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Goal Oriented Mesh Adaptivity for Mixed Control-State Constrained Elliptic Optimal Control Problems

  • Michael HintermüllerEmail author
  • Ronald H. W. Hoppe
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 15)

Abstract

Adaptive finite element methods for the numerical solution of partial differential equations consist of successive cycles of the loop
$${\rm SOLVE}\ \Longrightarrow\ {\rm ESTIMATE}\ \Longrightarrow\ {\rm MARK}\ \Longrightarrow\ {\rm REFINE}.$$

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 [31]). Finally, the goal oriented weighted dual approach extracts information on the error via the dual problem (cf. [4, 12]).

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References

  1. 1.
    M. Ainsworth and J. T. Oden. A posteriori error estimation in finite element analysis. Wiley, Chichester, 2000.zbMATHGoogle Scholar
  2. 2.
    N. Arada and J.-P. Raymond. Optimal control problems with mixed control-state constraints. SIAM J. Control Optim., 39(5):1391–1407, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    I. Babuska and T. Strouboulis. The finite element method and its reliability. Clarendon Press, Oxford, 2001.Google Scholar
  4. 4.
    W. Bangerth and R. Rannacher. Adaptive finite element methods for differential equations. Lectures in Mathematics, ETH Zürich. Birkhäuser, Basel, 2003.Google Scholar
  5. 5.
    R. Becker, H. Kapp, and R. Rannacher. Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J. Control Optim., 39(1):113–132, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. Bergounioux, M. Haddou, M. Hintermüller, and K. Kunisch. A comparison of a Moreau–Yosida-based active set strategy and interior point methods for constrained optimal control problems. SIAM J. Optim., 11(2):495–521, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Bergounioux, K. Ito, and K. Kunisch. Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim., 37(4):1176–1194, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods. Springer, Berlin, 2nd edition, 2003.Google Scholar
  9. 9.
    E. Casas, J.-P. Raymond, and H. Zidani. Pontryagin’s principle for local solutions of control problems with mixed control-state constraints. SIAM J. Control Optim., 39(4):1182–1203, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    K. Deckelnick and M. Hinze. Convergence of a finite element approximation to a state constrained elliptic control problem. SIAM J. Numer. Anal., 45(5):1937–1953, 2007.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    W. Dörfler. A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal., 33(3):1106–1124, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    K. Eriksson, D. Estep, P. Hansbo, and C. Johnson. Computational differential equations. Cambridge University Press, Cambridge, 1995.Google Scholar
  13. 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. 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. 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
  16. 16.
    A. Günther and M. Hinze. A posteriori error control of a state constrained elliptic control problem. J. Numer. Math., 16(4):307–322, 2008.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    M. Hintermüller and R. H. W. Hoppe. Goal-oriented adaptivity in control constrained optimal control of partial differential equations. SIAM J. Control Optim., 47(4):1721–1743, 2008.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 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. 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. 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
  21. 21.
    M. Hintermüller and K. Kunisch. Feasible and noninterior path-following in constrained minimization with low multiplier regularity. SIAM J. Control Optim., 45(4):1198–1221, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    M. Hintermüller and K. Kunisch. Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim., 17(1):159–187, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    R. H. W. Hoppe, Y. Iliash, C. Iyyunni, and N. H. Sweilam. A posteriori error estimates for adaptive finite element discretizations of boundary control problems. J. Numer. Math., 14(1):57–82, 2006.zbMATHMathSciNetGoogle Scholar
  24. 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
  25. 25.
    K. Kunisch and A. Rösch. Primal–dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optim., 13(2):321–334, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    R. Li, W. Liu, H. Ma, and T. Tang. Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim., 41(5):1321–1349, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    W. Liu and N. Yan. A posteriori error estimates for distributed optimal control problems. Adv. Comput. Math., 15(1–4):285–309, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    C. Meyer, U. Prüfert, and F. Tröltzsch. On two numerical methods for state-constrained elliptic control problems. Optim. Methods Softw., 22(6):871–899, 2007.zbMATHMathSciNetGoogle Scholar
  29. 29.
    C. Meyer, A. Rösch, and F. Tröltzsch. Optimal control problems of PDEs with regularized pointwise state constraints. Comput. Optim. Appl., 33(2–3):209–228, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    P. Morin, R. H. Nochetto, and K. G. Siebert. Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal., 38(2):466–488, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    P. Neittaanmäki and S. Repin. Reliable methods for computer simulation. Error control and a posteriori estimates. Elsevier, Amsterdam, 2004.Google Scholar
  32. 32.
    A. Rösch and F. Tröltzsch. Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints. SIAM J. Optim., 17(3):776–794, 2006.zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    F. Tröltzsch. A minimum principle and a generalized bang-bang principle for a distributed optimal control problem with constraints on control and state. Z. Angew. Math. Mech., 59(12):737–739, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    F. Tröltzsch. Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Optim., 15(2):616–634, 2005.zbMATHCrossRefGoogle Scholar
  35. 35.
    R. Verfürth. A review of a posteriori estimation and adaptive mesh-refinement techniques. Wiley/Teubner, New York, 1996.zbMATHGoogle Scholar
  36. 36.
    B. Vexler and W. Wollner. Adaptive finite elements for elliptic optimization problems with control constraints. SIAM J. Control Optim., 47(1):509–534, 2008.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Institute of MathematicsHumboldt UniversityBerlinGermany
  2. 2.Department of Mathematics and Scientific ComputingUniversity of GrazGrazAustria
  3. 3.Department of MathematicsUniversity of HoustonHoustonUSA
  4. 4.Institute of MathematicsUniversity of AugsburgAugsburgGermany

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