A priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems

In this paper, we discuss a priori error estimates and superconvergence of $P_0^2-P_1$ mixed finite element methods for elliptic boundary control problems. The state variables and co-state variables are approximated by a $P_0^2-P_1$ (velocity-pressure) pair and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates for the control variable, the state variables and the co-state variables. Then we obtain a superconvergence result for the control variable by using postprocessing projection operator.