Analysis for the space-time a posteriori error estimates for mixed finite element solutions of parabolic optimal control problems

This paper investigates the space-time residual-based a posteriori error bounds of the mixed finite element method for the optimal control problem governed by the parabolic equation in a bounded convex domain. For the spatial discretization of the state and co-state variables, the lowest-order Raviart-Thomas spaces are utilized, although for the control variable, variational discretization technique is used. The backward-Euler implicit method is applied for temporal discretization. To provide a posteriori error estimates for the state and control variables in the L-infinity(L-2)-norm, an elliptic reconstruction approach paired with an energy strategy is utilized. The reliability and efficiency of the a posteriori error estimators are discussed. The effectiveness of the estimators is finally confirmed through the numerical tests.