Superconvergence and a posteriori error estimates of splitting positive definite mixed finite element methods for elliptic optimal control problems

In this paper, we investigate superconvergence and a posteriori error estimates of splitting positive definite mixed finite element methods for elliptic optimal control problems. The presented scheme is independent symmetric and positive definite for the state variables and the adjoint state variables. Moreover, the matching relation (i.e., LBB-condition) between the mixed element spaces Vh and Wh is not necessary, thus, we can choose the approximation spaces more flexibly. In order to derive the superconvergence, we will use classical mixed finite element spaces. 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. At first, we derive some superconvergence properties for the control variable, the state variables and the adjoint state variables. Then, using a recovery operator, we obtain a superconvergence result for the control variable. Next, combining energy approach with postprocessing method, we derive a posteriori error estimates for optimal control problems. We will show that the method does not involve the jump residuals which make the analysis simpler. Finally, a numerical example is given to demonstrate the theoretical results on superconvergence.