ON THE SUPRACONVERGENCE OF ELLIPTIC FINITE-DIFFERENCE SCHEMES

This paper deals with the supraconvergence of elliptic finite differen ce schemes on variable grids for second order elliptic boundary value problems subject to Dirichlet boundary conditions in two-dimensional d omains. The assumptions in this paper are less restrictive than those considered so far in the literature allowing also variable coefficient s, mixed derivatives and polygonal domains. The nonequidistant grids w e consider are more flexible than merely rectangular ones such that, e .g., local grid refinements are covered. The results also develop a cl ose relation between supraconvergent finite difference schemes and pie cewise linear finite element methods. It turns out that the finite dif ference equation is a certain nonstandard finite element scheme on tri angular grids combined with a special form of quadrature. In extension to what is known for the standard finite element scheme, here also th e gradient is shown to be convergent of second order, and so our resul t is also a superconvergence result for the underlying finite element method, (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.