Numerische Mathematik

, Volume 106, Issue 3, pp 349–367 | Cite as

Optimal control of the convection-diffusion equation using stabilized finite element methods

Article

Abstract

In this paper we analyze the discretization of optimal control problems governed by convection-diffusion equations which are subject to pointwise control constraints. We present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case. Moreover, the in general different approaches “optimize-then- discretize” and “discretize-then-optimize” coincide for the proposed discretization scheme. This allows for a symmetric optimality system at the discrete level and optimal order of convergence.

Mathematics Subject Classification (2000)

Optimal control Stabilized finite elements Error estimates Pointwise inequality constraints 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arada N., Casas E. and Tröltzsch F. (2002). Error estimates for a semilinear elliptic optimal control problem. Comput. Optim. Approx. 23: 201–229 MATHCrossRefGoogle Scholar
  2. 2.
    Becker R. and Braack M. (2001). A finite element pressure gradient stabilization for the stokes equations based on local projections. Calcolo 38(4): 137–199 CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bergounioux M., Ito K. and Kunisch K. (1999). Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37(4): 1176–1194 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Braack, M.: Solution of complex flow problems: Mesh- and model adaptation with stabilized finite elements. Habilitationsschrift, Institut für Angewandte Mathematik, Universität (2005)Google Scholar
  5. 5.
    Braack M. and Burman E. (2006). Local projection stabilization for the Oseen problem and its interpretation as a variational multiscale method. SIAM J. Numer. Anal. 43(6): 2544–2566 MATHMathSciNetGoogle Scholar
  6. 6.
    Burman E. and Hansbo P (2004). Edge stabilization for Galerkin approximations of convection–diffusion–reaction problems. Comput. Methods Appl. Mech. Eng. 193(15–16): 1437–1453 MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Casas E. and Tröltzsch F. (2003). Error estimates for linear-quadratic elliptic control problems. In: Barbu, V. (eds) Analysis and Optimization of Differential Systems, pp 89–100. Kluwer, Boston Google Scholar
  8. 8.
    Ciarlet, P.G.: The finite element method for elliptic problems. In: Classics Appl. Math., vol. 40. SIAM, Philadelphia (2002)Google Scholar
  9. 9.
    Collis, S.S., Heinkenschloss, M.: Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems. CAAM TR02-01 (2002)Google Scholar
  10. 10.
    Dede L. and Quarteroni A. (2005). Optimal control and numerical adaptivity for advection-diffusion equations. M2AN 39(5): 1019–1040 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Falk R. (1973). Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44: 28–47 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fursikov, A.V.: Optimal control of distributed systems: theory and applications. In: Transl. Math. Monogr., vol. 187. AMS, Providence (1999)Google Scholar
  13. 13.
    Grisvard, P.: Singularities in Boundary Value Problems. Springer, Masson, Paris, Berlin (1992)Google Scholar
  14. 14.
    Guermond J.L. (1999). Stabilization of Galerkin approximations of transport equations by subgrid modeling. Modél. Math. Anal. Numér. 36(6): 1293–1316 CrossRefMathSciNetGoogle Scholar
  15. 15.
    Hinze M. (2005). A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30(1): 45–61 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Johnson C. (1987). Numerical Solution of Partial Differential Equations by Finite Element Method. Cambridge University Press, Cambridge MATHGoogle Scholar
  17. 17.
    Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Spectral problems associated with corner singularities of solutions to elliptic equations. In: Mathematical Surveys and Monographs, vol. 85. American Mathematical Society, Providence (2001)Google Scholar
  18. 18.
    Kunisch K. and Rösch A. (2002). Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J. Optim. 13(2): 321–334 MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lions, J.L.: Optimal control of systems governed by partial differential equations. In: Grundlehren Math. Wiss., vol. 170. Springer, Berlin (1971)Google Scholar
  20. 20.
    Meyer C. and Rösch A. (2004). Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43(3): 970–985 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations. In: Springer Series in Computational Mathematics, vol. 23. Springer, Berlin (1994)Google Scholar
  22. 22.
    Roos, H.G., Stynes, M., Tobiska, L.: Numerical methods for singularly perturbed differential equations. In: Springer Series in Computational Mathematics, vol. 24. Springer, Berlin (1996)Google Scholar
  23. 23.
    Rösch A. (2005). Error estimates for linear-quadratic control problems with control constraints.    Optim. Methods Softw. 21(1): 121–134 Google Scholar
  24. 24.
    Rösch A. and Vexler B. (2006). Optimal control of the stokes equations: a priori error analysis for finite element discretization with postprocessing. SIAM J. Numer. Anal. 44(5): 1903–1920 CrossRefMathSciNetGoogle Scholar
  25. 25.
    Tröltzsch F. (2005). Optimale Steuerung partieller Differentialgleichungen. Friedr. Vieweg & Sohn Verlag, Wiesbaden Google Scholar
  26. 26.
    Zhou G. (1997). How accurate is the streamline diffusion finite element method?. Math. Comp. 66(217): 31–44 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques AppliquéesUniversité de Pau et des Pays de l’AdourPAU CedexFrance
  2. 2.Johann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria

Personalised recommendations