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Numerische Mathematik

, Volume 138, Issue 2, pp 273–299 | Cite as

A priori \(L^2\)-discretization error estimates for the state in elliptic optimization problems with pointwise inequality state constraints

  • I. Neitzel
  • W. Wollner
Article

Abstract

In this paper, an elliptic optimization problem with pointwise inequality constraints on the state is considered. The main contributions of this paper are a priori \(L^2\)-error estimates for the discretization error in the optimal states. Due to the non separability of the space for the Lagrange multipliers for the inequality constraints, the problem is tackled by separation of the discretization error into two components. First, the state constraints are discretized. Second, with discretized inequality constraints, a duality argument for the error due to the discretization of the PDE is employed. For the second stage an a priori error estimate is derived with constants depending on the regularity of the dual problem. Finally, we discuss two cases in which these constants can be bounded in a favorable way; leading to higher order estimates than those induced by the known \(L^2\)-error in the control variable. More precisely, we consider a given fixed number of pointwise inequality constraints and a case of infinitely many but only weakly active constraints.

Mathematics Subject Classification

65K10 65N15 65N30 

Notes

Acknowledgements

The authors would like to thank Gerd Wachsmuth for pointing out an issue in a prior version of this manuscript. Moreover I. Neitzel and W. Wollner are grateful for the support of their former host institutions the Technische Universität München and the Universität Hamburg, respectively.

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics, vol. 140, 2nd edn. Elsevier, Amsterdam (2003)Google Scholar
  2. 2.
    Bangerth, W., Davydov, D., Heister, T., Heltai, L., Kanschat, G., Kronbichler, M., Maier, M., Turcksin, B., Wells, D.: The deal.II library, version 8.4. J. Numer. Math. 24, 135–141 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bangerth, W., Hartmann, R., Kanschat, G.: deal.II - a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33(4), 24/1–24/27 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benedix, O., Vexler, B.: A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints. Comput. Optim. Appl. 44(1), 3–25 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  6. 6.
    Casas, E.: Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24(6), 1309–1318 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Casas, E.: Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints. ESAIM Control Optim. Calc. Var. 8, 345–374 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Casas, E., Mateos, M.: Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21(1), 67–100 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Casas, E., Mateos, M.: Numerical approximation of elliptic control problems with finitely many pointwise constraints. Comput. Optim. Appl. 51(3), 1319–1343 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Casas, E., Mateos, M., Vexler, B.: New regularity results and improved error estimates for optimal control problems with state constraints. ESAIM Control Optim. Calc. Var. 20(3), 803–822 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chang, L., Gong, W., Yan, N.: Numerical analysis for the approximation of optimal control problems with pointwise observations. Math. Methods Appl. Sci. 38, 4502–4520 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cherednichenko, S., Krumbiegel, K., Rösch, A.: Error estimates for the Lavrentiev regularization of elliptic optimal control problems. Inverse Prob. 24(5), 055,003,21 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Deckelnick, K., Hinze, M.: Convergence of a finite element approximation to a state-constrained elliptic control problem. SIAM J. Numer. Anal. 45(5), 1937–1953 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Goll, C., Wick, T., Wollner, W.: DOpElib: differential equations and optimization environment. http://www.dopelib.net
  15. 15.
    Goll, C., Wick, T., Wollner, W.: DOpElib: differential equations and optimization environment; a goal oriented software library for solving PDEs and optimization problems with PDEs (2014). Preprint at http://www.dopelib.net/preprint_2014.pdf
  16. 16.
    Herzog, R., Rösch, A., Ulbrich, S., Wollner, W.: OPTPDE: a collection of problems in PDE-constrained optimization. In: Leugering, G., Benner, P., Engell, S., Griewank, A., Harbrecht, H., Hinze, M., Rannacher, R., Ulbrich, S. (eds.) Trends in PDE Constrained Optimization, International Series of Numerical Mathematics, vol. 165, pp. 539–543. Springer, Berlin (2014)Google Scholar
  17. 17.
    Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30(1), 45–61 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications., 1st edn. Springer, Berlin (2009)zbMATHGoogle Scholar
  19. 19.
    Hinze, M., Tröltzsch, F.: Discrete concepts versus error analysis in PDE constrained optimization. GAMM-Mitt. 33(2), 148–162 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Leykekhman, D., Meidner, D., Vexler, B.: Optimal error estimates for finite element discretization of elliptic optimal control problems with finitely many pointwise state constraints. Comput. Optim. Appl. 55, 769–802 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Merino, P., Neitzel, I., Tröltzsch, F.: Error estimates for the finite element discretization of semi-infinite elliptic optimal control problems. Discuss. Math. Differ. Incl. Control Optim. 30(2), 221–236 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Merino, P., Neitzel, I., Tröltzsch, F.: On linear-quadratic elliptic control problems of semi-infinite type. Appl. Anal. 90(6), 1047–1074 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Merino, P., Tröltzsch, F., Vexler, B.: Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space. M2AN. M2AN Math. Model. Numer. Anal. 44(1), 167–188 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Meyer, C.: Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybern. 37, 51–85 (2008)MathSciNetzbMATHGoogle Scholar
  25. 25.
    OPTPDE—a collection of problems in PDE-constrained optimization. http://www.optpde.net
  26. 26.
    Rannacher, R.: Zur \(L^{\infty }\)-Konvergenz linearer finiter Elemente beim Dirichlet-Problem. Math. Z. 149(1), 69–77 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schatz, A.H., Wahlbin, L.B.: Interior maximum-norm estimates for finite element methods. Math. Comput. 31(138), 414–442 (1977)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Institut für Numerische SimulationRheinische Friedrich-Wilhelms-Universität BonnBonnGermany
  2. 2.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

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