On the finite-volume approximation of regular solutions of the p-Laplacian
Boris Andreïanov, Franck Boyer, Florence Hubert · IMA Journal of Numerical Analysis · 2005
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 Ω̄, 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.