TY - JOUR T1 - Superconvergence and $L^∞$-Error Estimates of the Lowest Order Mixed Methods for Distributed Optimal Control Problems Governed by Semilinear Elliptic Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 479 EP - 498 PY - 2013 DA - 2013/06 SN - 6 DO - http://doi.org/10.4208/nmtma.2013.1133nm UR - https://global-sci.org/intro/article_detail/nmtma/5914.html KW - Semilinear elliptic equations, distributed optimal control problems, superconvergence, $L^∞$-error estimates, mixed finite element methods. AB -

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.