacian

We consider finite-volume schemes on rectangular meshes for the p-Laplacian with Dirichlet boundary conditions. In Andreianov et al. (2004a, Math. Model. Numer. Anal., 38, 931-959), we constructed a family of schemes and proved discrete W-1,W- p error estimates in the case of W-2,W- p solutions of the homogeneous problem. Here we improve these estimates in the case of W-4,W- 1 solutions on uniform meshes for p > 3, using symmetry properties of the schemes. The proof also works for the Laplace equation, giving O(R-2) convergence for a family of nine-point finite-volume schemes. With the same ideas, using the improved coercivity inequalities of Barrett and Liu, we obtain even better W-1,W- p, W-1,W- 1 and L-infinity convergence rates for special classes of regular solutions to the inhomogeneous problem-in particular, for solutions without critical points in (Omega) over bar, for all p is an element of (1, infinity). Numerical examples are given. They suggest the optimality of the L-infinity estimates, of order h(2), obtained for solutions without critical points.