Skip to main content
SpringerLink
Log in

Galerkin-Petrov least squares mixed element method for stationary incompressible magnetohydrodynamics

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

Abstract

The Galerkin-Petrov least squares method is combined with the mixed finite element method to deal with the stationary, incompressible magnetohydrodynamics system of equations with viscosity. A Galerkin-Petrov least squares mixed finite element format for the stationary incompressible magnetohydrodynamics equations is presented. And the existence and error estimates of its solution are derived. Through this method, the combination among the mixed finite element spaces does not demand the discrete Babuška-Brezzi stability conditions so that the mixed finite element spaces could be chosen arbitrartily and the error estimates with optimal order could be obtained.

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. Jackson J D. Classical electrodynamics[M]. New York: Wiley, 1975.

    Google Scholar 

  2. Gunzburger M D, Meir A J, Peterson J S. On the existence, uniqueness, and finite element approximation of solution of the equation of stationary, incompressible magnetohydrodynamics[J]. Mathematics of Computation, 1991, 56(194):523–563.

    Article  MATH  MathSciNet  Google Scholar 

  3. Wiedmer M. Finite element approximation for equation of magnetohydrodynamics[J]. Mathematics of Computation, 2000, 69(229):83–101.

    Article  MATH  MathSciNet  Google Scholar 

  4. Girault V, Raviart P. A. Finite element methods for Navier-Stokes equations, theory and algorithms[M]. Berlin: Springer-Verlag, 1986.

    Google Scholar 

  5. Temam R. Navier-Stokes equation[M]. Amsterdam: North-Holland, 1984.

    Google Scholar 

  6. France L P, Hughes T J. Two classes of mixed finite element methods[J]. Computer Methods in Applied Methanics and Engineering, 1988, 69(1):89–129.

    Article  Google Scholar 

  7. Hughes T J, France L P, Balestra M. A new finite element formulation for computational fluid dynamics, V. circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulations of stokes problem accommodating equal-order interpolation[J]. Computer Methods in Applied Mechanics and Engineering, 1986, 59(1):85–99.

    Article  MATH  MathSciNet  Google Scholar 

  8. Hughes T J, France L P. A new finite element formulation for computational fluid dynamics, VII. the stokes problem with various well posed boundary condition, symmetric formulations that converge for all velocity pressure space[J]. Computer Methods in Applied Mechanics and Engineering, 1987, 65(2):85–96.

    Article  MATH  MathSciNet  Google Scholar 

  9. Brezzi F, Douglas J Jr. Stabilized mixed method for the Stokes problem[J]. Numerische Mathematik, 1988, 53(2):225–235.

    Article  MathSciNet  Google Scholar 

  10. Douglas J Jr, Wang J P. An absolutely stabilized finite element method for the Stokes problem[J]. Mathematics of Computation, 1989, 52(186):495–508.

    Article  MATH  MathSciNet  Google Scholar 

  11. Hughes T J, Tezduyar T E. Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equation[J]. Computer Methods in Applied Methanics and Engineering, 1984, 45(3):217–284.

    Article  MATH  MathSciNet  Google Scholar 

  12. Johson C, Saranen J. Streamline diffusion methods for the incompressible euler and Navier-Stokes equation[J]. Mathematics of Computation, 1986, 47(175):1–18.

    Article  MathSciNet  Google Scholar 

  13. Ciarlet P G. The finite element method for elliptic problem[M]. Amsterdam: North-Holland, 1978.

    Google Scholar 

  14. Luo Zhendong. Bases and applications of mixed finite element methods[M]. Beijing: Chinese Science Press, 2006 (in Chinese).

    Google Scholar 

  15. Zhou T X, Feng M F. A least squares Petrov-Galerkin finite element method for the stationary Navier-Stokes equations[J]. Mathematics of Computation, 1993, 60(202):531–543.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Beijing Jiaotong University, Beijing, 100044, P. R. China

    Luo Zhen-dong  (罗振东) (Professor, Doctor) & Mao Yun-kui  (毛允魁)

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

    Luo Zhen-dong  (罗振东) (Professor, Doctor) & Zhu Jiang  (朱江)

Authors
  1. Luo Zhen-dong  (罗振东)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mao Yun-kui  (毛允魁)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhu Jiang  (朱江)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luo Zhen-dong  (罗振东).

Additional information

Communicated by ZHOU Zhe-wei

Project supported by the National Natural Science Foundation of China (Nos. 10471100 and 40437017) and the Science and Technology Foundation of Beijing Jiaotong University

Rights and permissions

Reprints and permissions

About this article

Cite this article

Luo, Zd., Mao, Yk. & Zhu, J. Galerkin-Petrov least squares mixed element method for stationary incompressible magnetohydrodynamics. Appl Math Mech 28, 395–404 (2007). https://doi.org/10.1007/s10483-007-0312-x

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-007-0312-x

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation