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We consider a control-constrained parabolic optimal control problem and use variational discretization for its time semidiscretization. The state equation is treated with a Petrov--Galerkin scheme using a piecewise constant Ansatz for the state and piecewise linear continuous test functions. This results in variants of the Crank--Nicolson scheme for the state and the adjoint state. Exploiting a superconvergence result, we prove second order convergence in time of the error in the controls. Moreover, the piecewise linear and continuous parabolic projection of the discrete state on the dual time grid provides a second order convergent approximation of the optimal state without further numerical effort. Numerical experiments confirm our analytical findings.

Crank--Nicolson Time Stepping and Variational Discretization of Control-Constrained Parabolic Optimal Control Problems