Skip to main content
SpringerLink
Log in

Local and parallel finite element algorithms for the time-dependent Oseen equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Based on two-grid discretizations, local and parallel finite element algorithms are proposed and analyzed for the time-dependent Oseen equations. Using conforming finite element pairs for the spatial discretization and backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Oseen equations using a coarse grid on the entire domain, and then correct the resulted residual using a fine grid on overlapped subdomains by some local and parallel procedures at each time step. By the theoretical tool of local a priori estimate for the fully discrete finite element solution, error bounds of the approximate solutions from the algorithms are estimated. Numerical results are also given to demonstrate the efficiency of the algorithms.

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

Similar content being viewed by others

References

  1. Adams, R.: Sobolev Spaces. Academic, New York (1975)

    MATH  Google Scholar 

  2. Arnold, D.N., Fortin, M., Brezzi, F.: A stable finite element method for the stokes equations. Calcolo 21(4), 337–344 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, D.N., Liu, X.: Local error estimates for finite element discretizations of the Stokes equations. RAIRO M2,AN 29, 367–389 (1995)

    MathSciNet  MATH  Google Scholar 

  4. Ciarlet, P.G., Oden, J.T.: The finite element method for elliptic problems. J. Appl. Mech. 1978 45(4), 968–969 (1978)

    Google Scholar 

  5. Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numer. 7(R-3), 33–76 (1973)

    MathSciNet  MATH  Google Scholar 

  6. Fortin, M.: Calcul numérique des ecoulements fluides de bingham et des fluides newtoniens incompressible par des méthodes d’eléments finis, Doctoral thesis, Université de Paris VI (1972)

  7. Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations: Theory and algorithms. Springer-Verlag, Berlin Heidelberg (1986)

    Book  MATH  Google Scholar 

  8. He, Y.: The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data. Math. Comput. 77(264), 2097–2124 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. He, Y., Lin, Y., Sun, W.: Stabilized finite element method for the non-staionary Navier-Stokes problem, Discrete Contin. Dyn. Syst. Ser. B 6(1), 41–68 (2006)

    MathSciNet  Google Scholar 

  10. He, Y., Xu, J., Zhou, A.: Local and parallel finite element algorithms for the Navier-Stokes problem. J. Comput. Math. 24(3), 227–238 (2006)

    MathSciNet  MATH  Google Scholar 

  11. He, Y., Xu, J., Zhou, A., Li, J.: Local and parallel finite element algorithms for the Stokes problem. Numer. Math. 109(3), 415–434 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20 (3-4), 251–265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hood, P., Taylor, C.: A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1, 73–100 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. John, V.: Finite element methods for incompressible flow problems. Springer International Publishing (2016)

  18. Liu, X., Li, J., Chen, Z.: A weak Galerkin finite element method for the Oseen equations. Adv. Comput. Math. 42(6), 1–18 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ma, F., Ma, Y., Wo, W.: Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations. Appl. Math. Mech. 28(1), 27–35 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ma, Y., Zhang, Z., Ren, C.: Local and parallel finite element algorithms based on two-grid discretization for the stream function form of Navier-Stokes equations. Appl. Math. Comput. 175, 786–813 (2006)

    MathSciNet  MATH  Google Scholar 

  21. Massing, A., Schott, B., Wall, W.A.: A stabilized Nitsche cut finite element method for the Oseen problem. Comput. Method Appl. Mech. Eng. 328, 262–300 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mansfield, L.: Finite element subspaces with optimal rates of convergence for stationary Stokes problem. RAIRO Anal. Numer. 16, 49–66 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Okamoto, H.: On the semi discrete finite element approximation for the nonstationary Navier-Stokes equations. J. Fac. Sci. Univ. Tokyo 29, 613–665 (1982)

    MathSciNet  MATH  Google Scholar 

  24. Shang, Y., Wang, K.: Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations. Numer. Algor. 54(2), 195–218 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shen, L.: Parallel adaptive finite element algorithms for electronic structure computing based on density functional theory, PhD Thesis. Academy of Mathematics and Systems Science, Chinese Academy of Sciences (2005)

  26. Schatz, A.H., Wahlbin, L.B.: Interior maximum-norm estimates for finite element methods. Math. Comput. 31, 414–442 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schatz, A. H., Wahlbin, L. B.: Interior maximum-norm estimates for finite element methods, part II. Math. Comput. 64, 907–928 (1995)

    MATH  Google Scholar 

  28. Toselli, A., Widlund, O.: Domain decomposition methods: Algorithms and theory. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  29. Wahlbin, L.B.: Superconvergence in Galerkin finite element methods, lecture notes in math., vol. 1605 springer (1995)

  30. Winter, M., Schott, B., Massing, A.: A Nitsche cut finite element method for the Oseen problem with general Navier boundary conditions. Comput. Method Appl. Mech. Eng. 330, 220–252 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69(231), 881–909 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xu, J., Zhou, A.: Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems. Adv. Comput. Math. 14(4), 293–327 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Xu, J., Zhou, A.: Local and parallel finite element algorithms for eigenvalue problems. Acta. Math. Appl. Sin. Engl. Ser. 18, 185–200 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  34. Xu, C., Shi, D., Liao, X.: A new streamline diffusion finite element method for the generalized Oseen problem. Appl. Math. Mech. 39(2), 1–14 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zheng, B., Shang, Y.: Parallel iterative stabilized finite element algorithms based on the lowest equal-order elements for the stationary Navier-Stokes equations. Appl. Math. Comput. 357, 35–56 (2019)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to express their deep gratitude to the anonymous reviewers for their valuable comments and suggestions, which led to an improvement of the paper.

Funding

This work was supported by the Natural Science Foundation of China (No.11361016), the Basic and Frontier Explore Program of Chongqing Municipality, China (No. cstc2018jcyjAX0305), the Fundamental Research Funds for the Central Universities (No. XDJK2018B032), and the Graduate Research Innovation Project of Chongqing Municipality, China (No. CYS19085).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, People’s Republic of China

    Qi Ding, Bo Zheng & Yueqiang Shang

Authors
  1. Qi Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yueqiang Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yueqiang Shang.

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

Ding, Q., Zheng, B. & Shang, Y. Local and parallel finite element algorithms for the time-dependent Oseen equations. Numer Algor 87, 1653–1677 (2021). https://doi.org/10.1007/s11075-020-01024-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-01024-2

Keywords

Mathematics Subject Classification (2010)

Search

Navigation