Superconvergence and $L^∞$-Error Estimates of the Lowest Order Mixed Methods for Distributed Optimal Control Problems Governed by Semilinear Elliptic Equations

Abstract

In this paper, we investigate the superconvergence property and the $L^∞$-error estimates of mixed finite element methods for a semilinear elliptic control problem. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive some superconvergence results for the control variable. Moreover, we derive $L^∞$-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.