%0 Generic
%A Wollner, Winnifried
%D 2010
%F heidok:10943
%K Optimal Control , Adaptive Finite Elements , Pointwise Inequality Constraints
%R 10.11588/heidok.00010943
%T Adaptive Methods for PDE-based Optimal Control with Pointwise Inequality Constraints
%U https://archiv.ub.uni-heidelberg.de/volltextserver/10943/
%X This work is devoted to the development of efficient numerical methods for a certain class of PDE-based optimization problems. The optimization is constraint by an elliptic PDE. In addition to prior work in this context pointwise inequality constraints on the control and state variable are considered. These problems are infinite dimensional and their solution can in general not be obtained exactly. Instead the solution of such problems means to find an approximate solution. This is done by (approximately) solving for some set of first order necessary optimality conditions. Hence an efficient algorithm has to find such an approximate solution with as little effort as possible while still being accurate enough for whatever the goal of the computation is. The work at hand contributes to this goal by deriving a posteriori error estimates with respect to a given functional. These estimates are required for two purposes, first, to generate efficient meshes for the solution of the PDEs required in the process of solving the necessary conditions. Second, to choose several parameters that occur in order to regularize the problems at hand in such a way that the regularization error is both small enough, to obtain a good result', and yet large enough to have easy to solve' problems. These a posteriori estimators are supplemented with a priori estimates in several cases where non have been available in the literature for the problem class under consideration. Finally, all theory and all heuristics will be substantiated with several numerical examples of different complexity.