Journal of Scientific Computing

, Volume 63, Issue 2, pp 426–451 | Cite as

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

  • Xiaojing Dong
  • Yinnian He


In this paper, Newton iteration and two-level finite element algorithm are combined for solving numerically the stationary incompressible magnetohydrodynamics (MHD) under a strong uniqueness condition. The method consists of solving the nonlinear MHD system by \(m\) Newton iterations on a coarse mesh with size \(H\) and then computing the Stokes and Maxwell problems on a fine mesh with size \(h\ll H\). The uniform stability and optimal error estimates of both Newton iterative method and two-level Newton iterative method are given. The error analysis shows that the two-level Newton iterative solution is of the same convergence order as the Newton iterative solution on a fine grid with \(h=O(H^2)\). However, the two-level Newton iterative method for solving the stationary incompressible MHD equations is simpler and more efficient than Newton iterative one. Finally, the effectiveness of the two-level Newton iterative method is illustrated by several numerical investigations.


Two-level method Newton iteration Finite element method Stationary incompressible magnetohydrodynamics 

Mathematical Subject Classification

35Q30 65M60 65N30 76D05 



The authors sincerely thank the reviewers and editor for their valuable comments and suggestions which led to a large improvement of the paper. The research was supported by the National Natural Science Foundation of China (Grant Nos.: 11271298, 11362021).


  1. 1.
    Moreau, R.: Magneto-Hydrodynamics. Kluwer, Boston (1990)Google Scholar
  2. 2.
    Gerbeau, J.-F., Bris, C.L., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2006)CrossRefGoogle Scholar
  3. 3.
    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. Comput. 56, 523–563 (1991)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Schötzau, D.: Mixed finite element methods for stationary incompressible magneto-hydrodynamics. Numer. Math. 96, 771–800 (2004)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Greif, C., Li, D., Schötzau, D., Wei, X.X.: A mixed finite element method with exactly divergence-free velocities for incompressible magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 199, 2840–2855 (2010)CrossRefMATHGoogle Scholar
  6. 6.
    Hasler, U., Schneebeli, A., Schötzau, D.: Mixed finite element approximation of incompressible MHD problems based on weighted regularization. Appl. Numer. Math. 51, 19–45 (2004)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Gerbeau, J.-F.: A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87, 83–111 (2000)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Salah, N.B., Soulaimani, A., Habashi, W.G.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867–5892 (2001)CrossRefMATHGoogle Scholar
  9. 9.
    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. Eng. 198, 1351–1359 (2009)CrossRefMATHGoogle Scholar
  10. 10.
    Xu, J.C.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Xu, J.C.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Layton, W.J.: A two-level discretization method for the Navier–Stokes equations. Comput. Math. Appl. 26, 33–38 (1993)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Layton, W.J., Tobiska, L.: A two-level method with backtraking for the Navier–Stokes equations. SIAM J. Numer. Anal. 35, 2035–2054 (1998)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Girault, V., Lions, J.L.: Two-grid finite element scheme for the transient Navier–Stokes problem. Math. Model. Numer. Anal. 35, 945–980 (2001)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    He, Y.N.: Two-level method based on finite element and Crank–Nicolson extrapolation for the time-dependent Navier–Stokes equations. SIAM J. Numer. Anal. 41, 1263–1285 (2003)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    He, Y.N., Wang, A.W.: A simplified two-level for the steady Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 197, 1568–1576 (2008)CrossRefMATHGoogle Scholar
  17. 17.
    He, Y.N., Zhang, Y., Xu, H.: Two-level Newton’s method for nonlinear elliptic PDEs. J. Sci. Comput. 57, 124–145 (2013)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Aydin, S.H., Nesliturk, A.I., Tezer-Sezgin, M.: Two-level finite element method with a stabilizing subgrid for the incompressible MHD equations. Int. J. Numer. Methods Fluids 62, 188–210 (2010)MATHMathSciNetGoogle Scholar
  19. 19.
    Layton, W.J., Meir, A.J., Schmidt, P.G.: A two-level discretization method for the stationary MHD equations. Electron. Trans. Numer. Anal. 6, 198–210 (1997)MATHMathSciNetGoogle Scholar
  20. 20.
    Layton, W.J., Lenferink, H.W.J., Peterson, J.S.: A two-level Newton, finite element algorithm for approximating electrically conducting incompressible fluid flows. Comput. Math. Appl. 28, 21–31 (1994)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    He, Y.N., Zhang, Y., Shang, Y.Q., Xu, H.: Two-level Newton iterative method for the 2D/3D steady Navier–Stokes equations. Numer. Methods PDEs 28, 1620–1642 (2012)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Dong, X.J., He, Y.N., 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)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Adams, R.A.: Sobolev Space. Academic Press, New York (1975)Google Scholar
  24. 24.
    Cirault, V., Raviart, P.A.: Finite Element Approximation of Navier–Stokes Equations. Springer, Berlin (1986)CrossRefGoogle Scholar
  25. 25.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003)CrossRefMATHGoogle Scholar
  26. 26.
    Heywood, J.G., Rannacher, R.: Finite element approximation of the nonstationary Navier–Stokes problem I: regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Sermane, M., Temam, R.: Some mathematics questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)CrossRefGoogle Scholar
  28. 28.
    Cattabriga, L.: Si un problem al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)MATHMathSciNetGoogle Scholar
  29. 29.
    Georgescu, V.: Some boundary value problems for differenttial forms on compact Riemannian manifolds. Annali di Matematica Pura ed. Applicata 4, 159–198 (1979)CrossRefMathSciNetGoogle Scholar
  30. 30.
    He, Y.N.: Euler implicit/explicit iterative scheme for the stationary Navier–Stokes equations. Numer. Math. 123, 67–96 (2013)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Brenner, S.C., Cui, J., Li, F., Sung, L.-Y.: A nonconforming finite element method for a two-dimensional curl–curl and grad–div problem. Numer. Math. 109, 509–533 (2008)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    He, Y.N.: Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations. IMA J. Numer. Anal. (2014). doi: 10.1093/imanum/dru015

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1. School of Mathematics and Statistics, State Key Laboratory of Multiphase Flow in Power EngineeringXi’an Jiaotong UniversityXi’an People’s Republic of China

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