Supraconvergent cell-centered scheme for two dimensional elliptic problems

In this paper we study the convergence properties of a cell-centered finite difference scheme for second order elliptic equations with variable coefficients subject to Dirichlet boundary conditions. We prove that the finite difference scheme on nonuniform meshes although not even being consistent are nevertheless second order convergent. More precisely, second order convergence with respect to a discrete version of L2(Ω)-norm is shown provided that the exact solution is in H4(Ω). Estimates for the difference between the pointwise restriction of the exact solution on the discretization nodes and the finite difference solution are proved. The convergence is studied with the aid of an appropriate negative norm. A numerical example support the convergence result.