Computational Optimization and Applications

, Volume 57, Issue 3, pp 703–729 | Cite as

A priori error analysis of the upwind symmetric interior penalty Galerkin (SIPG) method for the optimal control problems governed by unsteady convection diffusion equations

  • Tuğba Akman
  • Hamdullah Yücel
  • Bülent Karasözen
Article

Abstract

In this paper, we analyze the symmetric interior penalty Galerkin (SIPG) for distributed optimal control problems governed by unsteady convection diffusion equations with control constraint bounds. A priori error estimates are derived for the semi- and fully-discrete schemes by using piecewise linear functions. Numerical results are presented, which verify the theoretical results.

Keywords

Unsteady convection diffusion equations Optimal control Discontinuous Galerkin methods A priori error estimates 

References

  1. 1.
    Becker, R., Vexler, B.: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106, 349–367 (2007) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37(4), 1176–1194 (1999) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bergounioux, M., Haddou, M., Hintermueller, M., Kunisch, K.: A comparison of interior–point methods and a Moreau–Yosida based active set strategy for constrained optimal control problems. SIAM J. Optim. 11, 495–521 (2001) CrossRefGoogle Scholar
  4. 4.
    Brenner, S.: Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J. Numer. Anal. 41(1), 306–324 (2003) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Cangiani, A., Chapman, J., Georgoulis, E.H., Jensen, M.: On Local Super-Penalization of Interior Penalty Discontinuous Galerkin Methods. CoRR 1205.5672 [abs] (2012)
  6. 6.
    Collis, S.S., Heinkenschloss, M.: Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems. Tech. Rep. TR02–01, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892 (2002) Google Scholar
  7. 7.
    Evans, L.C.: Partial Differential Equations. Grad. Stud. Math., vol. 19. AMS, Providence (2002) Google Scholar
  8. 8.
    Feistauer, M., S̆vadlenka, K.: Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems. J. Numer. Math. 12, 97–117 (2004) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Fu, H.: A characteristic finite element method for optimal control problems governed by convection-diffusion equations. J. Comput. Appl. Math. 235, 825–836 (2010) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Fu, H., Rui, H.: A priori error estimates for optimal control problems governed by transient advection-diffusion equations. J. Sci. Comput. 38(3), 290–315 (2009) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Hinze, M.K., Turek, S.: A Space-Time Multigrid Method for Optimal Flow Control. International Series of Numerical Mathematics, vol. 160 (2011) Google Scholar
  12. 12.
    Hinze, M., Yan, N., Zhou, Z.: Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math. 27, 237–253 (2009) MATHMathSciNetGoogle Scholar
  13. 13.
    Houston, P., Schwab, C., Süli, E.: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Kunisch, K., Vexler, B.: Constrained Dirichlet boundary control in L 2 for a class of evolution equations. SIAM J. Control Optim. 46(5), 1726–1753 (2007) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Leykekhman, D.: Investigation of commutative properties of discontinuous Galerkin methods in PDE constrained optimal control problems. J. Sci. Comput. 53, 483–511 (2012) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Leykekhman, D., Heinkenschloss, M.: Local error analysis of discontinuous Galerkin methods for advection-dominated elliptic linear-quadratic optimal control problems. SIAM J. Numer. Anal. 50, 2012–2038 (2012) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Li, R., Liu, W., Ma, H., Tang, T.: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41, 1321–1349 (2002) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971) CrossRefMATHGoogle Scholar
  19. 19.
    Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems. II. Problems with control constraints. SIAM J. Control Optim. 47(3), 1301–1329 (2008) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Meidner, D., Vexler, B.: A priori error analysis of the Petrov–Galerkin Crank–Nicolson scheme for parabolic optimal control problems. SIAM J. Control Optim. 49(5), 2183–2211 (2011) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Rivìere, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Theory and Implementation. Frontiers in Applied Mathematics, vol. 35. Society for Industrial and Applied Mathematics, Philadelphia (2008) CrossRefMATHGoogle Scholar
  22. 22.
    Schötzau, D., Zhu, L.: A robust a-posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations. Appl. Numer. Math. 59, 2236–2255 (2009) CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Stoll, M., Wathen, A.: All-at-once solution of time-dependent Stokes control. J. Comput. Phys. 232, 498–515 (2013) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Sun, T.: Discontinuous Galerkin finite element method with interior penalties for convection diffusion optimal control problem. Int. J. Numer. Anal. Model. 7, 87–107 (2010) MathSciNetGoogle Scholar
  25. 25.
    Yan, N., Zhou, Z.: A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation. J. Comput. Appl. Math. 223, 198–217 (2009) CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Yücel, H., Karasözen, B.: Adaptive Symmetric Interior Penalty Galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints (2013). Optimization (electronic) Google Scholar
  27. 27.
    Yücel, H., Heinkenschloss, M., Karasözen, B.: An adaptive discontinuous Galerkin method for convection dominated distributed optimal control problems. Tech. Rep., Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005–1892 (2012) Google Scholar
  28. 28.
    Yücel, H., Heinkenschloss, M., Karasözen, B.: Distributed optimal control of diffusion-convection-reaction equations using discontinuous Galerkin methods. In: Numerical Mathematics and Advanced Applications, vol. 2011, pp. 389–397. Springer, Berlin (2013) Google Scholar
  29. 29.
    Zhou, Z., Yan, N.: The local discontinuous Galerkin method for optimal control problem governed by convection diffusion equations. Int. J. Numer. Anal. Model. 7, 681–699 (2010) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Tuğba Akman
    • 1
  • Hamdullah Yücel
    • 1
    • 2
  • Bülent Karasözen
    • 1
  1. 1.Department of Mathematics & Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Computational Methods in Systems and Control TheoryMax Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany

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