Goal Oriented Mesh Adaptivity for Mixed Control-State Constrained Elliptic Optimal Control Problems
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Here, SOLVE stands for the finite element solution of the problem with respect to a given triangulation of the computational domain. The following step ESTIMATE is devoted to the estimation of the global discretization error in some appropriate norm or a user specified quantity of interest by a cheaply computable a posteriori error estimator. The estimator is assumed to consist of local contributions whose actual magnitude is then used in the step MARK to specify elements of the triangulation for refinement. The final step REFINE deals with the generation of a new triangulation based on the refinement of the elements selected in the previous step according to specific refinement rules. Adaptive finite elements are by now well established. There are various approaches such as residual-type a posteriori error estimators which rely on the proper evaluation of the residuals with respect to a computed approximation in the norm of the dual space and hierarchical type estimators where the equation satisfied by the error is suitably localized along with a solution of the local problems by higher order finite elements (cf., e.g. [1, 3, 35]). Averaging-type estimators typically use some sort of gradient recovery on element-related patches (cf., e.g. [1, 35]), whereas the theory of guaranteed error majorants provides reliable upper bounds for the error (see ). Finally, the goal oriented weighted dual approach extracts information on the error via the dual problem (cf. [4, 12]).
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- 3.I. Babuska and T. Strouboulis. The finite element method and its reliability. Clarendon Press, Oxford, 2001.Google Scholar
- 4.W. Bangerth and R. Rannacher. Adaptive finite element methods for differential equations. Lectures in Mathematics, ETH Zürich. Birkhäuser, Basel, 2003.Google Scholar
- 8.S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods. Springer, Berlin, 2nd edition, 2003.Google Scholar
- 12.K. Eriksson, D. Estep, P. Hansbo, and C. Johnson. Computational differential equations. Cambridge University Press, Cambridge, 1995.Google Scholar
- 13.A. Gaevskaya, R. H. W. Hoppe, Y. Iliash, and M. Kieweg. A posteriori error analysis of control constrained distributed and boundary control problems. In O. Pironneau et al., editor, Proc. Conf. Advances in Scientific Computing (Moscow), pages 85–108, Moscow, 2006. Russian Academy of Sciences.Google Scholar
- 14.A. Gaevskaya, R. H. W. Hoppe, Y. Iliash, and M. Kieweg. Convergence analysis of an adaptive finite element method for distributed control problems with control constraints. In K. Kunisch, G. Leugering, J. Sprekels, and F. Tröltzsch, editors, Control of Coupled Partial Differential Equations (Oberwolfach, 2005), pages 47–68, Basel, 2007. Birkhäuser.CrossRefGoogle Scholar
- 15.A. Gaevskaya, R. H. W. Hoppe, and S. Repin. A posteriori estimates for cost functionals of optimal control problems. In A. Bermúdez de Castro, D. Gómez, P. Quintela, and P. Salgado, editors, Numerical Mathematics and Advanced Applications (ENUMATH 2005, Santiago de Compostela), pages 308–316, Berlin, 2006. Springer.CrossRefGoogle Scholar
- 18.M. Hintermüller and R. H. W. Hoppe. Adaptive finite element methods for control constrained distributed and boundary optimal control problems. In M. Heinkenschloss, L. N. Vicente, and L. M. Fernandes, editors, Numerical PDE Constrained Optimization, volume 73 of Lecture Notes in Computational Science and Engineering. Springer, Berlin, 2009. In press.Google Scholar
- 19.M. Hintermüller and R. H. W. Hoppe. Goal-oriented adaptivity in state constrained optimal control of partial differential equations. SIAM J. Control Optim., 2009. Submitted.Google Scholar
- 20.M. Hintermüller, R. H. W. Hoppe, Y. Iliash, and M. Kieweg. An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM Control Optim. Calc. Var., 14(3):540–560, 2008. DOI 10.1051/cocv:2007057.zbMATHCrossRefMathSciNetGoogle Scholar
- 24.R. H. W. Hoppe and M. Kieweg. A posteriori error estimation of finite element approximations of pointwise state constrained distributed control problems. Submitted, 2007.Google Scholar
- 31.P. Neittaanmäki and S. Repin. Reliable methods for computer simulation. Error control and a posteriori estimates. Elsevier, Amsterdam, 2004.Google Scholar