Skip to main content
SpringerLink
Log in

A nonlinear galerkin mixed element method and a posteriori error estimator for the stationary navier-stokes equations

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

Abstract

A nonlinear Galerkin mixed element (NGME) method and a posteriori error exstimator based on the method are established for the stationary Navier-Stokes equations. The existence and error estimates of the NGME solution are first discussed, and then a posteriori error estimator based on the NGME method is derived.

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. Foias C, Manley O P, Temam R. Modellization of the interaction of small and large eddies in two dimensional turbulent flows[J].Math Mod Numer Anal, 1988,22(1):93–114.

    MATH  MathSciNet  Google Scholar 

  2. Marion M, Temam R. Nonlinear Galerkin methods[J].SIAM J Numer Anal, 1989,2(5):1139–1157.

    Article  MathSciNet  Google Scholar 

  3. Foias C, Jolly M, Kevrekidis I G, et al. Dissipativy of numerical schemes[J].Nonlinearity, 1991,4(4):591–613.

    Article  MATH  MathSciNet  Google Scholar 

  4. Devulder C, Marion M, Titi E. On the rate of convergence of nonlinear Galerkin methods[J].Math Comp, 1992,59(200):173–201.

    Google Scholar 

  5. Marion M, Temam R. Nonlinear Galerkin methods: the finite elements case[J].Numer Math, 1990,57(3):205–226.

    Article  MATH  MathSciNet  Google Scholar 

  6. Marion M, Xu J C. Error estimates on a new nonlinear Galerkin method based on two-grid finite elements[J].SIAM J Numer Anal, 1995,32(4):1170–1184.

    Article  MATH  MathSciNet  Google Scholar 

  7. Ait Ou Ammi A, Marion M. Nonlinear Galerkin methods and mixed finite element: two-grid algorithms for the Navier-Stokes equations[J].Numer Math, 1994,68(2):189–213.

    Article  MathSciNet  Google Scholar 

  8. Li K T, Zhou L. Finite element nonlinear Galerkin methods for penalty Navier-Stokes equations [J].Math Numer Sinica, 1995,17(4):360–380.

    MATH  MathSciNet  Google Scholar 

  9. He Y, Li K T. Nonlinear Galerkin method and two-step method for the Navier-Stokes equations [J].Inc Numer Methods P D Eq, 1996,12(3):283–305.

    Article  MATH  MathSciNet  Google Scholar 

  10. Luo Z D, Wang L H. Nonlinear Galerkin mixed element methods for the non stationary conduction-convection problems (I): The continuous-time case[J].Chinese J Numer Math Appl, 1998,20 (4):71–94.

    MATH  MathSciNet  Google Scholar 

  11. Luo Z D, Wang L H. Nonlinear Galerkin mixed element methods for the non stationary conduction-convection problems (II): The backward one-step Euler fully discrete format[J].Chinese J Numer Math Appl, 1999,21(1):86–105.

    MathSciNet  Google Scholar 

  12. LUO Zhen-dong, ZHU Jiang, WANG Hui-jun. A nonlinear Galerkin/Petrov-least squares mixed element method for the stationary Navier-Stokes equations[J].Applied Mathematics and Mechanics (English Edition), 2002,23(7):697–708.

    MathSciNet  Google Scholar 

  13. Li G C, He Y N. Convergence of nonlinear Galerkin finite element algorithm for the steady incompressible equations of the Navier-Stokes type[J].Chinese J Comput Phys, 1997,14(1):83–89.

    MATH  Google Scholar 

  14. Bank R E, Welfert B. A posteriori error estimates for the Stokes equations: A comparison[J].Comput Methods Appl Mech Engrg, 1990,82(3):323–340.

    Article  MATH  MathSciNet  Google Scholar 

  15. Bank R E, Welfert B. A posteriori error estimates for the Stokes problems[J].SIAM J Numer Anal, 1991,28(3):591–623.

    Article  MATH  MathSciNet  Google Scholar 

  16. Oden J T, Demkowicz L.h-p adaptive finite element methods in computational fluid dynamics[J].Comput Methods Appl Mech Engrg, 1991,89(1):11–40.

    Article  MathSciNet  Google Scholar 

  17. Padra C, Buscaglia G C, Dari E A. Adaptivity in steady incompressible Navier-Stokes equations using discontinuous pressure interpolates[A]. In: H Alder, J C Heinrich, S Lavanchy, et al Eds.Num Meth Eng Appl Sci, Part I [C]. Barcelona: CIMNE, 1992,267–276.

    Google Scholar 

  18. Wu J, Zhu J Z, Szmelter J, et al. Error estimation and adaptivity in Navier-Stokes incompressible flows[J].Comput Mech, 1990,6(2):259–270.

    Article  MATH  Google Scholar 

  19. Verfürth R. A posteriori error estimators for the Stokes equations[J].Numer Math, 1989,55(3): 309–325.

    Article  MATH  MathSciNet  Google Scholar 

  20. Verfürth R. A posteriori error estimators and adaptive mesh-refinement techniques for the Navier-Stokes equations[A]. In: M Gunzburger, R A Nicolaides Eds.Incompressible Computational Fluid Dynamics, Trends and Advances[C]. Cambridge: Cambridge University Press, 1993, 447–477.

    Google Scholar 

  21. Verfürth R. A posteriori error estimates for non-linear problems: Finite element discretizations of elliptic equations[J].Math Comp, 1994,62(206):445–475.

    Article  MATH  MathSciNet  Google Scholar 

  22. Oden J T, Wu W, Ainsworth M. An a posteriori error estimate for finite element approximations of the Navier-Stokes equations[J].Comput Methods Appl Mech Engrg, 1994,111(2):185–220.

    Article  MATH  MathSciNet  Google Scholar 

  23. Arnica D, Padra C. A posteriori error estimators for steady incompressible Navier-Stokes equations [J].Inc Numer Methods P D Eq, 1997,13(5):561–574.

    Article  MATH  MathSciNet  Google Scholar 

  24. Ervin V, Layton W, Maubach J. A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations[J].Inc Numer Methods P D Eq, 1996,12(3):333–346.

    Article  MATH  MathSciNet  Google Scholar 

  25. Girault V, Raviart P A.Finite Element Approximations of the Navier-Stokes Equations, Theorem and Algorithms[M]. New York: Springer-Verlag, 1986.

    Google Scholar 

  26. Temam R.Navier-Stokes Equations[M]. Amsterdam: North-Holland, 1984.

    Google Scholar 

  27. Bernardi C, Raugel B. Analysis of some finite elements for the Stokes problem[J].Math Comp, 1985,44(169):71–79.

    Article  MATH  MathSciNet  Google Scholar 

  28. LUO Zhen-dong.Theory Bases and Applications of Finite Element and Mixed Finite Element Methods[M]. Jinan: Shandong Educational Press, 1996. (in Chinese)

    Google Scholar 

  29. Ciarlet P G.The Finite Element Method for Elliptic Problems[M]. Amsterdam: North-Holland, 1978.

    Google Scholar 

  30. Luo Z D. The third order estimate of mixed finite element for the Navier-Stokes problems[J].Chinese Quart J Math, 1995,10(3):9–12.

    MATH  Google Scholar 

  31. Clément P. Approximation by finite element function using local regularization[J].RAIRO, 1975,R-2(1):77–84.

    MATH  Google Scholar 

  32. Layton W. A two level discretization method for the Navier-Stokes equations[J].Comput Math Appl, 1993,26(1):33–45.

    Article  MATH  MathSciNet  Google Scholar 

  33. Brezzi F, Fortin M.Mixed and Hybrid Finite Element Methods[M]. New York: Berlin, Heidelberg: Springer-Verlag, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Capital Normal University, 100037, Beijing, P R China

    Luo Zhen-dong Professor, Doctor

  2. ICCES, Institute of Atmospheric Physics, Chinese Academy of Sciences, 100029, Beijing, P R China

    Luo Zhen-dong Professor, Doctor & Zhu Jiang

Authors
  1. Luo Zhen-dong Professor, Doctor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhu Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by XU Zheng-fan

Foundation items: the National Natural Science Foundation of China (10071052, 49776283); the Evolvement Plan of Science and Technology of Beijing Educational Council; the “Plan of Hundreds Persons” of Chinese Academy of Sciences (K2952-51-434); the Beijing Excellent Talent Special Foundation; the Municipal Science Foundation of Beijing

Biography: LUO Zhen-dong (1958-)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhen-dong, L., Jiang, Z. A nonlinear galerkin mixed element method and a posteriori error estimator for the stationary navier-stokes equations. Appl Math Mech 23, 1194–1206 (2002). https://doi.org/10.1007/BF02437668

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02437668

Key words

CLC number

Search

Navigation