Skip to main content
SpringerLink
Log in

Lowest order virtual element approximations for transient Stokes problem on polygonal meshes

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

In this paper, we discuss and analyze virtual element approximations for the nonstationary Stokes problem on polygonal meshes. The proposed scheme is based on pressure-velocity formulations, and the virtual element spaces associated with velocity and pressure are constructed in a way that they obey the discrete inf-sup (LBB) condition. The spatial discretization for velocity is based on piecewise linear polynomials as well as non-linear functions, and the pressure approximation is relied on discontinuous piecewise constant polynomials, whereas a backward Euler method is employed for the time discretization. By introducing suitable energy and \(L^2\) projection operators, the optimal error estimates are established in \(H^1\) and \(L^2\) norms for both semi and fully discrete schemes under the minimal regularity assumptions on continuous solutions. Moreover, several numerical experiments are conducted to validate the obtained theoretical rate of convergence and exhibit the performance of the proposed scheme.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66, 376–391 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Ahmed, N., Linke, A., Merdon, C.: On really locking-free mixed finite element methods for the transient incompressible Stokes equations. SIAM J. Numer. Anal. 56(1), 185–209 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Antonietti, P.F., da Veiga, L.B., Mora, D., Verani, M.: A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52, 386–404 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Bänsch, E., Karakatsani, F., Makridakis, C.G.: A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem. Calcolo 55(19), 1–32 (2018)

    MathSciNet  MATH  Google Scholar 

  5. da Veiga, L.B., Lipnikov, K., Manzini, G.: Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes. SIAM J. Numer. Anal. 48(4), 1419–1443 (2010)

    MathSciNet  MATH  Google Scholar 

  6. da Veiga, L.B., 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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. da Veiga, L.B., 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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. da Veiga, L.B., Mora, D., Vacca, G.: The Stokes Complex for Virtual Elements with Application to Navier-Stokes Flows. J. Sci. Comput. 81, 990–1018 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Bernardi, C., Raugel, G.: A conforming finite element method for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 22(3), 455–473 (1985)

    MathSciNet  MATH  Google Scholar 

  13. Bochev, P.B., Gunzburger, M.D., Shadid, J.N.: On inf-sup stabilized finite element methods for transient problems. Comput. Methods Appl. Mech. Engrg. 193, 1471–1489 (2004)

    MathSciNet  MATH  Google Scholar 

  14. Bochev, P.B., Dohrmann, C.R., Gunzburger, M.D.: Stabilization of low-order mixed finite elements for the Stokes equations. SIAM J. Numer. Anal. 44, 82–101 (2006)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  16. Brenner, S., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer Verlag, New York (2008)

    MATH  Google Scholar 

  17. Bürger, R., Kumar, S., Mora, D., Ruiz-Baier, R., Verma, N.: Virtual element methods for the three-field formulation of time-dependent linear poroelasticity. Adv. Comput. Math. (2021). https://doi.org/10.1007/s10444-020-09826-7

    Article  MathSciNet  MATH  Google Scholar 

  18. Burman, E., Fernández, M.A.: Continuous interior penalty finite element method for the time-dependent Navier-Stokes equations: space discretization and convergence. Numer. Math. 107, 39–77 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Frerichs, D., Merdon, C.: Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem. IMA J. Numer. Anal. (2020). https://doi.org/10.1093/imanum/draa073

    Article  Google Scholar 

  20. Gatica, G.N., Munar, M., Sequeira, F.A.: A mixed virtual element method for the Navier-Stokes equations. Math. Models Methods Appl. Sci. 28(14), 2719–2762 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics, Springer-Verlag, Berlin Heidelberg (1986)

    MATH  Google Scholar 

  22. He, Y.: A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem. IMA J. Numer. Anal. 23, 665–691 (2003)

    MathSciNet  MATH  Google Scholar 

  23. He, Y., Sun, W.: Stabilized finite element method based on the Crank-Nicolson Extrapolation scheme for the time-dependent Navier-Stokes equations. Math. Comput. 76, 115–136 (2007)

    MathSciNet  MATH  Google Scholar 

  24. He, G., He, Y., Chen, Z.: A penalty finite volume method for the transient Navier-Stokes equations. Appl. Numer. Math. 58(11), 1583–1613 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Heywood, J.G., Rannacher, R.: Finite Element Approximation of the Nonstationary Navier-Stokes Problem. I. Regularity of Solutions and Second-Order Error Estimates for Spatial Discretization. SIAM J. Numer. Anal. 19(2), 275–311 (1982)

    MathSciNet  MATH  Google Scholar 

  26. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem, part ii: stability of solutions and error estimates uniform in time. SIAM J. Numer. Anal. 23(4), 750–777 (1986)

    MathSciNet  MATH  Google Scholar 

  27. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes Problem, Part III. Smoothing property and higher order error estimates for spatial discretization. SIAM J. Numer. Anal. 25(3), 489–512 (1988)

    MathSciNet  MATH  Google Scholar 

  28. Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier Stokes problem. IV. Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)

    MathSciNet  MATH  Google Scholar 

  29. Hill, A.T., Süli, E.: Approximation of the global attractor for the incompressible Navier-Stokes equations. IMA J. Numer. Anal. 20(4), 633–667 (2000)

    MathSciNet  MATH  Google Scholar 

  30. Irisarri, D., Hauke, G.: Stabilized virtual element methods for the unsteady incompressible Navier-Stokes equations. Calcolo 56(38) (2019)

  31. Jiang, Y., Mei, L., Wei, H.: A stabilized finite element method for transient Navier-Stokes equations based on two local Gauss integrations. Int. J. Numer. Meth. Fluids 70, 713–723 (2011)

    MathSciNet  MATH  Google Scholar 

  32. Kumar, S., Ruiz-Baier, R.: Equal order discontinuous finite volume element methods for the Stokes problem. J. Sci. Comput. 65, 956–978 (2015)

    MathSciNet  MATH  Google Scholar 

  33. Adak, D., Natarajan, E., Kumar, S.: Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes. Numer. Methods Partial Differ. Equ. 35, 222–245 (2018)

    MathSciNet  MATH  Google Scholar 

  34. Adak, D., Natarajan, E., Kumar, S.: Virtual element method for semilinear hyperbolic problems on polygonal meshes. Int. J. Comput. Math. 96(5), 971–991 (2019)

    MathSciNet  MATH  Google Scholar 

  35. Li, J., He, Y., Chen, Z.: A new stabilized finite element method for the transient Navier-Stokes equations. Comput. Methods Appl. Mech. Engg. 197, 22–35 (2007)

    MathSciNet  MATH  Google Scholar 

  36. Liu, X., Chen, Z.: The nonconforming virtual element method for the Navier-Stokes equations. Adv. Comput. Math. 45, 51–74 (2019)

    MathSciNet  MATH  Google Scholar 

  37. Lu, X., Lin, P.: Error estimate of the P1 nonconforming finite element method for the penalized unsteady Navier-Stokes equations. Numer. Math. 115, 261–287 (2010)

    MathSciNet  MATH  Google Scholar 

  38. Mu, L., Ye, X.: A finite volume method for solving Navier-Stokes problems. Nonlinear Anal. 74(17), 6686–6695 (2011)

    MathSciNet  MATH  Google Scholar 

  39. Qiu, H., Xue, C., Xue, L.: Low-order stabilized finite element methods for the unsteady Stokes/Navier-Stokes equations with friction boundary conditions. Math Method. Appl. Sci. 41, 2119–2139 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Shang, Y.: New stabilized finite element method for time-dependent incompressible flow problems. Int. J. Numer. Meth. Fluids 62, 166–187 (2010)

    MathSciNet  MATH  Google Scholar 

  41. Shang, Yueqiang: Error analysis of a fully discrete finite element variational multiscale method for time-dependent incompressible Navier-Stokes equations. Numer. Methods Partial Differ. Equ. 29(6), 2025–2046 (2013)

    MathSciNet  MATH  Google Scholar 

  42. Talischi, C., Paulino, G.H., Pereira, A., Menezes, I.F.: Polymesher: a general-purpose mesh generator for polygonal elements written in matlab. Struct. Multidiscip. Optim. 45(3), 309–328 (2012)

    MathSciNet  MATH  Google Scholar 

  43. TEMAM, R.: Navier-Stokes: Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1977)

  44. Thomée, Vidar: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, Springer-Verlag, Berlin (2006)

    MATH  Google Scholar 

  45. Vacca, G., da Veiga, L.B.: Virtual element methods for parabolic problems on polygonal meshes. Numer. Methods Partial Differential Equations 31, 2110–2134 (2015)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  47. Verma, N., Gómez-Vargas, B., de Oliveira-Vilaca, L.M., Kumar, S., Ruiz-Baier, R.: Well-posedness and discrete analysis for advection-diffusion-reaction in poroelastic media. Appl. Anal. (2021). https://doi.org/10.1080/00036811.2021.1877677

    Article  MATH  Google Scholar 

  48. Xie, C., Feng, M.: New nonconforming finite element method for solving transient Naiver-Stokes equations. Appl. Math. Mech. Engl. Ed. 35(2), 237–258 (2014)

    MathSciNet  MATH  Google Scholar 

  49. Xu, C., Shi, D., Liao, X.: Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations. Appl. Math. Mech. Engl. Ed. 37(8), 1095–1112 (2016)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the valuable comments by the anonymous reviewer, whose suggestions lead to numerous improvements to the manuscript. We also would like to thank Prof. David Mora (Departamento de Matemática, Universidad del Bío-Bío, Concepción, Chile) and Prof. Ricardo Ruiz-Baier (Monash University, Australia) for their kind support and helping in the implementation. This work was partially supported by the Department of Science and Technology (DST-SERB), India through MATRICS Grant MTR/2019/000519 and Core Research Grant CRG/2019/003863).

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram, 695547, Kerala, India

    N. Verma & S. Kumar

Authors
  1. N. Verma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Kumar.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Verma, N., Kumar, S. Lowest order virtual element approximations for transient Stokes problem on polygonal meshes. Calcolo 58, 48 (2021). https://doi.org/10.1007/s10092-021-00440-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10092-021-00440-7

Keywords

Mathematics Subject Classification

Search

Navigation