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 aThis 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.show moreshow less

Download full text files

Export metadata

Statistics

Number of document requests

Additional Services

Share in Twitter Search Google Scholar
Metadaten
Author:Michael HintermüllerGND, Ronald H. W. HoppeGND
URN:urn:nbn:de:bvb:384-opus4-5140
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/652
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2008-17)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Contributing Corporation:University of Sussex, University of Houston
Release Date:2008/04/15
Tag:optimal control; adaptive finite element methods; distributed control; boundary control
GND-Keyword:Optimale Kontrolle; Finite-Elemente-Methode; A-posteriori-Abschätzung; Fehleranalyse
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Numerische Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik