Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces

• Published in: Journal of computational methods in applied mathematics Vol. 6; no. 2; pp. 154 - 177
• Main Authors: Emmrich, E., Grigorieff, R.D.
• Format: Journal Article
• Language: English
• Published: De Gruyter 2006
• Subjects: elliptic finite difference scheme; finite differences; fully discrete linear FEM; nonuniform grid; stability; supercloseness of gradient; supraconvergence
• ISSN: 1609-4840; 1609-9389
• DOI: 10.2478/cmam-2006-0008

In this paper, we study the convergence of the finite difference discretization of a second order elliptic equation with variable coefficients subject to general boundary conditions. We prove that the scheme exhibits the phenomenon of supraconvergence on nonuniform grids, i.e., although the truncation error is in general of the first order alone, one has second order convergence. All error estimates are strictly local. Another result of the paper is a close relationship between finite difference scheme and linear finite element methods combined with a special kind of quadrature. As a consequence, the results of the paper can be viewed as the introduction of a fully discrete finite element method for which the gradient is superclose. A numerical example is given.