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Optimal convergence of three iterative methods based on nonconforming finite element discretization for 2D/3D MHD equations

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Abstract

The main purpose of this paper is to analyze nonconforming iterative finite element methods for 2D/3D stationary incompressible magneto-hydrodynamics equations. First, the Crouzeix-Raviart–type finite element is used to approximate the velocity and the conforming piecewise linear element P1 is used for the pressure. Since the finite element method for the velocity field and the pressure is unstable, a simple locally stabilization term is added to satisfy the weak inf-sup condition. Then, the well-posedness and the optimal error estimates of the continuous and discrete problems are analyzed with the nonlinear terms being iteratively updated. Three effective iterative methods are proposed and their stability and convergence analyses are carried out. Finally, the theoretical analysis presented in this paper is verified by numerical experiments.

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References

  1. Alonso, A., Valli, A.: An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. Math. Com. 68, 607–631 (1999)

    Article  MathSciNet  Google Scholar 

  2. Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional nonsmooth domains. Math. Method. Appl. Sci. 21, 823–864 (1998)

    Article  Google Scholar 

  3. Aydin, S., Nesliturk, A., Tezer-Sezgin, M.: Two-level finite element method with a stabilizing subgrid for the incompressible MHD equations. Int. J. Numer. Meth. Fl. 62, 188–210 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Cai, Z., Douglas, J., Ye, X.: A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo 36, 215–232 (1999)

    Article  MathSciNet  Google Scholar 

  5. Crouzeix, M., Raviart, P.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. ESAIM: Math. Model. Num. 7, 33–75 (1973)

    MathSciNet  MATH  Google Scholar 

  6. Dong, X., He, Y.: A parallel finite element method for incompressible magnetohydrodynamics equations. Appl. Math. Lett. 102, 106076 (2019)

    Article  MathSciNet  Google Scholar 

  7. Dong, X., He, Y., Wei, H., Zhang, Y.: Local and parallel finite element algorithm based on the partition of unity method for the incompressible MHD flow. Adv. Comput. Math. 44, 1295–1319 (2018)

    Article  MathSciNet  Google Scholar 

  8. Dong, X., He, Y., Zhang, Y.: Convergence analysis of three finite element iterative methods for the 2D/3D stationary incompressible magnetohydrodynamics. Comput. Method. Appl. M. 276, 287–311 (2014)

    Article  MathSciNet  Google Scholar 

  9. Duvaut, G., Lions, J.: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. An. 46, 241–279 (1972)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  12. Girault, V., Raviart, P.: Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)

    Book  Google Scholar 

  13. Gunzburger, M., Meir, A., Peterson, J.: 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 

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

    Article  MathSciNet  Google Scholar 

  15. Hiptmair, R.: Finite elements in computational electromagnetism. Acta. Numerica. 11, 237–339 (2002)

    Article  MathSciNet  Google Scholar 

  16. Li, J., Chen, Z.: A new local stabilized nonconforming finite element method for the Stokes equations. Computing. 82, 157–170 (2008)

    Article  MathSciNet  Google Scholar 

  17. Li, J., He, Y.: A stabilized finite element method based on two local Gauss integrations for the Stokes equations. J. Comput. Appl. Math. 214, 58–65 (2008)

    Article  MathSciNet  Google Scholar 

  18. Li, J., He, Y., Chen, Z.: A new stabilized finite element method for the transient Navier-Stokes equations. Comput. Method. Appl. M. 197, 22–35 (2007)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  20. Planas, R., Badia, S., Codina, R.: Approximation of the inductionless MHD problem using a stabilized finite element method. J. Comput. Phys. 230, 2977–2996 (2011)

    Article  MathSciNet  Google Scholar 

  21. Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Meth. Part. D. E. 8, 97–112 (1992)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Shi, D., Ren, J.: Nonconforming mixed finite element method for the stationary conduction-convection problem. Int. J. Numer. Anal. Mod. 6, 293–310 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Shi, D., Yu, Z.: Low-order nonconforming fixed finite element methods for stationary incompressible magnetohydrodynamics equations. J. Appl. Math. 2012, 331–353 (2012)

    Google Scholar 

  25. Shi, D., Yu, Z.: Nonconforming mixed finite element methods for stationary incompressible magnetohydrodynamics. Int. J. Numer. Anal. Mod. 10, 904–919 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Shi, D., Zhang, Y.: A nonconforming anisotropic finite element approximation with moving grids for Stokes problem. J. Comput. Math. 24, 561–578 (2006)

    MathSciNet  MATH  Google Scholar 

  27. Zhang, G., He, Y., Yang, D.: Analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain. Comput. Math. Appl. 68, 770–788 (2014)

    Article  MathSciNet  Google Scholar 

  28. Zhu, L., Li, J., Chen, Z.: A new local stabilized nonconforming finite element method for solving stationary Navier-Stokes equations. J. Comput. Appl. Math. 235, 2821–2831 (2011)

    Article  MathSciNet  Google Scholar 

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Funding

This work is partly supported by the NSF of China (No. 11701493, 12061076, 12061075), the NSF of Xinjiang Province (No. 2016D01C073), Tianshan Youth Project of Xinjiang Province (No. 2017Q079), and Z. Li is partially supported by a Simons grant 633724.

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Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China

    Jiali Xu & Haiyan Su

  2. CRSC & Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Zhilin Li

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  1. Jiali Xu
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  2. Haiyan Su
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  3. Zhilin Li
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Correspondence to Haiyan Su.

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Xu, J., Su, H. & Li, Z. Optimal convergence of three iterative methods based on nonconforming finite element discretization for 2D/3D MHD equations. Numer Algor 90, 1117–1151 (2022). https://doi.org/10.1007/s11075-021-01224-4

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