Reduced-order finite element approximation based on POD for the parabolic optimal control problem

In this paper, we construct a reduced-order finite element (ROFE) method holding seldom unknowns for the parabolic optimal control problem. We apply the proper orthogonal decomposition (POD) technique to develop two unsteady systems about state and co-state approximations, which efficiently reduces the number of unknowns and computational costs. Optimal a priori error estimates for the state, co-state and control approximations are derived. Finally, numerical examples are presented to verify that the ROFE method is accurate and efficient for solving the parabolic optimal control problem.