Research article

A modified characteristics projection finite element method for unsteady incompressible Magnetohydrodynamics equations

  • Received: 18 February 2020 Accepted: 17 April 2020 Published: 26 April 2020
  • MSC : 76D05, 35Q30, 65M60, 65N30

  • This paper provides a modified characteristics projection finite element method for the unsteady incompressible magnetohydrodynamics(MHD) equations. In this method, modified characteristics finite element method and the projection method will be combined for solving the unsteady incompressible MHD equations. Both the stability and the optimal error estimates both in L2 and H1 norms for the modified characteristics projection finite element method will be shown. In order to demonstrate the effectiveness of our method, we will present some numerical results at the end.

    Citation: Shujie Jing, Jixiang Guan, Zhiyong Si. A modified characteristics projection finite element method for unsteady incompressible Magnetohydrodynamics equations[J]. AIMS Mathematics, 2020, 5(4): 3922-3951. doi: 10.3934/math.2020254

    Related Papers:

  • This paper provides a modified characteristics projection finite element method for the unsteady incompressible magnetohydrodynamics(MHD) equations. In this method, modified characteristics finite element method and the projection method will be combined for solving the unsteady incompressible MHD equations. Both the stability and the optimal error estimates both in L2 and H1 norms for the modified characteristics projection finite element method will be shown. In order to demonstrate the effectiveness of our method, we will present some numerical results at the end.


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    [1] R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, 1990.
    [2] W. Hughes, F. Young, The Electromagnetodynamics of Fluids, Wiley: New York, 1966.
    [3] M. D. Gunzburger, A. J. Meir, J. S. Peterson, On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics, Math. Comput., 56 (199), 523-563.
    [4] J. Gerbeau, C. Bris, T. Lelièvre, Mathematical methods for the Manetohydrodynamics of liquid metals, Oxford University Press, 2006.
    [5] M. Sermange, R. Temam, Some mathematical questions related to the magnetohydrodynamic equations, Comput. Compacts, 1 (1983), 212. doi: 10.1016/0167-7136(83)90286-X
    [6] U. Hasler, A. Schneebeli, D. Schötzau, Mixed finite element approximation of incompressible MHD problems based on weighted regularization, Appl. Numer. Math., 51 (2004), 19-45. doi: 10.1016/j.apnum.2004.02.005
    [7] J. L. Guermond, P. D. Minev, Mixed Finite Element approximation of an MHD problem involving conducting and insulating regions: The 2D case, ESAIM: Math. Modell. Numer. Analy., 36 (2002), 517-536. doi: 10.1051/m2an:2002024
    [8] J. L. Guermond, P. D. Minev, Mixed finite element approximation of an MHD problem involving conducting and insulating regions: The 3D case, Numer. Methods Partial Differential Equations, 19 (2002), 709-731. doi: 10.1002/num.10067
    [9] D. Schötzau, Mixed finite element methods for stationary incompressible magneto-hydrodynamics, Numer. Math., 96 (2004), 771-800. doi: 10.1007/s00211-003-0487-4
    [10] J. F. Gerbeau, A stabilized finite element method for the incompressible magnetohydrodynamic equations, Numerische Mathematik, 87 (2000), 83-111. doi: 10.1007/s002110000193
    [11] N. B. Salah, A. Soulaimani, W. G. Habashi, A finite element method for magnetohydrodynamics, Comput. Methods Appl. Mech. Eng., 190 (2001), 5867-5892. doi: 10.1016/S0045-7825(01)00196-7
    [12] A. I. Nesliturk, M. Tezer-Sezgin, Two-evel finite element method with a stabilizing subgrid for the incompressible MHD equations, Int. J. Numer. Methods Fluids, 62 (2010), 188-210.
    [13] X. J. Dong, Y. N. He, Y. Zhang, Convergence analysis of three finite element iterative methods for the 2D/3D stationary incompressible magnetohydrodynamics, Comput. Method. Appl. Math., 276 (2014), 287-311.
    [14] Y. N. He, Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations, IMA. J. Numer. Anal., 35 (2015), 767-801. doi: 10.1093/imanum/dru015
    [15] X. J. Dong, Y. N. He, Two-level newton iterative method for the 2D/3D stationary incompressible magnetohydrodynamics, J. Sci. Comput., 63 (2015), 426-451. doi: 10.1007/s10915-014-9900-7
    [16] R. Ingram, Numerical analysis of a finite element Crank-Nicolson discretization for MHD flows at small magnetic Reynolds numbers, Int. J. Numer. Analy. Model., 10 (2012), 74-98.
    [17] Y. F. Lei, Y. Yang, Z. Y. Si, Error estimate of fully discrete dc-fem for unsteady incompressible magnetohydrodynamics equations, Appl. Anal., 97 (2018), 2355-2376. doi: 10.1080/00036811.2017.1366990
    [18] Z. Y. Si, S. J. Jing, Y. X. Wang, Defect correction finite element method for the stationary incompressible magnetohydrodynamics equation, Appl. Math. Comput., 285 (2016), 184-194.
    [19] Z. Y. Si, C. Liu, Y. X. Wang, A semi-discrete defect correction finite element method for unsteady incompressible magnetohydrodynamics equations, Math. Method Appl. Sci., 40 (2017), 4179-4196. doi: 10.1002/mma.4296
    [20] J. E. Deng, Z. Y. Si, A decoupling penalty finite element method for the stationary incompressible magnetohydrodynamics equation, Int. J. Heat. Mass. Tran., 128 (2019), 601-612. doi: 10.1016/j.ijheatmasstransfer.2018.08.096
    [21] W. Layton, H. Tran, C. Trenchea, Numerical analysis of two parititioned methods for uncoupling evolutionary MHD flows, Numer. Methods Partial Differ. Equ., 30 (2014), 1083-1102. doi: 10.1002/num.21857
    [22] G. D. Zhang, Y. N. He, D. Yang, Analysis of coupling iterations based on the finite element method for stationary magnetohydrodynamics on a general domain, Comput. Math. Appl., 68 (2014), 770-788. doi: 10.1016/j.camwa.2014.07.025
    [23] G. D. Zhang, Y. N. He, Decoupled schemes for unsteady MHD equations. II: Finite element spatial discretization and numerical implementation, Comput. Math. Appl., 69 (2015), 1390-1406. doi: 10.1016/j.camwa.2015.03.019
    [24] R. A. Adams, Sobolev Space, In: Pure and Applied Mathematics, vol. 65, Academic Press, New York, 1975.
    [25] V. Girault, P. Raviart, Finite element methods for Navier-Stokes equations, 1986.
    [26] J. Heywood, G. John, The Navier-Stokes equations: On the existence, regularity and decay of solutions, Indiana University Math. J., 29 (1980), 639-681. doi: 10.1512/iumj.1980.29.29048
    [27] Y. N. He, Two-level method based on finite element and Crank-Nicolson extrapolation for the timedependent Navier-Stokes equations, SIAM J. Numer. Anal., 41 (2004), 1263-1285.
    [28] G. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations: Steady-state problems, Springer, New Yourk, 2011.
    [29] Y. Achdou, J. L. Guermond, Convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 37 (2000), 799-826. doi: 10.1137/S0036142996313580
    [30] R. Bermejo, P. Galán del Sastre, L. Saavedra, A second order in time modified Lagrange-Galerkin finite element method for the incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 50 (2012), 3084-3109. doi: 10.1137/11085548X
    [31] Z. Y. Si, J. L. Wang, W. W. Sun, Unconditional stability and error estimates of modified characteristics FEMs for the Navier-Stokes equations, Numer. Math., 134 (2016), 139-161. doi: 10.1007/s00211-015-0767-9
    [32] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.
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