Error estimates of variational discretization and fully discrete mixed finite element methods for semilinear parabolic optimal control problem

In this paper we study the variational discretization and fully discrete mixed finite element methods for optimal control problem governed by semilinear parabolic equations. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is not discreted. Then we derive a priori error estimates both for the coupled state and the control approximation. Finally, we present a numerical example which confirms our theoretical results.