Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids

Abstract

In this paper, we study the convergence of a finite difference scheme on nonuniform grids for the solution of second-order elliptic equations with mixed derivatives and variable coefficients in polygonal domains subjected to Dirichlet boundary conditions. We show that the scheme is equivalent to a fully discrete linear finite element approximation with quadrature. It exhibits the phenomenon of supraconvergence, more precisely, for s ∈ [1,2] order O(h s )-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H 1+s (Ω). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h 3/2−ε) with ε > 0 if u ∈ H 3(Ω). The second-order accuracy of the finite difference gradient is in the finite element context nothing else than the supercloseness of the gradient. For s ∈ {1,2}, the given error estimates are strictly local.