Computational Optimization and Applications

, Volume 61, Issue 2, pp 373–408 | Cite as

Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations



This paper is concerned with the discretization error analysis of semilinear Neumann boundary control problems in polygonal domains with pointwise inequality constraints on the control. The approximations of the control are piecewise constant functions. The state and adjoint state are discretized by piecewise linear finite elements. In a postprocessing step approximations of locally optimal controls of the continuous optimal control problem are constructed by the projection of the respective discrete adjoint state. Although the quality of the approximations is in general affected by corner singularities a convergence order of \(h^2|\ln h|^{3/2}\) is proven for domains with interior angles smaller than \(2\pi /3\) using quasi-uniform meshes. For larger interior angles mesh grading techniques are used to get the same order of convergence.


Semilinear elliptic Neumann boundary control problem Finite element error estimates  Graded meshes Postprocessing  Superconvergence 

Mathematics Subject Classification

65N30 49K20 49M25 65N15 65N50 


  1. 1.
    Apel, Th, Pfefferer, J., Rösch, A.: Finite element error estimates for Neumann boundary control problems on graded meshes. Comput. Opt. Appl. 52(1), 3–28 (2012)CrossRefMATHGoogle Scholar
  2. 2.
    Apel, Th., Pfefferer, J., Rösch, A.: Finite element error estimates on the boundary with application to optimal control. Math. Comp. 84, 33–70 (2015)Google Scholar
  3. 3.
    Apel, Th., Pfefferer, J., Winkler M.: Local mesh refinement for the discretisation of Neumann boundary control problems on polyhedra (2014, submitted)Google Scholar
  4. 4.
    Apel, Th, Rösch, A., Winkler, G.: Optimal control in non-convex domains: a priori discretization error estimates. CALCOLO 44(3), 137–158 (2007)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Arada, N., Casas, E., Tröltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Opt. Approx. 23(2), 201–229 (2002)CrossRefMATHGoogle Scholar
  6. 6.
    Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Opt. 37(4), 1176–1194 (1999)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Bonnans, J.F., Zidani, H.: Optimal control problems with partially polyhedric constraints. SIAM J. Control Opt. 37(6), 1726–1741 (1999)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Opt. 31(4), 993–1006 (1993)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Casas, E., Mateos, M.: Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Opt. 40(5), 1431–1454 (2002)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Casas, E., Mateos, M.: Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21(1), 67–100 (2002)MATHMathSciNetGoogle Scholar
  11. 11.
    Casas, E., Mateos, M.: Error estimates for the numerical approximation of Neumann control problems. Comput. Opt. Appl. 39(3), 265–295 (2008)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Casas, E., Mateos, M., Tröltzsch, F.: Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Opt. Appl. 31(2), 193–219 (2005)CrossRefMATHGoogle Scholar
  13. 13.
    Casas, E., Tröltzsch, F.: Error estimates for linear-quadratic elliptic control problems. In: Barbu, V., et al. (eds.) Analysis and Optimization of Differential Systems, pp. 89–100. Kluwer Academic Publishers, Boston, MA (2003)CrossRefGoogle Scholar
  14. 14.
    Casas, E., Tröltzsch, F.: Second order analysis for optimal control problems: improving results expected from abstract theory. SIAM J. Opt. 22(1), 261–279 (2012)CrossRefMATHGoogle Scholar
  15. 15.
    Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Finite Element Methods, Handbook of Numerical Analysis, vol. II, pp. 17–352. Elsevier, North-Holland (1991)Google Scholar
  16. 16.
    Dauge, M.: Elliptic Boundary Value Problems on Corner Domains-Smoothness and Asymptotics of Solutions. Lecture Notes in Mathematics. Springer, Berlin (1988)Google Scholar
  17. 17.
    Falk, M.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44(1), 28–47 (1973)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Geveci, T.: On the approximation of the solution of an optimal control problem governed by an elliptic equation. R.A.I.R.O. Analyse numeriqué 13(4), 313–328 (1979)MATHMathSciNetGoogle Scholar
  19. 19.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)MATHGoogle Scholar
  20. 20.
    Heinkenschloss, M., Tröltzsch, F.: Analysis of the Lagrange-SQP-Newton method for the control of a phase field equation. Control Cybern. 28(2), 178–211 (1999)Google Scholar
  21. 21.
    Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Opt. Appl. 30(1), 45–61 (2005)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Hinze, M., Matthes, U.: A note on variational discretization of elliptic Neumann boundary control. Control Cybern. 38, 577–591 (2009)MATHMathSciNetGoogle Scholar
  23. 23.
    Jerison, D., Kenig, C.: The Neumann problem on Lipschitz domains. Bull. (New Series) Am. Math. Soc. 4(2), 203–207 (1981)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Kelley, C.T., Sachs, E.: Approximate quasi-Newton methods. Math. Program. 48(1–3), 41–70 (1990)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, Waltham (1980)MATHGoogle Scholar
  26. 26.
    Kunisch, K., Rösch, A.: Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Opt. 13(2), 321–334 (2002)CrossRefMATHGoogle Scholar
  27. 27.
    Kunisch, K., Sachs, E.: Reduced SQP-methods for parameter identification problems. SIAM J. Numer. Anal. 29(6), 1793–1820 (1992)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Malanowski, K.: Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems. Appl. Math. Opt. 8(1), 69–95 (1982)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Mateos, M., Rösch, A.: On saturation effects in the Neumann boundary control of elliptic optimal control problems. Comput. Opt. Appl. 49(2), 359–378 (2011)CrossRefMATHGoogle Scholar
  30. 30.
    Maz’ya, V., Rossmann, J.: Elliptic Equations in Polyhedral Domains. American Mathematical Society, Providence (2010)Google Scholar
  31. 31.
    Meyer, C., Rösch, A.: \(L^\infty \)-estimates for approximated optimal control problems. SIAM J. Control Opt. 44(5), 1636–1649 (2005)CrossRefGoogle Scholar
  32. 32.
    Meyer, C., Rösch, A.: Superconvergence properties of optimal control problems. SIAM J. Control Opt. 43(3), 970–985 (2005)CrossRefGoogle Scholar
  33. 33.
    Nečas, J.: Direct Methods in the Theory of Elliptic Equations. Springer, Berlin (2012)MATHGoogle Scholar
  34. 34.
    Rösch, A.: Error estimates for linear-quadratic control problems with control constraints. Opt. Methods Softw. 21(1), 121–134 (2006)CrossRefMATHGoogle Scholar
  35. 35.
    Rösch, A., Simon, R.: Superconvergence properties for optimal control problems discretized by piecewise linear and discontinuous functions. Numer. Funct. Anal. Opt. 28(3), 425–443 (2007)CrossRefMATHGoogle Scholar
  36. 36.
    Roßmann, J.: Gewichtete Sobolev–Slobodetskij-Räume und Anwendungen auf elliptische Randwertprobleme in Gebieten mit Kanten. Universität Rostock, Habilitationsschrift (1988)Google Scholar
  37. 37.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence (2010)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute for Applied Mathematics and Stochastics, Nonlinear Optimization and Inverse ProblemsBerlinGermany
  2. 2.Universität der Bundeswehr MünchenNeubibergGermany

Personalised recommendations