New a posteriori $L^{infty }(L^2) $ and $L^2(L^2)$-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

摘要

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in $L^{infty }(J;L^2(Omega )) $-norm and $L^2(J;L^2(Omega ))$-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.