Skip to main content
Springer Nature Link
Log in

The Arrow–Hurwicz Iterative Finite Element Method for the Stationary Thermally Coupled Incompressible Magnetohydrodynamics Flow

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

Abstract

In this paper, we construct the Arrow–Hurwicz iterative finite element method for solving the stationary thermally coupled incompressible magnetohydrodynamics system, where a decoupled discrete system is obtained and no saddle point problem is required to deal with at each iterative step. Under several conditions, it is proved that the iterative solution solved by the proposed iterative method is convergent. Finally, the effectiveness of the considered iterative method is illustrated with some numerical examples.

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

Similar content being viewed by others

Data Availability

Not applicable.

References

  1. Arrow, K., Hurwicz, L., Uzawa, H.: Studies in Nonlinear Programming. Standford University Press, Standford (1958)

    MATH  Google Scholar 

  2. Badia, S., Hernández, N.: Approximation of the thermally coupled MHD problem using a stabilized finite element method. J. Comput. Phys. 230, 1281–1303 (2011)

    Article  MathSciNet  Google Scholar 

  3. Badia, S., Martína, A.F., Planas, R.: Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem. J. Comput. Phys. 274, 562–591 (2014)

    Article  MathSciNet  Google Scholar 

  4. Bermúdez, A., Muñoz-Sola, R., Vázquez, R.: Analysis of two stationary magnetohydrodynamics systems of equations including Joule heating. J. Math. Anal. Appl. 368, 444–468 (2010)

    Article  MathSciNet  Google Scholar 

  5. Boland, J., Layton, W.: Error analysis for finite element methods for steady natural convection problems. Numer. Funct. Anal. Optimiz. 11, 449–483 (1990)

    Article  MathSciNet  Google Scholar 

  6. Bramble, J.H., Pasciak, J.E., Vassilev, A.T.: Analysis of the inexact Uzawa algorithm for saddle point problem. SIAM J. Numer. Anal. 34, 1072–1092 (1997)

    Article  MathSciNet  Google Scholar 

  7. Chen, P., Huang, J.: On the geometric convergence of the Arrow-Hurwicz algorithm for steady incompressible Navier-Stokes equations. J. Comput. Anal. Appl. 18, 628–635 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Chen, P., Huang, J., Sheng, H.: Some Uzawa methods for steady incompressible Navier-Stokes equations discretized by mixed element methods. J. Comput. Appl. Math. 273, 313–325 (2015)

    Article  MathSciNet  Google Scholar 

  9. Chen, P., Huang, J., Sheng, H.: Solving steady incompressible Navier-Stokes equations by the Arrow-Hurwicz method. J. Comput. Appl. Math. 311, 100–114 (2017)

    Article  MathSciNet  Google Scholar 

  10. Degond, P., Ferreira, M.A., Motsch, S.: Damped Arrow-Hurwicz algorithm for sphere packing. J. Comput. Phys. 332, 47–65 (2017)

    Article  MathSciNet  Google Scholar 

  11. Ding, Q.Q., Long, X.N., Mao, S.P.: Convergence analysis of Crank-Nicolson extrapolated fully discrete scheme for thermally coupled incompressible magnetohydrodynamic system. Appl. Numer. Math. 157, 522–543 (2020)

    Article  MathSciNet  Google Scholar 

  12. Dong, X.J., He, Y.N.: Convergence of some finite element iterative methods related to different Reynolds numbers for the 2D/3D stationary incompressible magnetohydrodynamics. Sci. China Math. 59, 589–608 (2016)

    Article  MathSciNet  Google Scholar 

  13. Du, B.B., Huang, J.G.: The generalized Arrow-Hurwicz method with applications to fluid computation, Commun. Comput. Phys. 25, 752–780 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Du, B.B., Huang, J.G., Zheng, H.B.: Two-Grid Arrow-Hurwicz methods for the steady incompressible Navier-Stokes equations. J. Sci. Comput. 89, 24 (2021)

    Article  MathSciNet  Google Scholar 

  15. Gerbeau, J.F., Le Bris, C., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Oxford University Press, Oxford (2006)

    Book  Google Scholar 

  16. Girault, V., Raviart, P.A.: Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, New York (1986)

    Book  Google Scholar 

  17. Gunzburger, M.D., Meir, A.J., Peterson, J.S.: On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics. Math. Comp. 56, 523–563 (1991)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Huang, P.Z., He, Y.N.: A Uzawa-type algorithm for the coupled Stokes equations. Appl. Math. Mech. 41, 1095–1104 (2020)

    Article  MathSciNet  Google Scholar 

  20. Huang, P.Z.: Convergence of the Uzawa method for the Stokes equations with damping. Complex Var. Elliptic Equ. 62, 876–886 (2017)

    Article  MathSciNet  Google Scholar 

  21. Li, X., Huang, P.: An Uzawa iterative method for the natural convection problem based on mixed finite element method. Math. Methods Appl. Sci. 44, 13326–13343 (2021)

    Article  MathSciNet  Google Scholar 

  22. Meir, A.J.: Thermally coupled magnetohydynamics flow. Appl. Math. Comput. 65, 79–94 (1994)

    MathSciNet  MATH  Google Scholar 

  23. Meir, A.J.: Thermally coupled, stationary, incompressible MHD flow; existence uniqueness, and finite element approximation. Numer. Meth. Part. Differ. Equs. 11, 311–337 (1995)

    Article  MathSciNet  Google Scholar 

  24. Meir, A.J., Schmidt, P.G.: On electronmagnetically and thermally driven liquid-metal flows. Nonliear Anal. 47, 3281–3294 (2001)

    Article  Google Scholar 

  25. Nochetto, R.H., Pyo, J.H.: Optimal relaxation parameter for the Uzawa method. Numer. Math. 98, 695–702 (2004)

    Article  MathSciNet  Google Scholar 

  26. Olshanskii, M.A.: An iterative solver for the Oseen problem and numerical solution of incompressible Navier-Stokes equations. Numer. Linear Algebra Appl. 6, 353–378 (1999)

    Article  MathSciNet  Google Scholar 

  27. Queck, W.: The convergence factor of preconditioned algorithms of the Arrow-Hurwicz algorithms. SIAM J. Numer. Anal. 26, 1016–1030 (1989)

    Article  MathSciNet  Google Scholar 

  28. Ravindran, S.S.: A decoupled Crank-Nicolson time-stepping scheme for thermaly coupled magneto-hydrodynamic system. Int. J. Optimiz. Control Theories Appl. 8, 2146–5703 (2018)

    Google Scholar 

  29. Ravindran, S.S.: Partitioned time-stepping scheme for an MHD system with temperature-dependent coefficients. IMA J. Numer. Anal. 39, 1860–1887 (2019)

    Article  MathSciNet  Google Scholar 

  30. Temam, R.: Navier-Stokes Equations. North Holland, Amsterdam (1979)

    MATH  Google Scholar 

  31. Xu, H., He, Y.N.: Some iterative finite element methods for steady Navier-Stokes equations with different viscosities. J. Comput. Phys. 232, 136–152 (2013)

    Article  MathSciNet  Google Scholar 

  32. Yang, Y.B., Jiang, Y.L., Kong, Q.X.: The Arrow-Hurwicz iterative finite element method for the stationary magnetohydrodynamics flow. Appl. Math. Comput. 356, 347–361 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Yang, J.T., Zhang, T.: Stability and convergence of iterative finite element methods for the thermally coupled incompressible MHD flow. Int. J. Numer. Methods Heat fluid flow 30, 5103–5141 (2020)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the editor and anonymous reviewers for their helpful comments and suggestions which lead to a considerably improved presentation.

Funding

This work is sponsored by Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant Number 2021D01E11).

Author information

Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, 830017, Urumqi, People’s Republic of China

    Aytura Keram & Pengzhan Huang

Authors
  1. Aytura Keram
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Pengzhan Huang
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Pengzhan Huang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

This work is sponsored by Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant Number 2021D01E11)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Keram, A., Huang, P. The Arrow–Hurwicz Iterative Finite Element Method for the Stationary Thermally Coupled Incompressible Magnetohydrodynamics Flow. J Sci Comput 92, 11 (2022). https://doi.org/10.1007/s10915-022-01867-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-01867-y

Keywords