Skip to main content
SpringerLink
Log in

A new stabilized method for quasi-Newtonian flows

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

Abstract

For a generalized quasi-Newtonian flow, a new stabilized method focused on the low-order velocity-pressure pairs, (bi)linear/(bi)linear and (bi)linear/constant element, is presented. The pressure projection stabilized method is extended from Stokes problems to quasi-Newtonian flow problems. The theoretical framework developed here yields an estimate bound, which measures error in the approximate velocity in the W 1,r(Ω) norm and that of the pressure in the L r′(Ω) (1/r + 1/r′ = 1). The power law model and the Carreau model are special ones of the quasi-Newtonian flow problem discussed in this paper. Moreover, a residual-based posterior bound is given. Numerical experiments are presented to confirm the theoretical results.

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. Barrett, J. W. and Liu, W. B. Finite element error anaysis of a quasi-Newtonian flow obeying the Carreau or power law. Numer. Math. 64, 433–453 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barrett, J. W. and Liu, W. B. Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow. Numer. Math. 68, 437–456 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brezzi, F. and Douglas, J. Stabilized mixed methods for the Stokes problem. Numer. Math. 53, 225–235 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Hansbo, P. and Szepessy, A. A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 84, 175–192 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5. Zhou, T. X. and Feng, M. F. A least squares Petro-Galerkin finite element method for the stationary Navier-Stokes equations. Math. Comp. 60, 531–543 (1993)

    MATH  MathSciNet  Google Scholar 

  6. Zhou, L. and Zhou, T. X. Finite element method for a three-fields model for quasi-Newtonian flow. Mathematica Numerica Sinica 3, 305–312 (1997)

    Google Scholar 

  7. Hughes, T. J. R., Mazzei, L., Oberai, A. A., and Wray, A. A. The multiscale formulation of large eddy simulation: decay of homogeneous isotropic turbulence. Phys. Fluids 13, 505–512 (2001)

    Article  Google Scholar 

  8. Li, J. and He, Y. N. A stabilized finite element method based on two local Gauss integrations for the Stokes equations. J. Comput. Appl. Math. 214, 58–65 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. He, Y. N. and Li, J. A stabilized finite element method based on local polynomial pressure projection for the stationary Navier-Stokes equations. Appl. Numer. Math. 58, 1503–1514 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Horgan, C. O. Korn’s inequalities and their applications in continuum mechanics. SIAM Review 37, 491–511 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mosolov, P. P. and Myasnikov, V. P. A proof of Korn’s inequality. Soviet Math. Dokl. 12, 1618–1622 (1971)

    MATH  Google Scholar 

  14. Baranger, J. and Najib, K. Analyse numérique des écoulements quasi-Newtoniens dont la viscosité obéit á la loi puissance ou la loi de Carreau. Numer. Math. 58, 35–49 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  15. Berrone, S. and Süli, E. Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows. IMA J. Numer. Anal. 28, 382–421 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dohrmann, C. R. and Bochev, P. B. A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Meth. Fluids 46, 183–201 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Baranger, J., Najib, K., and Sandri, D. Numerical anlysis of a three-fields model for a quasi-Newtonian flow. Comput. Methods Appl. Mech. Engrg. 109, 281–292 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Mu, J. and Feng, M. F. Numerical anlysis of an FEM for a transient viscoelastic flow. Numerical Mathematics: A Journal of Chinese Universities 2, 150–165 (2004)

    MathSciNet  Google Scholar 

  19. Zhou, L. and Zhou, T. X. Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows nonlinear model. J. Comput. Appl. Math. 81, 19–28 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ge, Z. H., Feng, M. F., and He, Y. N. A stabilized nonconfirming finite element method based on multiscale enrichment for the stationary Navier-Stokes equations. Appl. Math. Comput. 202, 700–707 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, 610054, P. R. China

    Chun-mei Xie  (谢春梅) & Min-fu Feng  (冯民富)

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

    Min-fu Feng  (冯民富)

Authors
  1. Chun-mei Xie  (谢春梅)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Min-fu Feng  (冯民富)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Min-fu Feng  (冯民富).

Additional information

Communicated by Xing-ming GUO

Project supported by the Key Technology Research and Development Program of Sichuan Province of China (No. 05GG006-006-2)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xie, Cm., Feng, Mf. A new stabilized method for quasi-Newtonian flows. Appl. Math. Mech.-Engl. Ed. 31, 1081–1096 (2010). https://doi.org/10.1007/s10483-010-1344-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-010-1344-z

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation