Skip to main content
SpringerLink
Log in

Two-Level Penalty Newton Iterative Method for the 2D/3D Stationary Incompressible Magnetohydrodynamics Equations

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

Abstract

This work is concerned with the study of two-level penalty finite element method for the 2D/3D stationary incompressible magnetohydrodynamics equations. The new method is an interesting combination of the Newton iteration and two-level penalty finite element algorithm with two different finite element pairs \(P_{1}b\)-\(P_{1}\)-\(P_{1}b\) and \(P_{1}\)-\(P_{0}\)-\(P_{1}\). Moreover, the rigorous analysis of stability and error estimate for the proposed method are given. Numerical results verify the theoretical results and show the applicability and effectiveness of the presented scheme.

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
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Gunzburger, M., Meir, A., Peterson, J.: On the existence, uniquess and finite element approximation of solutions of the equations of sationary, incompressible magnetohydrodynamic. Math. Comput. 56, 523–563 (1991)

    Article  MATH  Google Scholar 

  2. Moreau, R.: Magneto-hydrodynamics. Kluwer Academic Publishers, Dordrecht (1990)

    Google Scholar 

  3. Cao, C., Wu, J.: Two regularity criteria for the 3D MHD equations. J. Differ. Equ. 248, 2263–2274 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. He, C., Wang, Y.: On the regularity criteria for weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 238, 1–17 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Schonbek, M., Schonbek, T., Solli, E.: Large-time behaviour of solutions to the magnetohydrodynamics equations. Math. Ann. 304, 717–756 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  7. Salah, N., Soulaimani, A., Habashi, W.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867–5892 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Codina, R., Silva, N.: Stabilized finite element approximation of the stationary magnetohydrodynamics equations. Comput. Mech. 38, 344–355 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sermange, M., Temam, R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gerbeau, J.: A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87, 83–111 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ravindran, S.: Linear feedback control and approximation for a system governed by unsteady MHD equations. Comput. Methods Appl. Mech. Eng. 198, 524–541 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Salah, N., Soulaimani, A., Habashi, W.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867–5892 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Meir, A., Schmidt, P.: Analysis and numerical approximation of a stationary MHD flow problem with nonideal boundary. SIAM J. Numer. Anal. 36, 1304–1332 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. He, Y., Li, J.: Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 198, 1351–1359 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Xu, H., He, Y.: Some iterative finite element methods for steady Navier–Stokes equations with different viscosities. J. Comput. Phys. 232, 123–152 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dong, X., He, Y., Zhang, Y.: Convergence analysis of three finite element iterative methods for the 2D/3D stationary incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 276, 287–311 (2014)

    Article  MathSciNet  Google Scholar 

  17. Dong, X., He, Y.: Two-level Newton iterative method for the 2D/3D stationary incompressible magnetohydrodynamics. J. Sci. Comput. 63, 426–451 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Su, H., Feng, X., Huang, P.: Iterative methods in penalty finite element discretization for the steady MHD equations. Comput. Methods Appl. Mech. Eng. 304, 521–545 (2016)

    Article  MathSciNet  Google Scholar 

  19. Shen, J.: On error estimates of some higher order projection and penalty-projection methods for Navier–Stokes equations. Numer. Math. 62, 49–73 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shen, J.: On error estimates of the penalty method for unsteady Navier–Stokes equations. SIAM J. Numer. Anal. 32, 386–403 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. He, Y.: Optimal error estimate of the penalty finite element method for the time-dependent Navier–Stokes equations. Math. Comput. 74, 1201–1216 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. He, Y., Li, J., Yang, X.: Two-level penalized finite element methods for the stationary Navier–Stoke equations. Int. J. Inf. Syst. Sci. 2, 131–143 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Xu, J.: A novel two two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Layton, W., Lenferink, H., Peterson, J.: A two-level Newton finite element algorithm for approximating electrically conducting incompressible fluid flows. Comput. Math. Appl. 28, 21–31 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Layton, W., Meir, A., Schmidtz, P.: A two-level discretization method for the stationary MHD equations. Electron. Trans. Numer. Anal. 6, 198–210 (1997)

    MathSciNet  MATH  Google Scholar 

  27. Zhang, G., Zhang, Y., He, Y.: Two-level coupled and decoupled parallel correction methods for stationary incompressible magnetohydrodynamics. J. Sci. Comput. 65, 920–939 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. He, Y.: Two-level method based on fnite element and Crank–Nicolson extrapolation for the time-dependent Navier–Stokes equations. SIAM J. Numer. Anal. 41, 1263–1285 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gerbeau, J., Le Bris, C., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2006)

    Book  MATH  Google Scholar 

  30. He, Y.: Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations. IMA J. Numer. Anal. 35, 767–801 (2014)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors sincerely thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this paper.

Author information

Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China

    Haiyan Su, Xinlong Feng & Jianping Zhao

Authors
  1. Haiyan Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xinlong Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianping Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinlong Feng.

Additional information

This work is supported by the Natural Science Foundation of Xinjiang Province (No. 2016D01C073).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Su, H., Feng, X. & Zhao, J. Two-Level Penalty Newton Iterative Method for the 2D/3D Stationary Incompressible Magnetohydrodynamics Equations. J Sci Comput 70, 1144–1179 (2017). https://doi.org/10.1007/s10915-016-0276-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-016-0276-8

Keywords

Mathematics Subject Classification

Search

Navigation