Adaptive finite element methods for control constrained distributed and boundary optimal control problems

This contribution is concerned with the development, analysis and implementation of Adaptive Finite Element Methods (AFEMs) for distributed and boundary control problems with control constraints. AFEMs consist of successive loops of the cycle ’SOLVE’, ’ESTIMATE’, ’MARK’, and ’REFINE’. Emphasis will be on the steps ’SOLVE’ and ’ESTIMATE’. In this context, ’SOLVE’ stands for the efficient solution of the finite element discretized problems and the following step ’ESTIMATE’ is devoted to a residual-type a posteriori error estimation of the global discretization errors in the state, the co-state, the control and the co-control. A bulk criterion is the core of the step ’MARK’ to indicate selected edges and elements for refinement. The final step ’REFINE’ deals with the technical realization of the refinement process itself. The efficient solution of the underlying constrained minimization problems is achieved by employing a primal-dual active set strategy. This method is equivalent to a class of semismooth Newton algorithms and converges locally at a superlinear rate. The a posteriori error analysis features convergence in the states, co-states, controls, and co-controls and addresses the issue of a guaranteed error reduction. Finally, numerical results for distributed as well as boundary control problems will be discussed. The test problems under consideration also include the case of lack of strict complementarity.