New a posteriori error estimates of mixed finite element methods for quadratic optimal control problems governed by semilinear parabolic equations with integral constraint

In this paper, we investigate new L∞(L2) and L2(L2)-posteriori error estimates of mixed finite element solutions for quadratic optimal control problems governed by semilinear parabolic equations. The state and the co-state are discretized by the order one Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L∞(J;L2(Ω))-norm and L2(J;L2(Ω))-norm for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the optimal control problem.