Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problems

In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order $m=1$ Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results.