Open Access
Volume 56, Number 3, May-June 2022
Page(s) 767 - 789
Published online 25 April 2022
  1. F. Armero and J.C. Simo, Long-term dissipativity of time-stepping algorithms for an abstract evolution equation with applications to the incompressible MHD and Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 131 (1996) 41–90. [CrossRef] [Google Scholar]
  2. S. Asai, Electromagnetic Processing of Materials: Fluid Mechanics and Its Applications. Springer, Netherlands (2012). [CrossRef] [Google Scholar]
  3. J.B. Bell, P. Colella and H.M. Glaz, A second order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys. 85 (1989) 257–283. [CrossRef] [MathSciNet] [Google Scholar]
  4. S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, NY, (2002). [CrossRef] [Google Scholar]
  5. W. Chen, C. Wang, X. Wang and S.M. Wise, A linear iteration algorithm for energy stable second order scheme for a thin film model without slope selection. J. Sci. Comput. 59 (2014) 574–601. [CrossRef] [MathSciNet] [Google Scholar]
  6. K. Cheng, C. Wang, S.M. Wise and X. Yue, A second-order, weakly energy-stable pseudo-spectral scheme for the Cahn-Hilliard equation and its solution by the homogeneous linear iteration method. J. Sci. Comput. 69 (2016) 1083–1114. [CrossRef] [MathSciNet] [Google Scholar]
  7. A.J. Chorin, Numerical solution of the Navier-Stokes equations. Math. Comp. 22 (1968) 745–762. [CrossRef] [MathSciNet] [Google Scholar]
  8. A. Diegel, C. Wang and S.M. Wise, Stability and convergence of a second order mixed finite element method for the Cahn-Hilliard equation. IMA J. Numer. Anal. 36 (2016) 1867–1897. [CrossRef] [MathSciNet] [Google Scholar]
  9. H. Gao and W. Qiu, A semi-implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations. Comput. Methods Appl. Mech. Eng. 346 (2019) 982–1001. [CrossRef] [Google Scholar]
  10. J.F. Gerbeau, A stabilized finite element method for the incompressible magnetohydrodynamic equations. Numer. Math. 87 (2000) 83–111. [CrossRef] [MathSciNet] [Google Scholar]
  11. V. Girault and P. Raviart, Finite Element Method for Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin, Herdelberg (1987). [Google Scholar]
  12. J.-L. Guermond, Un résultat de convergence d’ordre deux en temps pour l’approximation des équations de Navier-Stokes par une technique de projection incrémentale. ESAIM: M2AN 33 (1999) 169–189. [CrossRef] [EDP Sciences] [Google Scholar]
  13. J.L. Guermond and P.D. Minev, Mixed finite element approximation of an MHD problem involving conducting and insulating regions: the 3D case. Numer. Methods Part. Differ. Equ. 19 (2003) 709–731. [CrossRef] [Google Scholar]
  14. J.L. Guermond, P.D. Minev and J. Shen, An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195 (2006) 6011–6045. [CrossRef] [Google Scholar]
  15. M. Gunzburger, A.J. Meir and J.P. Peterson, On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics. Math. Comp. 56 (1991) 523–563. [CrossRef] [MathSciNet] [Google Scholar]
  16. J. Guo, C. Wang, S.M. Wise and X. Yue, An H2 convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation. Commu. Math. Sci. 14 (2016) 489–515. [CrossRef] [Google Scholar]
  17. C. He and Y. Wang, On the regularity criteria for weak solutions to the magnetohydrodynamic equations. J. Differ. Equ. 238 (2007) 1–17. [CrossRef] [Google Scholar]
  18. Y. He, Unconditional convergence of the Euler semi-implicit scheme for the 3D incompressible MHD equations. IMA J. Numer. Anal. 35 (2015) 767–801. [CrossRef] [MathSciNet] [Google Scholar]
  19. Y. He, J. Zou, A priori estimates and optimal finite element approximation of the MHD flow in smooth domains. ESAIM: M2AN 52 (2018) 181–206. [Google Scholar]
  20. T. Heister, M. Mohebujjaman and L.G. Rebholz, Decoupled, unconditionally stable, higher order discretizations for MHD flow simulation. J. Sci. Comput. 71 (2017) 21–43. [CrossRef] [MathSciNet] [Google Scholar]
  21. R. Hiptmair, L. Li, S. Mao and W. Zheng, A fully divergence-free finite element method for magnetohydrodynamic equations. Math. Models Methods Appl. Sci. 28 (2018) 659–695. [CrossRef] [MathSciNet] [Google Scholar]
  22. J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations. J. Comput. Phys. 59 (1985) 308–323. [CrossRef] [MathSciNet] [Google Scholar]
  23. W. Layton, H. Tran and C. Trenchea, Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows. Numer. Methods Part. Differ. Equ. 30 (2014) 1083–1102. [CrossRef] [Google Scholar]
  24. B. Li, J. Wang and L. Xu, A convergent linearized lagrange finite element method for the magneto-hydrodynamic equations in 2D nonsmooth and nonconvex domains. SIAM J. Numer. Anal. 58 (2020) 430–459. [CrossRef] [MathSciNet] [Google Scholar]
  25. F. Lin and P. Zhang, Global small solutions to an MHD-type system: the three-dimensional case. Comm. Pure Appl. Math. 67 (2014) 531–580. [CrossRef] [MathSciNet] [Google Scholar]
  26. F. Lin, L. Xu and P. Zhang, Global small solutions to 2-D incompressible MHD system. J. Differ. Equ. 259 (2015) 5440–5485. [CrossRef] [Google Scholar]
  27. C. Liu, J. Shen and X. Yang, Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density. J. Sci. Comput. 62 (2015) 601–622. [CrossRef] [MathSciNet] [Google Scholar]
  28. J.-G. Liu and W. Wang, An energy-preserving MAC–Yee scheme for the incompressible MHD equation. J. Comput. Phys. 174 (2001) 12–37. [CrossRef] [MathSciNet] [Google Scholar]
  29. A. Prohl, Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system. ESAIM: M2AN 42 (2008) 1065–1087. [CrossRef] [EDP Sciences] [Google Scholar]
  30. J. Ridder, Convergence of a finite difference scheme for two-dimensional incompressible magnetohydrodynamics. SIAM J. Numer. Anal. 54 (2016) 3550–3576. [CrossRef] [MathSciNet] [Google Scholar]
  31. R. Samelson, R. Temam, C. Wang and S. Wang, Surface pressure poisson equation formulation of the primitive equations: numerical schemes. SIAM J. Numer. Anal. 41 (2003) 1163–1194. [CrossRef] [MathSciNet] [Google Scholar]
  32. M.E. Schonbek, T.P. Schonbek and E. Süli, Large-time behavior of solutions to the magnetohydrodynamics equations. Math. Ann. 304 (1996) 717–756. [CrossRef] [MathSciNet] [Google Scholar]
  33. M. Sermange and R. Temam, Some mathematical questions related to the MHD equations. Comm. Pure Appl. Math. 36 (1983) 635–664. [CrossRef] [MathSciNet] [Google Scholar]
  34. J. Shen, On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes. Math. Comp. 65 (1996) 1039–1065. [CrossRef] [MathSciNet] [Google Scholar]
  35. J.A. Shercliff, A Textbook of Magnetohydrodyamics. Pergamon Press, Oxford-New York-Paris (1965). [Google Scholar]
  36. R. Temam, Sur l’approximation de la solution des équation de Navier-Stokes par la méthode des pas fractionnaires (II). Arch. Ration. Mech. Anal. 33 (1969) 377–385. [CrossRef] [Google Scholar]
  37. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin (2006). [Google Scholar]
  38. Y. Unger, M. Mond and H. Branover, Liquid Metal Flows: Magnetohydrodynamics and Application. American Institute of Aeronautics and Astronautic (1988). [CrossRef] [Google Scholar]
  39. J. Van Kan, A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Statist. Comput. 7 (1986) 870–891. [CrossRef] [MathSciNet] [Google Scholar]
  40. C. Wang and J.-G. Liu, Convergence of gauge method for incompressible flow. Math. Comp. 69 (2000) 1385–1407. [CrossRef] [MathSciNet] [Google Scholar]
  41. E. Weinan and J.-G. Liu, Projection method I: convergence and numerical boundary layers. SIAM J. Numer. Anal. 32 (1995) 1017–1057. [CrossRef] [MathSciNet] [Google Scholar]
  42. E. Weinan and J.-G. Liu, Projection method III: spatial discretization on the staggered grid. Math. Comp. 71 (2002) 27–47. [MathSciNet] [Google Scholar]
  43. X. Yang, G. Zhang and X. He, On an efficient second order backward difference Newton scheme for MHD system. J. Math. Anal. Appl. 458 (2018) 676–714. [CrossRef] [MathSciNet] [Google Scholar]
  44. J. Zhao, X. Yang, J. Shen and Q. Wang, A decoupled energy stable scheme for a hydrodynamic phase-field model of mixtures of nematic liquid crystals and viscous fluids. J. Comput. Phys. 305 (2016) 539–556. [CrossRef] [MathSciNet] [Google Scholar]

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