Skip to main content
SpringerLink
Log in

Adaptive Virtual Element Method for Optimal Control Problem Governed by Stokes Equations

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

Abstract

In this paper, adaptive virtual element method (VEM) approximation of optimal control problem governed by Stokes equations with control constraints is discussed. The virtual element discrete scheme of the optimal control problem is constructed by polynomial projections and variational discretization of the control variable. Based on the a posteriori error estimates of VEM for Stokes equations and approximated error equivalence between the solutions of the optimal control problem and the solutions of the state and adjoint equations, we build up upper and lower bounds for the a posteriori error estimates of the optimal control problem. It proves that the a posteriori error indicator is reliable and efficient. The theoretical findings are illustrated by the numerical experiments.

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

Algorithm 1
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Data Availability

Enquiries about data availability should be directed to the authors.

References

  1. Allendes, A., Fuica, F., Otarola, E., Quero, D.: An adaptive FEM for the pointwise tracking optimal control problem of the Stokes equations. SIAM J. Sci. Comput. 41(5), A2967–A2998 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allendes, A., Fuica, F., Otarola, E., Quero, D.: A posteriori error estimates for a distributed optimal control problem of the stationary Navier–Stokes equations. SIAM J. Control. Optim. 59(4), 2898–2923 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Becker, R., Kapp, R., 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  MathSciNet  MATH  Google Scholar 

  4. Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beirão da Veiga, L., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: Virtual element method for general second order elliptic problems on polygonal meshes. Math. Models Methods Appl. Sci. 26(4), 729–750 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beirão da Veiga, L., Lovadina, C., Vacca, G.: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM Math. Model. Numer. Anal. 51(2), 509–535 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Beirão da Veiga, L., Lovadina, C., Vacca, G.: Virtual elements for the Navier–Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 56(3), 1210–1242 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  10. Braack, M., Richter, T.: Solutions of 3D Navier–Stokes benchmark problems with adaptive finite elements. Comput. Fluids 35(4), 372–392 (2006)

    Article  MATH  Google Scholar 

  11. Brenne, S.C., Guan, Q.G., Sung, L.Y.: Some estimates for virtual element methods. Comput. Methods Appl. Math. 17(4), 553–574 (2017)

    Article  MathSciNet  Google Scholar 

  12. Brezzi, F., Marini, L.D.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253, 455–462 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cangiani, A., Georgoulis, E.H., Pryer, T., Sutton, O.J.: A posteriori error estimates for the virtual element method. Numer. Math. 137(4), 857–893 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Canuto, C., Rosso, D.: A posteriori error analysis and adaptivity for a VEM discretization of the Navier–Stokes equations (2022). Preprint at https://doi.org/10.48550/arXiv.2212.14414

  15. Chen, L., Huang, J.G.: Some error analysis on virtual element methods. Calcolo 55(1), 5 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. de los Reyes, J.C., Kunisch, K.: A semi-smooth Newton method for control constrained boundary optimal control of the Navier–Stokes equations. Nonlinear Anal. 62(7), 1289–1316 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Deckelnick, K., Hinze, M.: Semidiscretization and error estimates for distributed control of the instationary Navier–Stokes equations. Numer. Math. 97(2), 297–320 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30(1), 45–61 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, H.P., Yan, N.N.: Recovery type superconvergence and a posteriori error estimates for control problems governed by Stokes equations. J. Comput. Appl. Math. 209(2), 187–207 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, W.B., Yan, N.N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Science Press, Beijing (2008)

    Google Scholar 

  22. Mora, D., Rivera, G., Rodríguez, R.: A posteriori error estimates for a virtual element method for the Steklov eigenvalue problem. Comput. Math. Appl. 74(9), 2172–2190 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nicaise, S., Sirch, D.: Optimal control of the Stokes equations: conforming and non-conforming finite element methods under reduced regularity. Comput. Optim. Appl. 49(3), 567–600 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Niu, H.F., Yang, D.P.: Finite element analysis of optimal control problem governed by Stokes equations with \({L}^2\)-norm state-constraints. J. Comput. Math. 29(5), 589–604 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Niu, H.F., Yuan, L., Yang, D.P.: Adaptive finite element method for an optimal control problem of Stokes flow with \({L}^2\)-norm state constraint. SIAM J. Numer. Anal. 69(3), 534–549 (2012)

    Google Scholar 

  26. Rösch, A., Vexler, B.: Optimal control of the Stokes equations, a priori error analysis for finite element discretization with postprocessing. SIAM J. Numer. Anal. 44(5), 1903–1920 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tröltzsch, F., Wachsmuth, D.: Second-order sufficient optimality conditions for the optimal control of Navier–Stokes equations. Nonlinear Anal. 12(1), 93–119 (2006)

    MathSciNet  MATH  Google Scholar 

  28. Vacca, G.: An \({H}^1\)-conforming virtual element for Darcy and Brinkman equations. Math. Models Methods Appl. Sci. 28(1), 159–194 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Verfürth, R.: A posteriori error estimators for the Stokes equations. Numer. Math. 55(3), 309–325 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang, Q.M., Zhou, Z.J.: Adaptive virtual element method for optimal control problem governed by general elliptic equation. J. Sci. Comput. 88(1), 14 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, Q.M., Zhou, Z.J.: A priori and a posteriori error analysis for virtual element discretization of elliptic optimal control problem. Numer. Algor. 90, 989–1015 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, J.P., Wang, Y.Q., Ye, X.: A posteriori error estimation for an interior penalty type method employing H(div) elements for the Stokes equations. SIAM J. Sci. Comput. 33(1), 131–152 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, G., Wang, Y., He, Y.N.: A posteriori error estimates for the virtual element method for the Stokes problem. J. Sci. Comput. 84(2), 37 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The research was supported by National Natural Science Foundation of China (No. 11971276, 12171287).

Author information

Authors and Affiliations

  1. Shandong Normal University, Jinan, China

    Yanwei Li, Qiming Wang & Zhaojie Zhou

Authors
  1. Yanwei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiming Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhaojie Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhaojie Zhou.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Y., Wang, Q. & Zhou, Z. Adaptive Virtual Element Method for Optimal Control Problem Governed by Stokes Equations. J Sci Comput 97, 63 (2023). https://doi.org/10.1007/s10915-023-02377-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-023-02377-1

Keywords

Mathematics Subject Classification

Search

Navigation