Superconvergence of quadratic optimal control problems by triangular mixed finite element methods

The aim of this work is to investigate the discretization of a quadratic convex optimal control problem using the mixed finite element method. The state and co-state are approximated by the order k <= 1 Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise constant functions. We construct an interpolation of the exact control and a projection of the discrete scalar co-state to be the approximated Solution of the control variable for the continuous optimal control problem. As a result, it can be proved that the difference between the interpolation and the piecewise constant approximation has superconvergence property for the control of order h(3/2) for k = 0 and of order h(2) for k = 1. Moreover, only for the order k = 1 Raviart-Thornas mixed finite element approximation does the postprocessing C, technique possess the superconvergence property of order h(2). Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence. Copyright (C) 2008 John Wiley & Sons, Ltd.