Skip to main content
Springer Nature Link
Log in

On Two-Level Oseen Penalty Iteration Methods for the 2D/3D Stationary Incompressible Magnetohydronamics

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

Abstract

This paper studies several decoupled penalty methods to overcome the saddle point system of the steady state 2D/3D incompressible magnetohydronamics (MHD). These approaches combine the Oseen iteration and two-level technique with strong uniqueness condition \(0<\frac{\sqrt{2}C_{0}^{2}\max \{1,\sqrt{2}S_{c}\}\Vert {\mathbf{F }}\Vert _{-1}}{(\min \{R_{e}^{-1},S_{c}C_{1}R_{m}^{-1}\})^2}\le 1-\left( \frac{\Vert \mathbf{F }\Vert |_{-1}}{\Vert |\mathbf{F }\Vert _{0}}\right) ^{\frac{1}{2}}<1\) satisfied. For the convenience of implementation, we employ two different simple Lagrange finite element pairs \(P_{1}b-P_{1}-P_{1}b\) and \(P_{1}-P_{0}-P_{1}\) for velocity field, pressure and magnetic field, respectively. Rigorous analysis of the optimal error estimate and stability are provided. We present comprehensive numerical experiments, which indicate the effectiveness of the proposed methods for both two dimensional and three-dimensional problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

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

    Book  Google Scholar 

  2. 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  Google Scholar 

  3. Moreau, R.: Magneto-Hydrodynamics. Kluwer Academic Publishers, Berlin (1990)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Hu, K., Ma, Y., Xu, J.: Stable finite element methods preserving \(\nabla \cdot {B}=0\) exactly for MHD models. Numer. Math. 135, 371–396 (2017)

    Article  MathSciNet  Google Scholar 

  6. Zhao, J., Mao, S., Zheng, W.: Anisotropic adaptive finite element method for magnetohydrodynamic flow at high Hartmann numbers. Appl. Math. Mech. 37, 1479–1500 (2016)

    Article  MathSciNet  Google Scholar 

  7. Li, L., Zheng, W.: A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D. J. Comput. Phys. 351, 254–270 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. 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  Google Scholar 

  12. Qiu, W., Shi, K.: A mixed DG method and an HDG method for incompressible magnetohydrodynamics. arXiv:1702.01473 [math.NA]

  13. Gao, H., Qiu, W.: A linearized energy preserving finite element method for the dynamical incompressible magnetohydrodynamics equations. arXiv:1801.01252 [math.NA]

  14. Badia, S., Codina, R., Planas, R.: Analysis of an unconditionally convergent stabilized finite element formulation for incompressible magnetohydrodynamics. Arch. Comput. Methods Eng. 22, 621–636 (2015)

    Article  MathSciNet  Google Scholar 

  15. Greif, C., Li, D., Schötz, D., Wei, X.: A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Comput. Method Appl. Mech. Eng. 199, 45–48 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Su, H., Feng, X., Zhao, J.: Two-level penalty Newton iterative method for the 2D/3D stationary incompressible magnetohydrodynamics equations. J. Sci. Comput. 70(3), 1144–1179 (2017)

    Article  MathSciNet  Google Scholar 

  18. Su, H., Mao, S., Feng, X.: Optimal error estimates of penalty based iterative methods for steady incompressible magnetohydrodynamics equations with different viscosities. J. Sci. Comput. 79(2), 1078–1110 (2019)

    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  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  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  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. An, R., Li, Y.: Error analysis of first-order projection method for time-dependent magnetohydrodynamics equations. Appl. Numer. Math. 112, 167–181 (2017)

    Article  MathSciNet  Google Scholar 

  24. Lu, X., Huang, P.: A modular grad-div stabilization for the 2D/3D nonstationary incompressible magnetohydrodynamic equations. J. Sci. Comput. (2020). https://doi.org/10.1007/s10915-019-01114-x

    Article  MathSciNet  MATH  Google Scholar 

  25. Wu, J., Liu, D., Feng, X., Huang, P.: An efficient two-step algorithm for the stationary incompressible magnetohydrodynamic equations. Appl. Math. Comput. 302, 21–33 (2017)

    MathSciNet  MATH  Google Scholar 

  26. Wang, L., Li, J., Huang, P.: An efficient two-level algorithm for the 2D/3D stationary incompressible magnetohydrodynamics based on the finite element method. Int. Commun. Heat Mass Transfer 98, 183–190 (2018)

    Article  Google Scholar 

  27. Zhu, T., Su, H., Feng, X.: Some Uzawa-type finite element iterative methods for the steady incompressible magnetohydrodynamic equations. Appl. Math. Comput. 302, 34–47 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Zhang, Q., Su, H., Feng, X.: A partitioned finite element scheme based on Gauge–Uzawa method for time-dependent MHD equations. Numer. Algorithms 78(1), 277–295 (2018)

    Article  MathSciNet  Google Scholar 

  29. Ping, Y., Su, H., Feng, X.: Parallel two-step finite element algorithm for the stationary incompressible magnetohydrodynamic equations. Int. J. Numer. Methods Heat Fluid Flow 29(8), 2709–2727 (2019)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  32. 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 

  33. 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  Google Scholar 

  34. Girault, V., Raviart, P.A.: Finite Element Method for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1987)

    MATH  Google Scholar 

  35. 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  Google Scholar 

Download references

Acknowledgements

This work is in part supported by the NSF of China (Grant Nos. 11701493) and the GRF of Hong Kong (Grant Nos. 9041980, 9042081)” shoud be replaced by “This work is in part supported by the NSF of Xinjiang Uygur Autonomous Region (Nos. XJEDU2020I001, 2016D01C073, 2019D01C047), the NSF of China (Nos. 11701493, 61962056)”, Tianshan Youth Project of Xinjiang Uygur Autonomous Region (No.2017Q079), 2019 Autonomous Region University Research Program (No.XJEDU2019Y002), the Xinjiang Provincial University Research Foundation of China (No.XJEDU2018I002) and the Matching Projects for Study Abroad of People’s Goverment of Xinjiang Uygur Autonomous Region (No. 2219-51160000313).

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 inPubMed Google Scholar

  2. Xinlong Feng
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Jianping Zhao
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Xinlong Feng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is in part supported by the NSF of China (Grant Nos. 11701493) and the GRF of Hong Kong (Grant Nos. 9041980, 9042081).

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. On Two-Level Oseen Penalty Iteration Methods for the 2D/3D Stationary Incompressible Magnetohydronamics. J Sci Comput 83, 11 (2020). https://doi.org/10.1007/s10915-020-01186-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01186-0

Keywords

Mathematics Subject Classification