Superconvergence Analysis of Finite Element Methods for Optimal Control Problems of the Stationary Bénard Type

Abstract

In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Bénard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in $L^\infty$-norm and optimal error estimates in $L^2$-norm.