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(hs)-convergence of the finite difference solution, and its gradient is shown if the exact solution is in the Sobolev space H1+s(O). In the case of an equation with mixed derivatives in a domain containing oblique boundary sections, the convergence order is reduced to O(h3/2-e) with e > 0 if u? H3(O). 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? , the given error estimates are strictly local.