[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Local and parallel finite element methods based on two-grid discretizations for the nonstationary Navier-Stokes equations

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

Abstract

In this paper, some local and parallel finite element methods based on two-grid discretizations are proposed and investigated for the unsteady Navier-Stokes equations. The backward Euler scheme is considered for the temporal discretization, and two-grid method is used for the space discretization. The key idea is that for a solution to the unsteady Navier-Stokes problem, we could use a relatively coarse mesh to approximate low-frequency components and use some local fine mesh to compute high-frequency components. Some local a priori estimate is obtained. With that, theoretical results are derived. Finally, some numerical results are reported to support the theoretical findings.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MATH  Google Scholar 

  2. Ciarlet, P. G., Lions, J. L.: Handbook of Numerical Analysis, vol. 2. Finite Element Methods (Part, vol. 1. Elsevier, Amsterdam (1991)

    Google Scholar 

  3. Du, G., Zuo, L.: Local and parallel finite element methods for the coupled Stokes/Darcy model. Numer. Algorithms. https://doi.org/10.1007/s11075-020-01021-5

  4. Du, G., Zuo, L.: Local and parallel finite element method for the mixed Navier-Stokes/Darcy model with Beavers-Joseph interface conditions. Acta. Math. Sci. 37, 1331–1347 (2017)

    Article  MathSciNet  Google Scholar 

  5. Du, G., Zuo, L.: Local and parallel finite element post-processing scheme for the Stokes problem. Comput. Math. Appl. 73, 129–140 (2017)

    Article  MathSciNet  Google Scholar 

  6. Du, G., Zuo, L.: A parallel partition of unity scheme based on Two-Grid discretizations for the Navier-Stokes problem. J. Sci. Comput. 75, 1445–1462 (2018)

    Article  MathSciNet  Google Scholar 

  7. Du, G., Zuo, L.: A two-grid parallel partition of unity finite element scheme. Numer. Algorithm. 80, 429–445 (2019)

    Article  MathSciNet  Google Scholar 

  8. Girault, V., Raviart, P. A.: Finite Element Approximation of the Navier-Stokes Equations. Springer, Berlin (1981)

    MATH  Google Scholar 

  9. Heywood, J. G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem 1: regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19(2), 275–311 (1982)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Heywood, J. G., Rannacher, R.: Finite element approximation of the nonstationary Navier-Stokes problem 4: error analysis for second-order time discretization. SIAM. J. Numer. Anal. 27(2), 353–384 (1990)

    Article  MathSciNet  Google Scholar 

  12. He, Y., Li, K.: Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations. Numer. Math. 79, 77–106 (1998)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. He, Y.: A two-level finite element Galerkin method for the nonstationary Navier-Stokes equations 2: time discretization. J. Comput. Math. 22(1), 33–54 (2004)

    Article  MathSciNet  Google Scholar 

  15. He, Y., Lin, Y., Sun, W.: Stabilized finite element method for the non-stationary Navier-Stokes problem. Discret. Contin. Dyn. Syst. Ser. B6(1), 41–68 (2006)

    MathSciNet  MATH  Google Scholar 

  16. 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 

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

    Article  MathSciNet  Google Scholar 

  18. Hou, Y., Du, G.: An expandable local and parallel Two-Grid finite element scheme. Comput. Math. Appl. 71, 2541–2556 (2016)

    Article  MathSciNet  Google Scholar 

  19. Layton, W., Ye, X.: Two-level discretizations of stream functional form of the Navier-Stokes equations. Numer. Funct. Anal. Optim. 20, 909–916 (1999)

    Article  MathSciNet  Google Scholar 

  20. Li, J., He, Y., Chen, Z.: A new stabilized finite element method for the transient Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 197(1), 22–35 (2007)

    Article  MathSciNet  Google Scholar 

  21. Liu, Q., Hou, Y.: Local and parallel finite element algorithms for the time-dependent convection diffusion equations. Appl. Math. Mech. -Engl. Edit. 30, 787–794 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Schatz, A. H., Wahlbin, L. B.: Interior maximum-norm estimates for finite element methods, part 2. Math. Comput. 64, 907–928 (1995)

    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. Algorithm. 54, 195–218 (2010)

    Article  MathSciNet  Google Scholar 

  25. Shang, Y., He, Y.: Parallel iterative finite element algorithms based on full domain partition for the stationary Navier-Stokes equations. Appl. Numer. Math. 60, 719–737 (2010)

    Article  MathSciNet  Google Scholar 

  26. Shang, Y.: A parallel finite element algorithm for simulation of the generalized stokes problem. B. Korean. Math. Soc. 53, 853–874 (2016)

    Article  MathSciNet  Google Scholar 

  27. Temam, R.: Navier-Stokes Equations, North-Holland (1984)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Yu, J., Shi, F., Zheng, H.: Local and parallel finite element algorithms based on the partition of unity for the stokes problem. SIAM J. Sci. Comput. 36, C547–C567 (2014)

    Article  MathSciNet  Google Scholar 

  31. Zheng, H., Yu, J., Shi, F.: Local and parallel finite element method based on the partition of unity for incompressible flow. J. Sci. Comput. 65, 512–532 (2015)

    Article  MathSciNet  Google Scholar 

  32. Zheng, H., Song, L., Hou, Y., et al.: The partition of unity parallel finite element algorithm. Adv. Comput. Math. 41, 937–951 (2015)

    Article  MathSciNet  Google Scholar 

  33. Zheng, H., Shi, F., Hou, Y., et al.: New local and parallel finite element algorithm based on the partition of unity. J. Math. Anal. Appl. 435, 1–19 (2016)

    Article  MathSciNet  Google Scholar 

  34. Zuo, L., Du, G.: A parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem. Numer. Algorithm. 77, 151–165 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work is subsidized by NSFC (Grant No. 11701343).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Qingtao Li & Guangzhi Du

Authors
  1. Qingtao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guangzhi Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guangzhi Du.

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

Li, Q., Du, G. Local and parallel finite element methods based on two-grid discretizations for the nonstationary Navier-Stokes equations. Numer Algor 88, 1915–1936 (2021). https://doi.org/10.1007/s11075-021-01100-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-021-01100-1

Keywords

Search

Navigation