Skip to main content
SpringerLink
Log in

Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations

  • Published:
Numerical Analysis and Applications Aims and scope Submit manuscript

Abstract

In this paper, we investigate the L ∞(L 2)-error estimates and superconvergence of the semidiscrete mixed finite elementmethods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k ≥ 0). We derive error estimates for approximation of both state and control. Moreover, we present the superconvergence analysis for mixed finite element approximation of the optimal control problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alt, W. and Machenroth, U., Convergence of Finite Element Approximations to State Constrained Convex Parabolic Boundary Control Problems, SIAM J. Control Optim., 1989, no. 27, pp. 718–736.

  2. Arada, N., Casas, E., and Tröltzsch, F., Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem, Comp. Optim. Appl., 2002, no. 23, pp. 201–229.

  3. Babuska, I. and Strouboulis, T., The Finite Element Method and Its Reliability, Oxford: Univ. Press, 2001.

    Google Scholar 

  4. Becker, R., Kapp, H., and Rancher, R., Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept, SIAM J. Control Optim., 2000, no. 39, pp. 113–132.

  5. Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, New York: Springer-Verlag, 1991.

    Book  MATH  Google Scholar 

  6. Brunner, H. and Yan, N., Finite Element Methods for Optimal Control Problems Governed by Integral Equations and Integro-Differential Equations, Num. Math., 2005, no. 101, pp. 1–27.

  7. Chen, Y., Superconvergence of Quadratic Optimal Control Problems by Triangular Mixed Finite Elements, Int. J. Num. Methods Eng., 2008, no. 75, pp. 881–898.

  8. Chen, Y., Huang, Y., Liu, W.B., and Yan, N.N., Error Estimates and Superconvergence of Mixed Finite ElementMethods for Convex Optimal Control Problems, J. Sci. Comp., 2009, no. 42, pp. 382–403.

  9. Chen, Y. and Liu, W.B., A Posteriori Error Estimates for Mixed Finite Element Solutions of Convex Optimal Control Problems, J. Comp. Appl.Math., 2008, no. 211, pp. 76–89.

  10. Douglas, J. and Roberts, J.E., Global Estimates for Mixed Finite Element Methods for Second-Order Elliptic Equations, Math. Comp., 1985, no. 44, pp. 39–52.

  11. Ewing, R.E., Liu, M.M., and Wang, J., Superconvergence of Mixed Finite Element Approximations over Quadrilaterals, SIAM J. Num. Anal., 1999, no. 36, pp. 772–787.

  12. Garcia, S.M., Improved Error Estimates for Mixed Finite Element Approximations for Nonlinear Parabolic Equations: The Continuous-Time Case, Num. Meth. Partial Diff. Eq., 1994, no. 10, pp. 129–147.

  13. Gong, W. and Yan, N., A Posteriori Error Estimate for Boundary Control Problems Governed by the Parabolic Partial Differential Equations, J. Comp. Math., 2009, no. 27, pp. 68–88.

  14. Hou, L. and Turner, J.C., Analysis and Finite Element Approximation of an Optimal Control Problem in Electrochemistry with Current Density Controls, Num. Math., 1995, no. 71, pp. 289–315.

  15. Haslinger, J. and Neittaanmaki, P., Finite Element Approximation for Optimal Shape Design, Chichester, UK: Wiley, 1989.

    Google Scholar 

  16. Knowles, G., Finite Element Approximation of Parabolic Time Optimal Control Problems, SIAM J. Control Optim., 1982, no. 20, pp. 414–427.

  17. Lions, J., Optimal Control of Systems Governed by Partial Differential Equations, Berlin: Springer, 1971.

    Book  MATH  Google Scholar 

  18. Lions, J. and Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, 1972.

  19. Liu, W., Ma, H., Tang, T., and Yan, N., A Posteriori Error Estimates for Discontinuous Galerkin Time-Stepping Method for Optimal Control Problems Governed by Parabolic Equations, SIAM J. Num. Anal., 2004, no. 42, pp. 1032–1061.

  20. Liu, W. and Yan, N., A Posteriori Error Estimates for Convex Boundary Control Problems, SIAM J. Num. Anal., 2001, no. 39, pp. 73–99.

  21. Liu, W. and Yan, N., A Posteriori Error Estimates for Optimal Control Problems Governed by Stokes Equations, SIAM J. Num. Anal., 2003, no. 40, pp. 1850–1869.

  22. Li, R., Liu, W., Ma, H., and Tang, T., Adaptive Finite Element Approximation of Elliptic Control Problems, SIAM J. Control Optim., 2002, no. 41, pp. 1321–1349.

  23. Mcknight, R. and Bosarge, W., Jr., The Ritz-Galerkin Procedure for Parabolic Control Problems, SIAM J. Control Optim., 1973, no. 11, pp. 510–524.

  24. Neittaanmaki, P. and Tiba, D., Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications, New York: M. Dekker, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan, 411105, Hunan, P.R. China

    T. Hou

Authors
  1. T. Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Hou.

Additional information

Original Russian Text © T. Hou, 2012, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2012, Vol. 15, No. 4, pp. 425–440.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hou, T. Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations. Numer. Analys. Appl. 5, 348–362 (2012). https://doi.org/10.1134/S1995423912040076

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995423912040076

Keywords

Search

Navigation