Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems
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Mixed control-state constraints are used as a relaxation of originally state constrained optimal control problems for partial differential equations to avoid the intrinsic difficulties arising from measure-valued multipliers in the case of pure state constraints. In particular, numerical solution techniques known from the pure control constrained case such as active set strategies and interior-point methods can be used in an appropriately modified way. However, the residual-type a posteriori error estimators developed for the pure control constrained case can not be applied directly. It is the essence of this paper to show that instead one has to resort to that type of estimators known from the pure state constrained case. Up to data oscillations and consistency error terms, they provide efficient and reliable estimates for the discretization errors in the state, a regularized adjoint state, and the control. A documentation of numerical results is given to illustrate the performance of the estimators.
KeywordsDistributed optimal control problems Mixed control-state constraints Adaptive finite elements Posteriori error analysis
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- 3.Babuska, I., Strouboulis, T.: The Finite Element Method and Its Reliability. Clarendon Press, Oxford (2001) Google Scholar
- 10.Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Computational Differential Equations. Cambridge University Press, Cambridge (1995) Google Scholar
- 11.Gaevskaya, A., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: A posteriori error analysis of control constrained distributed and boundary control problems. In: Fitzgibbon, W., et al. (eds.) Proceedings of Conference on Advances in Scientific Computing, Moscow, Russia, pp. 85–108. Russian Academy of Sciences, Moscow (2006) Google Scholar
- 12.Gaevskaya, A., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: Convergence analysis of an adaptive finite element method for distributed control problems with control constraints. In: Leugering, G., et al. (eds.) Proceedings of Conference on Optimal Control for PDEs, Oberwolfach, Germany, pp. 47–68. Birkhäuser, Basel (2007) Google Scholar
- 13.Günther, A., Hinze, M.: A posteriori error control of a state constrained elliptic control problem. J. Numer. Math. 16 (2008, in press) Google Scholar
- 16.Hintermüller, M., Kunisch, K.: Stationary state constrained optimal control problems. Preprint IFB, Report No. 3, Karl-Franzens-University of Graz (2006) Google Scholar
- 18.Hoppe, R.H.W., Kieweg, M.: A posteriori error estimation of finite element approximations of pointwise state constrained distributed control problems. Preprint No. 16, Inst. of Math., Univ. of Augsburg (2007) Google Scholar
- 21.Liu, W., Yan, N.: A posteriori error estimates for convex boundary control problems. Preprint, Institute of Mathematics and Statistics, University of Kent, Canterbury (2003) Google Scholar
- 22.Meyer, C., Prüfert, U., Tröltzsch, F.: On two numerical methods for state-constrained elliptic control problems. Preprint, Department of Mathematics, Berlin University of Technology (2005) Google Scholar
- 25.Neittaanmäki, P., Repin, S.: Reliable Methods for Mathematical Modelling. Error Control and a Posteriori Estimates. Elsevier, New York (2004) Google Scholar