Skip to main content
SpringerLink
Log in

A Priori Error Estimates and Superconvergence of Splitting Positive Definite Mixed Finite Element Methods for Pseudo-Hyperbolic Integro-Differential Optimal Control Problems

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

ABSTRACT

In this paper we discuss a priori error estimates and superconvergence of splitting positive definite mixed finite element methods for optimal control problems governed by pseudo-hyperbolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element functions, and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates for the control variable, state variables, and co-state variables. Second, we obtain a superconvergence result for the control variable.

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

  2. Brunner, H. and Yan, N., Finite Element Methods for Optimal Control Problems Governed by Integral Equations and Integro-Differential Equations, Numerische Mathematik, 2005, vol. 101, pp. 1–27.

  3. Chen, Y., Superconvergence of Quadratic Optimal Control Problems by Triangular Mixed Finite Element Elements, Int. J. Num. Meth. Engin., 2008, vol. 75, no. 8, pp. 881–898.

  4. Chen, Y., Superconvergence of Mixed Finite Element Methods for Optimal Control Problems, Math. Comp., 2008, vol. 77, pp. 1269–1291.

  5. Chen, Y. and Dai, Y., Superconvergence for Optimal Control Problems Governed by Semi-Linear Elliptic Equations, J. Sci. Comp., 2009, vol. 39, pp. 206–221.

  6. Chen, Y., Huang, Y., Liu, W.B., and Yan, N., Error Estimates and Superconvergence of Mixed Finite Element Methods for Convex Optimal Control Problems, J. Sci. Comp., 2010, vol. 42, no. 3, pp. 382–403.

  7. Douglas, J. and Roberts, J.E., Global Estimates for Mixed Methods for Second Order Elliptic Equations, Math. Comp., 1985, vol. 44, pp. 39–52.

  8. Ewing, R.E., Liu, M.M., and Wang, J., Superconvergence of Mixed Finite Element Approximations over Quadrilaterals, SIAM J. Num. An., 1999, vol. 36, pp. 772–787.

  9. Guo, H., Fu, H., and Zhang, J., A Splitting Positive Definite Mixed Finite Element Method for Elliptic Optimal Control Problem,Appl. Math. Comp., 2013, vol. 219, pp. 11178–11190.

  10. Hou, T., Zhang, J., Li, Y., and Yang, Y., New Elliptic Projections and A Priori Error Estimates of \(H^{1}\)-Galerkin Mixed Finite Element Methods for Optimal Control Problems Governed by Parabolic Integro-Differential Equations, Appl. Math. Comp., 2017, vol. 311, pp. 29–46.

  11. Hou, T., Error Estimates of Expanded Mixed Methods for Optimal Control Problems Governed by Hyperbolic Integro-Differential Equations, Num. Meth. Part. Diff. Eqs., 2013, vol. 29, pp. 1675–1693.

  12. Li, R., Liu, W.B., and Yan, N., A Posteriori Error Estimates of Recovery Type for Distributed Convex Optimal Control Problems,J. Sci. Comp., 2002, vol. 41, pp. 1321–1349.

  13. Lions, J.L., Optimal Control of Systems Governed by Partial Differential Equations, Berlin: Springer-Verlag, 1971.

  14. Liu, Y., Li, H., and Wang, J., Splitting Positive Definite Mixed Element Method for Pseudo-Hyperbolic Equations, Num. Meth. Part. Diff. Eqs., 2012, vol. 28, pp. 670–688.

  15. Meyer, C. and Rösch, A., Superconvergence Properties of Optimal Control Problems, SIAM J. Control Optim., 2004, vol. 433, pp. 970–985.

  16. Meidner, D. and Vexler, B., A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part I: Problems without Control Constraints, SIAM J. Control Optim., 2008, vol. 47, pp. 1150–1177.

  17. Meidner, D. and Vexler, B., A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems Part II: Problems with Control Constraints, SIAM J. Control Optim., 2008, vol. 47, pp. 1301–1329.

  18. Raviart, P.A. and Thomas, J.M., A Mixed Finite Element Method for 2nd Order Elliptic Problems, in Mathematical Aspects of the Finite Element Method, Berlin: Springer, 1977, pp. 292–315.

  19. Shen, W., Ge, L., Yang, D., and Liu, W., Sharp A Posteriori Error Estimates for Optimal Control Governed by Parabolic Integro-Differential Equations, J. Sci. Comp., 2015, vol. 65, pp. 1–33.

  20. Wang, F., Chen, Y., and Tang, Y., Superconvergence of Fully Discrete Splitting Positive Definite Mixed FEM for Hyperbolic Equations, Num. Meth. Part. Diff. Eqs., 2014, vol. 30, pp. 175–186.

  21. Xing, X. and Chen, Y., Superconvergence of Mixed Methods for Optimal Control Problems Governed by Parabolic Equations, Adv. Appl. Math. Mech., 2011, vol. 3, pp. 401–419.

  22. Yang, D., Chang, Y., and Liu, W.B., A Priori Error Estimate and Superconvergence Analysis for an Optimal Control Problem of Bilinear Type, J. Comp. Math., 2008, vol. 4, pp. 471–487.

  23. Yang, D., A Splitting Positive Definite Mixed Element Method for Miscible Displacement of Compressible Flow in Porous Media,Num. Meth. Part. Diff. Eqs., 2011, vol. 17, pp. 229–249.

  24. Zhang, J. and Yang, D., A Splitting Positive Definite Mixed Element Method for Second-Order Hyperbolic Equations, Num. Meth. Part. Diff. Eqs., 2009, vol. 25, pp. 622–636.

Download references

Funding

This work was supported by the NNSF of China (grant no. 11601014) and by Beihua University Youth Research and Innovation Team Development Project.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Beihua University, 132013, Jilin, China

    C. Xu

Authors
  1. C. Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Xu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, C. A Priori Error Estimates and Superconvergence of Splitting Positive Definite Mixed Finite Element Methods for Pseudo-Hyperbolic Integro-Differential Optimal Control Problems. Numer. Analys. Appl. 13, 17–33 (2020). https://doi.org/10.1134/S1995423920010024

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995423920010024

Keywords

Search

Navigation