On the finite-volume approximation of regular solutions of the $p$-Laplacian
Résumé
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 W1,p error estimates in the case of W2,p solutions of the homogeneous problem. Here we improve these estimates in the case of W4,1 solutions on uniform meshes for p > 3, using symmetry properties of the schemes. The proof also works for the Laplace equation, giving O(R2) 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 W1,p, W1,1 and L∞ convergence rates for special classes of regular solutions to the inhomogeneous problem—in particular, for solutions without critical points in Ω, for all p ∈ (1,∞). Numerical examples are given. They suggest the optimality of the L∞ estimates, of order h2, obtained for solutions without critical points.