Browsing by Author "Kohls, Kristina"

Filter results by typing the first few letters
Now showing 1 - 1 of 1
  • Thumbnail Image
    ItemOpen Access
    An adaptive finite element method for control-constrained optimal control problems
    (2012) Kohls, Kristina; Siebert, Kunibert G. (Prof. Dr.)
    Many problems from physics like heat conduction and energy conservation lead to partial differential equations (PDEs). Only some of them can be solved directly; in general one has to rely on approximation techniques like the Finite Element Method (FEM). Adaptive Finite Elements intend to only increase accuracy in those parts of the domain where the error is large relative to the rest of the domain. The gain in accuracy that can be achieved by this, in comparison to the classical FEM, depends on the exact solution itself. In this thesis the weak formulation of a PDE constitutes the side-constraint of an optimizing problem. Usually this consists of a convex functional that is minimized with respect to two variables - control and state - which are connected via the side-constraint. Additionally the control has to satisfy further constraints. To be able to apply Adaptive Finite Elements one needs to construct error- estimators that satisfy certain properties. In contrast to previous results in this field, this thesis uses a general approach to find error-estimators. This approach includes distributed and boundary-control as well as the cases of discretized and non-discretized control. The particularities of the involved PDE are only of interest when choosing the appropriate estimators for the linear subproblems from the toolbox. The other main contribution of this thesis consists of three convergence results. One for non-discretized control, one for discontinuous and one for continuous control-discretizations. We not only prove the convergence of the solution but also of the estimator which implies that the algorithm terminates for any given tolerance TOL > 0. Finally, a few numerical examples with boundary-control are investigated for varying marking strategies and estimators.