Superconvergence and a posteriori error estimation for triangular mixed finite elements

Abstract

In this paper,we prove superconvergence results for the vector variable when lowest order triangular mixed finite elements of Raviart-Thomas type [17] on uniform triangulations are used, i.e., that the $H(\mbox{div;} \Omega)$ -distance between the approximate solution and a suitable projection of the real solution is of higher order than the $H(\mbox{div;} \Omega)$ -error. We prove results for both Dirichlet and Neumann boundary conditions. Recently, Duran [9] proved similar results for rectangular mixed finite elements, and superconvergence along the Gauss-lines for rectangular mixed finite elements was considered by Douglas, Ewing, Lazarov and Wang in [11], [8] and [18]. The triangular case however needs some extra effort. Using the superconvergence results, a simple postprocessing of the approximate solution will give an asymptotically exact a posteriori error estimator for the $L^2(\Omega)$ -error in the approximation of the vector variable.