Error estimates and superconvergence of a mixed finite element method for elliptic optimal control problems

Abstract

In this paper, we investigate error estimates and superconvergence of a mixed finite element method for elliptic optimal control problems. The gradient for our method belongs to the square integrable space instead of the classical H(div;Ω) space. The state and co-state are approximated by the P02–P1 (velocity–pressure) pair and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates in H1-norm for the state and the co-state scalar functions, a priori error estimates in (L2)2-norm for the state and the co-state vector functions and a priori error estimates in L2-norm for the control function. Then, using postprocessing projection operator, we derive a superconvergence result for the control variable. Next, we get a priori error estimates in L2-norm for the state and the co-state scalar functions. Finally, a numerical example is given to demonstrate the theoretical results.