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Adaptive Finite Elements for Optimally Controlled Elliptic Variational Inequalities of Obstacle Type

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Optimization with PDE Constraints

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 101))

Abstract

We are concerned with the numerical solution of distributed optimal control problems for second order elliptic variational inequalities by adaptive finite element methods. Both the continuous problem as well as its finite element approximations represent subclasses of Mathematical Programs with Equilibrium Constraints (MPECs) for which the optimality conditions are stated by means of stationarity concepts in function space (Hintermüller and Kopacka, SIAM J. Optim. 20:868–902, 2009) and in a discrete, finite dimensional setting (Scheel and Scholtes, Math. Oper. Res. 25:1–22, 2000) such as (\(\varepsilon\)-almost, almost) C- and S-stationarity. With regard to adaptive mesh refinement, in contrast to the work in (Hintermüller, ESAIM Control Optim. Calc. Var., 2012, submitted) which adopts a goal oriented dual weighted approach, we consider standard residual-type a posteriori error estimators. The first main result states that for a sequence of discrete C-stationary points there exists a subsequence converging to an almost C-stationary point, provided the associated sequence of nested finite element spaces is limit dense in its continuous counterpart. As the second main result, we prove the reliability and efficiency of the residual-type a posteriori error estimators. Particular emphasis is put on the approximation of the reliability and efficiency related consistency errors by heuristically motivated computable quantities and on the approximation of the continuous active, strongly active, and inactive sets by their discrete counterparts. A detailed documentation of numerical results for two representative test examples illustrates the performance of the adaptive approach.

Mathematics Subject Classification (2000). Primary 65K15; Secondary 49M99; 65K10; 90C56.

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Acknowledgements

A.G. has been partially supported by a grant from the European Science Foundation within the Networking Programme ‘OPTPDE’. M.H. acknowledges support by the German Research Fund (DFG) through the Research Center MATHEON Project C28 and C31 and the SPP 1253 “Optimization with Partial Differential Equations”, and the Austrian Science Fund (FWF) through the START Project Y 305-N18 Interfaces and Free Boundaries and the SFB Project F32 04-N18 “Mathematical Optimization and Its Applications in Biomedical Sciences”. R.H.W.H. has been supported by the DFG Priority Programs SPP 1253 and SPP 1506, by the NSF grants DMS-0914788, DMS-1115658, by the Federal Ministry for Education and Research (BMBF) within the projects ‘FROPT’ and ‘MeFreSim’, and by the European Science Foundation within the Networking Programme ‘OPTPDE’.

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Augsburg, Universitätsstr. 14, D-86159, Augsburg, Germany

    A. Gaevskaya & R. H. W. Hoppe

  2. Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099, Berlin, Germany

    M. Hintermüller & C. Löbhard

  3. Department of Mathematics, University of Houston, 659 P.G. Hoffman, Houston, TX, 77204-3008, USA

    R. H. W. Hoppe

Authors
  1. A. Gaevskaya
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  2. M. Hintermüller
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  3. R. H. W. Hoppe
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  4. C. Löbhard
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Correspondence to R. H. W. Hoppe .

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Editors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, Augsburg, Germany

    Ronald Hoppe

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© 2014 Springer International Publishing Switzerland

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Gaevskaya, A., Hintermüller, M., Hoppe, R.H.W., Löbhard, C. (2014). Adaptive Finite Elements for Optimally Controlled Elliptic Variational Inequalities of Obstacle Type. In: Hoppe, R. (eds) Optimization with PDE Constraints. Lecture Notes in Computational Science and Engineering, vol 101. Springer, Cham. https://doi.org/10.1007/978-3-319-08025-3_4

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