Discontinuous Galerkin approximations for an optimal control problem of three-dimensional Navier–Stokes–Voigt equations


We analyze a fully discrete scheme based on the discontinuous (in time) Galerkin approach, which is combined with conforming finite element subspaces in space, for the distributed optimal control problem of the three-dimensional Navier–Stokes–Voigt equations with a quadratic objective functional and box control constraints. The space-time error estimates of order \(O(\sqrt{\tau }+h)\), where \(\tau \) and h are respectively the time and space discretization parameters, are proved for the difference between the locally optimal controls and their discrete approximations.

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


  1. 1.

    Abergel, F., Temam, R.: On some control problems in fluid mechanics. Theor. Comput. Fluid Dyn. 1, 303–325 (1990)

    Article  Google Scholar 

  2. 2.

    Anh, C.T., Nguyet, T.M.: Optimal control of the instationary 3D Navier–Stokes–Voigt equations. Numer. Funct. Anal. Optim. 37, 415–439 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Anh, C.T., Nguyet, T.M.: Time optimal control of the unsteady 3D Navier–Stokes–Voigt equations. Appl. Math. Optim. 79, 397–426 (2019)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Anh, C.T., Trang, P.T.: Pull-back attractors for three-dimensional Navier–Stokes–Voigt equations in some unbounded domains. Proc. R. Soc. Edinb. Sect. A 143, 223–251 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Anh, C.T., Trang, P.T.: Decay rate of solutions to the 3D Navier–Stokes–Voigt equations in \(H^m\) space. Appl. Math. Lett. 61, 1–7 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cao, Y., Lunasin, E.M., Titi, E.S.: Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models. Commun. Math. Sci. 4, 823–848 (2006)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Casas, E., Chrysafinos, K.: A discontinuous Galerkin time-stepping scheme for the velocity tracking problem. SIAM J. Numer. Anal. 50, 2281–2306 (2012)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Casas, E., Chrysafinos, K.: Error estimates for the discretization of the velocity tracking problem. Numer. Math. 130, 615–643 (2015)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Casas, E., Chrysafinos, K.: Analysis of the velocity tracking control problem for the 3D evolutionary Navier–Stokes equations. SIAM J. Control Optim. 54, 99–128 (2016)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Casas, E., Chrysafinos, K.: Error estimates for the approximation of the velocity tracking problem with bang-bang controls. ESAIM Control Optim. Calc. Var. 23, 1267–1291 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Casas, E., Mateos, M., Raymond, J.-P.: Error estimates for the numerical approximation of a distributed control problem for the steady-state Navier–Stokes equations. SIAM J. Control Optim. 46, 952–982 (2007)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Chrysafinos, K., Walkington, N.J.: Discontinuous Galerkin approximations of the Stokes and Navier–Stokes equations. Math. Comput. 79, 2135–2167 (2010)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Conti-Zelati, M., Gal, C.G.: Singular limits of Voigt models in fluid dynamics. J. Math. Fluid Mech. 17, 233–259 (2015)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Contantin, P., Foias, C.: Navier–Stokes Equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1988)

    Google Scholar 

  15. 15.

    Damázio, P.D., Manholi, P., Silvestre, A.L.: \(L^q\)-theory of the Kelvin–Voigt equations in bounded domains. J. Differ. Equ. 260, 8242–8260 (2016)

    Article  Google Scholar 

  16. 16.

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

    MathSciNet  Article  Google Scholar 

  17. 17.

    Ebrahimi, M.A., Holst, M., Lunasin, E.: The Navier–Stokes–Voight model for image inpainting. IMA J. Appl. Math. 78, 869–894 (2013)

    MathSciNet  Article  Google Scholar 

  18. 18.

    García-Luengo, J., Marín-Rubio, P., Real, J.: Pullback attractors for three-dimensional non-autonomous Navier–Stokes–Voigt equations. Nonlinearity 25, 905–930 (2012)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Girault, P., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)

    Google Scholar 

  20. 20.

    Gunzburger, M.D.: Perspectives in Flow Control and Optimization. Advances in Design and Control. SIAM, Philadelphia (2003)

    Google Scholar 

  21. 21.

    Gunzburger, M.D., Manservisi, S.: Analysis and approximation of the velocity tracking problem for Navier–Stokes flows with distributed control. SIAM J. Numer. Anal. 37, 1481–1512 (2000)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Gunzburger, M.D., Manservisi, S.: The velocity tracking problem for Navier–Stokes flows with boundary controls. SIAM J. Control Optim. 39, 594–634 (2000)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Hinze, M.: Optimal and instantaneous control of the instationary Navier–Stokes equations, Habilitationsschrift, Fachbereich Mathematik, Technische Universit’at Berlin (2000)

  24. 24.

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

    MathSciNet  Article  Google Scholar 

  25. 25.

    Hinze, M., Kunisch, K.: Second order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40, 925–946 (2001)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Holst, M., Lunasin, E., Tsogtgerel, G.: Analysis of a general family of regularized Navier–Stokes and MHD models. J. Nonlinear Sci. 20, 523–567 (2010)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Kalantarov, V.K., Titi, E.S.: Global attractor and determining modes for the 3D Navier–Stokes–Voight equations. Chin. Ann. Math. Ser. B 30, 697–714 (2009)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Niche, C.J.: Decay characterization of solutions to Navier–Stokes–Voigt equations in term of the initial datum. J. Differ. Equ. 260, 4440–4453 (2016)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Oskolkov, A.P.: The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers. Nauchn. Semin. LOMI 38, 98–136 (1973)

    MathSciNet  Google Scholar 

  30. 30.

    Qin, Y., Yang, X., Liu, X.: Averaging of a 3D Navier–Stokes–Voigt equations with singularly oscillating forces. Nonlinear Anal. Real World Appl. 13, 893–904 (2012)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  32. 32.

    Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. 146, 65–96 (1987)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Sritharan, S.S.: Optimal Control of Viscous Flow. SIAM, Philadelphia (1998)

    Google Scholar 

  34. 34.

    Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis, 2nd edn. North-Holland, Amsterdam (1979)

    Google Scholar 

  35. 35.

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

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Wachsmuth, D.: Optimal control of the unsteady Navier–Stokes equations. PhD thesis, TU Berlin (2006)

  37. 37.

    Yue, G., Zhong, C.K.: Attractors for autonomous and nonautonomous 3D Navier–Stokes–Voight equations. Discret. Cont. Dyn. Syst. Ser. B 16, 985–1002 (2011)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Zhao, C., Zhu, H.: Upper bound of decay rate for solutions to the Navier–Stokes–Voigt equations in \(\mathbb{R}^3\). Appl. Math. Comput. 256, 183–191 (2015)

    MathSciNet  MATH  Google Scholar 

Download references


The authors would like to thank the reviewers for the helpful comments and suggestions, which improved the presentation of the paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.02-2018.303.

Author information



Corresponding author

Correspondence to Cung The Anh.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Anh, C.T., Nguyet, T.M. Discontinuous Galerkin approximations for an optimal control problem of three-dimensional Navier–Stokes–Voigt equations. Numer. Math. (2020). https://doi.org/10.1007/s00211-020-01132-0

Download citation

Mathematics Subject Classification

  • 49J20
  • 49K20
  • 35Q35
  • 65N30