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Journal of Scientific Computing

, Volume 70, Issue 1, pp 149–174 | Cite as

Local and Parallel Finite Element Algorithm Based on Oseen-Type Iteration for the Stationary Incompressible MHD Flow

  • Qili Tang
  • Yunqing HuangEmail author
Article

Abstract

In this work, we are concerned with the local and parallel finite element algorithm based on the Oseen-type iteration for solving the stationary incompressible magnetohydrodynamics. Under the uniqueness condition, the error estimates with respect to iterative step m and small mesh sizes H and \(h\ll H\) of the proposed method are derived. Finally, some numerical experiments are provided to show the high efficiency of our algorithm.

Keywords

Local and parallel algorithm Finite element Oseen iteration Stationary incompressible magnetohydrodynamics 

Mathematics Subject Classification

35Q30 65M60 65N30 76D05 

Notes

Acknowledgments

The authors sincerely thank the reviewers and editor for their helpful suggestions. The first author is supported by NSFC (No. 11401174). The second author is supported by the Major Research Plan of NSFC (No. 91430213).

References

  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Moreau, R.: Magneto-hydrodynamics. Kluwer Academic Publishers, Dordrecht (1990)Google Scholar
  3. 3.
    Davidson, P.A.: An Introduction to Magnetohydrodynamics. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  4. 4.
    Schötzau, D.: Mixed finite element methods for stationary incompressible magneto-hydrodynamics. Numer. Math. 96, 771–800 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Badia, S., Codina, R., Planas, R.: On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics. J. Comput. Phys. 234, 399–416 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Li, F.Y., Xu, L.W.: Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations. J. Comput. Phys. 231, 2655–2675 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Li, F.Y., Xu, L.W., Yakovlev, S.: Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field. J. Comput. Phys. 230, 4828–4847 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dong, X.J., He, Y.N.: Two-level newton iterative method for the 2D/3D stationary incompressible magnetohydrodynamics. J. Sci. Comput. 63, 426–451 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Salah, N.B., Soulaimani, A., Habashi, W.G.: A finite element method for magnetohydrodynamics. Comput. Methods Appl. Mech. Eng. 190, 5867–5892 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    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)CrossRefzbMATHGoogle Scholar
  13. 13.
    Yuksel, G., Ingram, R.: Numerical analysis of a finite element, Crank–Nicolson discretization for MHD flows at small magnetic Reynolds numbers. Int. J. Numer. Anal. Model. 10, 74–98 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Prohl, A.: Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system. M2AN. Math. Model. Numer. Anal. 42, 1065–1087 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Huang, Y.Q., Shi, Z.C., Tang, T., Xue, W.M.: A multilevel successive iteration method for nonlinear elliptic problems. Math. Comput. 73, 525–539 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Huang, Y.Q., Chen, Y.P.: A multi-level iterative method for solving finite element equations of nonlinear singular two-point boundary value problems. Nat. Sci. J. Xiangtan Univ. 16, 23–26 (1994)zbMATHGoogle Scholar
  17. 17.
    Huang, Y.Q., Xue, W.M.: Convergence of finite element approximations and multilevel linearization for Ginzburg–Landau model of \(d\)-wave superconductors. Adv. Comput. Math. 17, 309–330 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms based on two-grid discretizations. Math. Comput. 69, 881–909 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    He, Y.N., Xu, J.C., Zhou, A.H., Li, J.: Local and parallel finite element algorithms for the Stokes problem. Numer. Math. 109, 415–434 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shang, Y.Q., He, Y.N.: A parallel Oseen-linearized algorithm for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 209, 172–183 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    He, Y.N., Mei, L.Q., Shang, Y.Q., Cui, J.: Newton iterative parallel finite element algorithm for the steady Navier–Stokes equations. J. Sci. Comput. 44, 92–106 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Xu, J.C.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Chen, Y.P., Liu, H.W., Liu, S.: Analysis of two-grid methods for reaction-diffusion equations by expanded mixed finite element methods. Int. J. Numer. Methods Eng. 69, 408–422 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Huang, Y.Q., Kornhuber, R., Widlund, O., Xu, J.C.: Domain Decomposition Methods in Sicience and Engineering XIX. Springer, Berlin (2011)CrossRefGoogle Scholar
  25. 25.
    Huang, Y.Q., Xu, J.C.: A conforming finite element method for overlapping and nonmatching grids. Math. Comput. 72, 1057–1066 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hasler, U., Schneebeli, A., Schötzau, D.: Mixed finite element approximation of incompressible MHD problems based on weighted regularization. Appl. Numer. Math. 51, 19–25 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sermane, M., Temam, R.: Some mathematics questions related to the MHD equations. Commun. Pure Appl. Math. XXXIV, 635–664 (1984)Google Scholar
  29. 29.
    Brenner, S.C., Cui, J., Li, F.Y., Sung, L.-Y.: A nonconforming finite element method for a two-dimensional curl–curl and grad–div problem. Numer. Math. 109, 509–533 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Girault, V., Raviart, P.A.: Finite Element Approximation of Navier–Stokes Equations. Springer, Berlin (1986)CrossRefzbMATHGoogle Scholar
  31. 31.
    Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo 21, 337–344 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Heywood, J., 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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    He, Y.N.: Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations. IMA J. Numer. Anal. 35, 767–801 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    He, Y.N., Xu, J.C., Zhou, A.H.: Local and parallel finite element algorithms for the Navier–Stokes problem. J. Comput. Math. 24, 227–238 (2006)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Hecht, F., Pironneau, O., Hyaric, A., Ohtsuka, K.: http://www.freefem.org

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational ScienceXiangtan UniversityXiangtanChina
  2. 2.School of Mathematics and StatisticsHenan University of Science and TechnologyLuoyangChina

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