A new discretization concept for optimal control problems with control constraints is introduced which utilizes for the discretization of the control variable the relation between adjoint state and control. Its key feature is not to discretize the space of admissible controls but to implicitly utilize the first order optimality conditions and the discretization of the state and adjoint equations for the discretization of the control. For discrete controls obtained in this way an optimal error estimate is proved. The application to control of elliptic equations is discussed. Finally it is shown that the new concept is numerically implementable with only slight increase in program management. A numerical test confirms the theoretical investigations.

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An adaptive finite element semi-smooth Newton solver for the Cahn-Hilliard model with double obstacle free energy is proposed. For this purpose, the governing system is discretised in time using a semi-implicit scheme, and the resulting time-discrete system is formulated as an optimal control problem with pointwise constraints on the control. For the numerical solution of the optimal control problem, we propose a function space based algorithm which combines a Moreau-Yosida regularization technique for handling the control constraints with a semi-smooth-Newton method for solving the optimality systems of the resulting sub-problems. Further, for the discretization in space and in connection with the proposed algorithm, an adaptive finite element method is considered. The performance of the overall algorithm is illustrated by numerical experiments.