Skip to main content
SpringerLink
Log in

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming (P 1)2 − P 0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.

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

Similar content being viewed by others

References

  1. Hughes, T. J. R. and Brooks, A. N. A multi-dimensional up-wind scheme with no crosswind diffusion. Finite Element Methods for Convection-Dominated Flows (ed. Hughes, T. J. R.), ASME Monograph AMD-34, ASME, New York, 19–35 (1979)

    Google Scholar 

  2. Nävert, U. A Finite Element for Convection-Diffusion Problems, Ph. D. dissertation, Chalmers University of Technology, Goteborg (1982)

    Google Scholar 

  3. Johnson, C. Finite element methods for convection-diffusion problems. Computing Methods in Engineering and Applied Sciences (eds. Glowinski, R. and Lions, J. L.), North-Holland, Amsterdam (1981)

    Google Scholar 

  4. Johnson, C. and Nävert, U. An analysis of some finite element methods for advection-diffusion. Analytical and Numerical Approaches to Asymptotic Problems in Analysis (eds. Axelsson, O., Frank, L. S., and Van der Sluis, A.), North-Holland, Amsterdam, 99–118 (1981)

    Chapter  Google Scholar 

  5. Johnson, C., Nävert, U., and Pitkaranta, J. Finite element methods for linear hyperbolic problems. Comput. Methods Appl. Mech. Engrg. 45(1–3), 285–312 (1985)

    Google Scholar 

  6. Johnson, C. and Saranen, J. Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations. Mathematics of Computation 47(175), 1–18 (1986)

    MATH  MathSciNet  Google Scholar 

  7. Tobiska, L. and Verfürth, R. Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations. SIAM J. Numer. Anal. 33(1), 107–127 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Zhou, G. H. How accurate is the streamline diffusion finite element method? J. Comput. Math. 66(217), 31–44 (1997)

    Article  MATH  Google Scholar 

  9. Sun, C. and Shen, H. The finite difference streamline diffusion method for time-dependent convection-diffusion equations. Numerical Mathematics: A Journal of Chinese Universities, English Series 7(1), 72–85 (1998)

    MATH  MathSciNet  Google Scholar 

  10. Zhang, Q. Finite difference streamline diffusion method for incompressible N-S equations (in Chinese). Mathematica Numerica Sinica 25(3), 311–320 (2003)

    MathSciNet  Google Scholar 

  11. Zhang, Q. and Sun, C. Finite difference streamline diffusion method for nonlinear convection-diffusion equation (in Chinese). Mathematica Numerica Sinica 20(2), 211–224 (1998)

    Google Scholar 

  12. Sun, T. J. and Ma, K. Y. The finite difference streamline diffusion method for the incompressible Navier-Stokes equations. Appl. Math. Comput. 149(2), 493–505 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. John, V., Maubach, J., and Tobiska, L. Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer. Math. 78(2), 165–188 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. John, V., Matthies, G., Schiewech, F., and Tobiska, L. A streamline-diffusion method for nonconforming finite element approximations applied to convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 166(1–2), 85–97 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Matthies, G. and Tobiska, L. The streamline-diffusion method for conforming and nonconforming finite elements of lowesr order applied to convection-diffusion peoblems. Computing 66(4), 343–364 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Knobloch, P. and Tobiska, L. A streamline diffusion method for nonconforming finite element approximations applied to the linearized incompressible Navier-Stokes equations. Proceedings of the 4th International Conference on Numerical Methods and Applications (eds. Iliev, O. P., Kaschiev, M. S., Margenov, S. D., Sendov, B. H., and Vassilevski, P. S.), World Scientific, Singapore, 530–538 (1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Sichuan University, Chengdu, 610064, P. R. China

    Yu-mei Chen  (陈豫眉) & Xiao-ping Xie  (谢小平)

  2. College of Mathematics and Information, China West Normal University, Nanchong, 637002, Sichuan Province, P. R. China

    Yu-mei Chen  (陈豫眉)

Authors
  1. Yu-mei Chen  (陈豫眉)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiao-ping Xie  (谢小平)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-ping Xie  (谢小平).

Additional information

Communicated by Zhe-wei ZHOU

Project supported by the National Natural Science Foundation of China (No. 10771150), the National Basic Research Program of China (No. 2005CB321701), the Program for New Century Excellent Talents in University (No. NCET-07-0584), and the Natural Science Foundation of Sichuan Province (No. 07ZB087)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, Ym., Xie, Xp. A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations. Appl. Math. Mech.-Engl. Ed. 31, 861–874 (2010). https://doi.org/10.1007/s10483-010-1320-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-010-1320-z

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation