Skip to main content
SpringerLink
Log in

A Posteriori Error Estimates of Recovery Type for Distributed Convex Optimal Control Problems

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we derive a posteriori error estimates of recovery type, and present the superconvergence analysis for the finite element approximation of distributed convex optimal control problems. We provide a posteriori error estimates of recovery type for both the control and the state approximation, which are generally equivalent. Under some stronger assumptions, they are further shown to be asymptotically exact. Such estimates, which are apparently not available in the literature, can be used to construct adaptive finite element approximation schemes and as a reliability bound for the control problems. Numerical results demonstrating our theoretical results are also presented in this paper.

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. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley Interscience, New York (2000)

    MATH  Google Scholar 

  2. Becker, R., Rannacher, R.: An optimal control approach to a-posteriori error estimation. In: Iserles, A. (ed.) Acta Numerica 2001, pp. 1–102. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  3. Becker, R., Kapp, H., Rannacher, R.: Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 39(1), 113–132 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Carstensen, C., Verfürth, R.: Edge residual dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36, 1571–1587 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  6. Falk, F.S.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44, 28–47 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  7. Haslinger, J., Neittaanmaki, P.: Finite Element Approximation for Optimal Shape Design. Wiley, Chichester (1989)

    Google Scholar 

  8. Huang, Y., Li, R., Liu, W., Yan, N.: Multi-adaptive mesh discretization for finite element approximation of constrained optimal control problems. SIAM. J. Control Optim. (to appear)

  9. Li, R., Lin, W., Ma, H., Tang, T.: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41(5), 1321–1349 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Krizek, M., Neitaanmaki, P., Stenberg, R.: Finite Element Methods: Superconvergence, Post-processing, and A Posteriori Estimates. Lectures Notes in Pure and Applied Mathematics, vol. 196. Dekker, New York (1998)

    MATH  Google Scholar 

  11. Lin, Q., Yan, N.: Construction and Analysis of High Efficient Finite Element. Hebei University Press (1996) (in Chinese)

  12. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    MATH  Google Scholar 

  13. Liu, W., Yan, N.: A posteriori error analysis for convex distributed optimal control problems. Adv. Comput. Math. 15, 285–309 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu, W., Yan, N.: A posteriori error estimates for convex boundary control problems. SIAM J. Numer. Anal. 39, 73–99 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, W., Yan, N.: A posteriori error estimates for control problems governed by Stokes equations. SIAM J. Numer. Anal. 40(5), 1850–1869 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Liu, W., Yan, N.: A posteriori error estimates for optimal control problems governed by parabolic equations. Numer. Math. 93(3), 497–521 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Malanowski, K.: Convergence of approximations vs. regularity of solutions for convex, control constrained, optimal control systems. Appl. Math. Optim. 8 (1982)

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

    Google Scholar 

  19. Oganesyan, L.A., Rukhovetz, L.A.: Study of the rate of convergence of variational difference scheme for second order elliptic equation in two-dimensional field with a smooth boundary. V.S.S.R. Comput. Math. Math. Phys. 9, 158–183 (1969)

    Google Scholar 

  20. Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984)

    MATH  Google Scholar 

  21. Tiba, D., Troltzsch, F.: Error estimates for the discretization of state constrained convex control problems. Numer. Funct. Anal. Optim. 17, 1005–1028 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  22. Verfurth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement. Wiley-Teubner, London (1996)

    Google Scholar 

  23. Yan, N., Zhou, A.: Gradient recovery type a posteriori error estimates for finite filament approximations on irregular meshes. Comput. Methods Appl. Mech. Eng. 190, 4289–4299 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Zhang, Z., Yan, N.: Recovery type a posteriori error estimates in finite element methods. Korean J. Comput. Appl. Math. 8, 235–251 (2001)

    MATH  MathSciNet  Google Scholar 

  25. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. Int. J. Numer. Methods Eng. 33, 1331–1382 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  26. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergence patch recovery (SPR) and adaptive finite element refinement. Comput. Methods Appl. Mech. Eng. 101, 207–224 (1992)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Ruo Li

  2. KBS & Institute of Mathematics and Statistics, The University of Kent, Canterbury, CT2 7NF, England

    Wenbin Liu

  3. Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Ningning Yan

Authors
  1. Ruo Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenbin Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ningning Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ruo Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, R., Liu, W. & Yan, N. A Posteriori Error Estimates of Recovery Type for Distributed Convex Optimal Control Problems. J Sci Comput 33, 155–182 (2007). https://doi.org/10.1007/s10915-007-9147-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-007-9147-7

Keywords

Search

Navigation