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

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Abstract

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.

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References

  1. Moreau, R.: Magneto-Hydrodynamics. Kluwer, Boston (1990)

    Google Scholar 

  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)

    Book  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Schötzau, D.: Mixed finite element methods for stationary incompressible magneto-hydrodynamics. Numer. Math. 96, 771–800 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Layton, W.J.: A two-level discretization method for the Navier–Stokes equations. Comput. Math. Appl. 26, 33–38 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  17. He, Y.N., Zhang, Y., Xu, H.: Two-level Newton’s method for nonlinear elliptic PDEs. J. Sci. Comput. 57, 124–145 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  23. Adams, R.A.: Sobolev Space. Academic Press, New York (1975)

    Google Scholar 

  24. Cirault, V., Raviart, P.A.: Finite Element Approximation of Navier–Stokes Equations. Springer, Berlin (1986)

    Book  Google Scholar 

  25. Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003)

    Book  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  28. Cattabriga, L.: Si un problem al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)

    MATH  MathSciNet  Google Scholar 

  29. Georgescu, V.: Some boundary value problems for differenttial forms on compact Riemannian manifolds. Annali di Matematica Pura ed. Applicata 4, 159–198 (1979)

    Article  MathSciNet  Google Scholar 

  30. He, Y.N.: Euler implicit/explicit iterative scheme for the stationary Navier–Stokes equations. Numer. Math. 123, 67–96 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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

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Acknowledgments

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).

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Authors and Affiliations

  1. School of Mathematics and Statistics, State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an , 710049, People’s Republic of China

    Xiaojing Dong & Yinnian He

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  1. Xiaojing Dong
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  2. Yinnian He
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Correspondence to Yinnian He.

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Dong, X., He, Y. Two-Level Newton Iterative Method for the 2D/3D Stationary Incompressible Magnetohydrodynamics. J Sci Comput 63, 426–451 (2015). https://doi.org/10.1007/s10915-014-9900-7

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  • DOI: https://doi.org/10.1007/s10915-014-9900-7

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