A priori and a posteriori error estimates of H1H1-Galerkin mixed finite element methods for elliptic optimal control problems

In this paper, we investigate numerical approximations of H1H1-Galerkin 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 WhWh is not necessary, thus, we can choose the approximation spaces more flexibly. The state and co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the standard finite element spaces, the control variable is approximated by piecewise constant functions. We derive a priori and a posteriori error estimates for the control variable, the state variables and the adjoint state variables. Finally, some numerical examples are given to demonstrate the theoretical results about a priori error estimates.