A Priori Mesh Grading for Distributed Optimal Control Problems

  • Thomas ApelEmail author
  • Dieter Sirch
Part of the International Series of Numerical Mathematics book series (ISNM, volume 160)


This paper deals with L2-error estimates for finite element approximations of control constrained distributed optimal control problems governed by linear partial differential equations. First, general assumptions are stated that allow to prove second-order convergence in control, state and adjoint state. Afterwards these assumptions are verified for problems where the solution of the state equation has singularities due to corners or edges in the domain or nonsmooth coefficients in the equation. In order to avoid a reduced convergence order, graded finite element meshes are used.


Linear-quadratic optimal control problems control constraints finite element method error estimates graded meshes anisotropic elements. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Th. Apel and A. Rösch and G. Winkler, Optimal control in non-convex domains: a priori discretization error estimates. Calcolo, 44 (2007), 137–158.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Th. Apel and D. Sirch, 𝐿 2 -error estimates for the Dirichlet and Neumann problem on anisotropic finite element meshes. Appl. Math. 56 (2011), 177–206.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Th. Apel, D. Sirch and G. Winkler, Error estimates for control constrained optimal control problems: Discretization with anisotropic finite element meshes. DFGPriority Program 1253, 2008, preprint, SPP1253-02-06, Erlangen.Google Scholar
  4. 4.
    Th. Apel and G. Winkler, Optimal control under reduced regularity. Appl. Numer. Math., 59 (2009), 2050–2064.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, 1978.Google Scholar
  6. 6.
    M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case. Comput. Optim. Appl., 30 (2005), 45–61.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints. Springer, 2008.Google Scholar
  8. 8.
    M. Hinze and A. Rösch, Discretization of optimal control problems. International Series of Numerical Mathematics, Vol. 160 (this volume), Springer, Basel, 2011, 391–430.Google Scholar
  9. 9.
    A. Kufner and A.-M. Sändig, Some Applications of Weighted Sobolev Spaces. Teubner, Leipzig, 1987.zbMATHGoogle Scholar
  10. 10.
    J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer, 1971. Translation of the French edition “Contrôle optimal de systèmes gouvernés par de équations aux dérivées partielles”, Dunod and Gauthier-Villars, 1968.Google Scholar
  11. 11.
    K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems. Appl. Math. Optim., 8 (1981), 69–95.CrossRefMathSciNetGoogle Scholar
  12. 12.
    V.G. Maz’ya and B.A. Plamenevsky, The first boundary value problem for classical equations of mathematical physics in domains with piecewise smooth boundaries, part I, II. Z. Anal. Anwend., 2 (1983), 335–359, 523–551. In Russian.zbMATHGoogle Scholar
  13. 13.
    V.G. Maz’ya and J. Rossmann, Schauder estimates for solutions to boundary value problems for second-order elliptic systems in polyhedral domains. Applicable Analysis, 83 (2004), 271–308.CrossRefMathSciNetGoogle Scholar
  14. 14.
    C. Meyer and A. Rösch, Superconvergence properties of optimal control problems. SIAM J. Control Optim., 43 (2004), 970–985.CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    S. Nicaise, Polygonal Interface Problems. Peter Lang GmbH, Europäischer Verlag der Wissenschaften, volume 39 of Methoden und Verfahren der mathematischen Physik, Frankfurt/M., 1993.zbMATHGoogle Scholar
  16. 16.
    S. Nicaise and D. Sirch, Optimal control of the Stokes equations: Conforming and non-conforming finite element methods under reduced regularity. Comput. Optim. Appl. 49 (2011), 567–600.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    D. Sirch, Finite element error analysis for PDE-constrained optimal control problems: The control constrained case under reduced regularity. TU München, 2010, Google Scholar
  18. 18.
    G. Winkler, Control constrained optimal control problems in non-convex threedimensional polyhedral domains. PhD thesis, TU Chemnitz, 2008. Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Institut für Mathematik und BauinformatikUniversität der Bundeswehr MünchenNeubibergGermany

Personalised recommendations