Skip to main content
SpringerLink
Log in

Adaptive Local Postprocessing Finite Element Method for the Navier-Stokes Equations

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

Abstract

An adaptive local postprocessing finite element method for the Navier-Stokes equations is presented in this paper. We firstly solve the problem on a relative coarse grid to get a rough approximation. Then, we correct the rough approximation by solving a series of approximate local residual equations defined on some local fine grids, which can be implemented in parallel. In addition, we also propose a reliable local a posteriori error estimator and construct an adaptive algorithm based on the corresponding a posterior error estimate. Finally, some numerical examples are presented to verify the algorithm.

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
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. García-Archilla, B., Novo, J., Titi, E.S.: Postprocessing the Galerkin method: a novel approach to approximate inertial manifolds. SIAM J. Numer. Anal. 35, 941–972 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. García-Archilla, B., Novo, J., Titi, E.S.: An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations. Math. Comput. 68, 893–911 (1999)

    Article  MATH  Google Scholar 

  3. García-Archilla, B., Titi, E.S.: Postprocessing the Galerkin method: the finite-element case. SIAM J. Numer. Anal. 37, 470–499 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hou, Y., Li, K.: Tangent space correction method for the Galerkin approximation based on two-grid finite element. Appl. Math. Comput. 175, 413–429 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hou, Y., Li, K.: Postprocessing Fourier Galerkin method for the Navier-Stokes equations. SIAM J. Numer. Anal. 47, 1909–1922 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Li, K., Hou, Y.: An AIM and one-step Newton method for the Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 190, 6141–6155 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. He, Y., Mei, L., Shang, Y., Cui, J.: Newton iterative parallel nite element algorithm for the steady Navier-Stokes equations. J. Sci. Comput. 44, 92–106 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Shang, Y., He, Y., Luo, Z.: A comparison of three kinds of local and parallel nite element algorithms based on two-grid discretizations for the stationary Navier-Stokes equations. Comput. Fluids 40, 249–257 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Larson, M., Målqvist, A.: Adaptive variational multiscale methods based on a posteriori error estimation: energy norm estimates for elliptic problems. Comput. Methods Appl. Mech. Eng. 196, 2313–2324 (2007)

    Article  MATH  Google Scholar 

  12. Larson, M., Målqvist, A.: An adaptive variational multiscale method for convection-diffusion problems. Commun. Numer. Methods Eng. 25, 65–79 (2009)

    Article  MATH  Google Scholar 

  13. Hughes, T., Engel, G., Mazzei, L., Larson, M.: The continuous Galerkin method is locally conservative. J. Comput. Phys. 163, 467–488 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. FreeFem++. http://www.freefem.org/ff++/ftp/

  15. Zheng, H., Hou, Y., Shi, F., Song, L.: A finite element variational multiscale method for incompressible flows based on two local Gauss integrations. J. Comput. Phys. 228, 5961–5971 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ghia, U., Ghia, K.N., Shin, C.T.: High-resolutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48, 387–411 (1982)

    Article  MATH  Google Scholar 

  17. Gravemeier, V., Wall, W.A., Ramm, E.: A three-level finite element method for the instationary incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 193, 1323–1366 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Supported by NSF of China (Grant No. 11171269 and 11001216) and PhD Programs Foundation of Ministry of Education of China (Grant No. 20110201110027).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Lina Song, Yanren Hou & Haibiao Zheng

  2. Center for Computational Geosciences, Xi’an Jiaotong University, Xi’an, 710049, China

    Yanren Hou

Authors
  1. Lina Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yanren Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haibiao Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanren Hou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Song, L., Hou, Y. & Zheng, H. Adaptive Local Postprocessing Finite Element Method for the Navier-Stokes Equations. J Sci Comput 55, 255–267 (2013). https://doi.org/10.1007/s10915-012-9631-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9631-6

Keywords

Search

Navigation