Skip to main content
Springer Nature Link
Log in

Unconditionally Optimal Convergent Zero-Energy-Contribution Scheme for Two Phase MHD Model

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

This paper focuses on the unconditionally optimal error estimates of a fully discrete decoupled scheme for two-phase magnetohydrodynamic (MHD) model with different viscosities and electric conductivities, by using the zero-energy-contribution (ZEC) method for the temporal discretization and mixed finite elements for the spatial discretization. Based on the ZEC property of the nonlinear and coupled terms of the model, an ordinary differential equation is designed to introduce a nonlocal scalar auxiliary variable which will play a key role in the design and the energy stability of the decoupled scheme. Combining fully explicit treatment on the nonlinear and coupled terms with the stabilization method for nonlinear potential, a decoupled temporal discrete scheme is proposed. Utilizing mixed finite elements for the spatial discretization in this temporal discrete scheme, a fully discrete scheme is proposed. Both schemes are proved to be mass-conservative and unconditionally energy stable. The unconditionally optimal error estimates of the temporal discrete scheme are derived for two dimensional and three dimensional (2D/3D) cases. Utilizing a modified Maxwell projection with variable electric conductivities, the superconvergence of its negative norm estimates, mathematical induction, and the unconditional stability of the numerical scheme, we also derive the optimal error estimates in \(L^2\)-norm for the fully discrete scheme in 2D/3D cases, without any restriction on the time step size and mesh size. Finally, numerical experiments are provided to verify the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Data availability

The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Gerbeau, J.F., Le Bris, C., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford (2006)

    MATH  Google Scholar 

  2. Davidson, P.A.: An Introduction to Magnetohydrodynamics. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  3. Morley, N.B., Smolentsev, S., Barleon, L., Kirillov, I.R., Takahashi, M.: Liquid magnetohydrodynamics-recent progress and future directions for fusion. Fusion Eng. Design 51–52, 701–713 (2000)

    Google Scholar 

  4. Szekely, J.: Fluid Flow Phenomena in Metals Processing. Academic Press, New York (1979)

    MATH  Google Scholar 

  5. Zhang, J., Ni, M.J.: Direct simulation of multi-phase MHD flows on an unstructured Cartesian adaptive system. J. Comput. Phys. 270, 345–365 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Yang, J., Mao, S., He, X.M., Yang, X., He, Y.: A diffuse interface model and semi-implicit energy stable finite element method for two-phase magnetohydrodynamic flows. Comput. Methods Appl. Mech. Engrg. 356, 435–464 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Jacqmin, D.: Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 155(1), 96–127 (1999)

    MathSciNet  MATH  Google Scholar 

  8. Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D 179(3–4), 211–228 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Diegel, A.E., Wang, C., Wang, X., Wise, S.M.: Convergence analysis and error estimates for a second order accurate finite element method for the Cahn-Hilliard-Navier-Stokes system. Numer. Math. 137(3), 495–534 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Xu, J., Zhao, J., Zhao, Y.: Numerical Approximations of the Allen-Cahn-Ohta-KawasakiEquation with Modified Physics-Informed Neural Networks (PINNs). Int. J. Numer. Anal. Model. 20, 693–708 (2023)

    MathSciNet  MATH  Google Scholar 

  11. Gao, Y., He, X.M., Mei, L., Yang, X.: Decoupled, linear, and energy stable finite element method for the Cahn-Hilliard-Navier-Stokes-Darcy phase field model. SIAM J. Sci. Comput. 40(1), B110–B137 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Diegel, A.E., Feng, X., Wise, S.M.: Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system. SIAM J. Num. Anal. 53(1), 127–152 (2015)

    MathSciNet  MATH  Google Scholar 

  13. Feng, X.: Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44(3), 1049–1072 (2006)

    MathSciNet  MATH  Google Scholar 

  14. Feng, X., He, Y., Liu, C.: Analysis of finite element approximations of a phase field model for two-phase fluids. Math. Comput. 76(258), 539–571 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Gao, Y., Han, D., He, X.M., Rüde, U.: Unconditionally stable numerical methods for Cahn-Hilliard-Navier-Stokes-Darcy system with different densities and viscosities. J. Comput. Phys. 454, 110968 (2022)

    MathSciNet  MATH  Google Scholar 

  16. Bai, F., He, X.M., Zhou, R., Yang, X., Wang, C.: Three dimensional phase-field investigation of droplet formation in microfluidic flow focusing devices with experimental validation. Int. J. Multiphase Flow 93, 130–141 (2017)

    MathSciNet  Google Scholar 

  17. Bai, F., Han, D., He, X.M., Yang, X.: Deformation and coalescence of ferrodroplets in Rosensweig model using the phase field and modified level set approaches under uniform magnetic fields. Commun. Nonlinear Sci. Numer. Simul. 85, 105213 (2020)

    MathSciNet  MATH  Google Scholar 

  18. Han, D., He, X.M., Wang, Q., Wu, Y.: Existence and weak-strong uniqueness of solutions to the Cahn-Hilliard-Navier-Stokes-Darcy system in superposed free flow and porous media. Nonlinear Anal. 211, 112411 (2021)

    MathSciNet  MATH  Google Scholar 

  19. Li, R., Gao, Y., Chen, J., Zhang, L., He, X.M., Chen, Z.: Discontinuous finite volume element method for a coupled Navier-Stokes-Cahn-Hilliard phase field model. Adv. Comput. Math. 46, 25 (2020)

    MathSciNet  MATH  Google Scholar 

  20. Gu, Y., He, X.M., Han, D.: On the phase-field modeling of rapid solidification. Comput. Mater. Sci. 199, 110812 (2021)

    MATH  Google Scholar 

  21. Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. In: Computational and Mathematical Models of Microstructural Evolution (San Francisco CA, 1998). Mater. Res. Soc. Sympos. Proc., vol. 529, pp. 39–46. MRS, Warrendale, PA (1998)

    MATH  Google Scholar 

  22. Guo, J., Wang, C., Wise, S.M., Yue, X.: An \(H^2\) convergence of a second-order convex-splitting, finite difference scheme for the three-dimensional Cahn-Hilliard equation. Commun. Math. Sci. 14(2), 489–515 (2016)

    MathSciNet  MATH  Google Scholar 

  23. Chen, X., Wang, C., Wise, S.M.: A preconditioned steepest descent solver for the cahn-hilliard equation with variable mobility. Int. J. Numer. Anal. Model. 19(6), 839–863 (2022)

    MathSciNet  MATH  Google Scholar 

  24. Shen, J., Yang, X.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. 28(4), 1669–1691 (2010)

    MathSciNet  MATH  Google Scholar 

  25. Huang, Q., Yang, X., He, X.M.: Numerical approximations for a smectic-A liquid crystal flow model: first-order, linear, decoupled and energy stable schemes. Discrete Contin. Dyn. Syst. Ser. B 23(6), 2177–2192 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Gao, Y., He, X.M., Lin, T., Lin, Y.: Fully decoupled energy-stable numerical schemes for two-phase coupled porous media and free flow with different densities and viscosities. ESAIM Math. Model. Numer. Anal. 57(3), 1323–1354 (2023)

    MathSciNet  MATH  Google Scholar 

  27. Yang, X., Zhang, G.D.: Convergence analysis for the invariant energy quadratization (IEQ) schemes for solving the Cahn-Hilliard and Allen-Cahn equations with general nonlinear potential. J. Sci. Comput. 82(3), 55 (2020)

    MathSciNet  MATH  Google Scholar 

  28. Yang, X., He, X.M.: Numerical approximations of flow coupled binary phase field crystal system: fully discrete finite element scheme with second-order temporal accuracy and decoupling structure. J. Comput. Phys. 467, 111448 (2022)

    MathSciNet  MATH  Google Scholar 

  29. Yang, X., Zhao, J., He, X.M.: Linear, second order and unconditionally energy stable schemes for the viscous Cahn-Hilliard equation with hyperbolic relaxation using the invariant energy quadratization method. J. Comput. Appl. Math. 343(1), 80–97 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Xu, C., Chen, C., Yang, X., He, X.M.: Numerical approximations for the hydrodynamics coupled binary surfactant phase field model: second order, linear, unconditionally energy stable schemes. Commun. Math. Sci. 17(3), 835–858 (2019)

    MathSciNet  MATH  Google Scholar 

  31. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407–416 (2018)

    MathSciNet  MATH  Google Scholar 

  32. Hou, D., Azaiez, M., Xu, C.: A variant of scalar auxiliary variable approaches for gradient flows. J. Comput. Phys. 395, 307–332 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474–506 (2019)

    MathSciNet  MATH  Google Scholar 

  34. Lin, F., He, X.M., Wen, X.: Fast, unconditionally energy stable large time stepping method for a new Allen-Cahn type square phase-field crystal model. Appl. Math. Lett. 92, 248–255 (2019)

    MathSciNet  MATH  Google Scholar 

  35. Layton, W., Tran, H., Trenchea, C.: Numerical analysis of two partitioned methods or uncoupling evolutionary MHD flows. Numer. Methods Partial Differ. Equ. 30(4), 1083–1102 (2014)

    MathSciNet  MATH  Google Scholar 

  36. He, Y.: Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations. IMA J. Numer. Anal. 35(2), 767–801 (2015)

    MathSciNet  MATH  Google Scholar 

  37. Hu, K., Ma, Y., Xu, J.: Stable finite element methods preserving \(\nabla \cdot B=0\) exactly for MHD models. Numer. Math. 135(2), 371–396 (2017)

    MathSciNet  MATH  Google Scholar 

  38. Hiptmair, R., Li, L., Mao, S., Zheng, W.: A fully divergence-free finite element method for magnetohydrodynamic equations. Math. Models Methods Appl. Sci. 28(4), 659–695 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Zhang, G.D., Yang, J., Bi, C.: Second order unconditionally convergent and energy stable linearized scheme for MHD equations. Adv. Comput. Math. 44(2), 505–540 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Fu, P., Li, F., Xu, Y.: Globally divergence-free discontinuous Galerkin methods for ideal magnetohydrodynamic equations. J. Sci. Comput. 77(3), 1621–1659 (2018)

    MathSciNet  MATH  Google Scholar 

  41. He, Y., Dong, X., Feng, X.: Uniform stability and convergence with respect to \((\nu , \mu , s, 1-\sigma )\) of the three iterative finite element solutions for the 3D steady MHD equations. J. Sci. Comput. 90(1), 17 (2022)

    MathSciNet  MATH  Google Scholar 

  42. Gao, H., Qiu, W.: A semi-implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations. Comput. Methods Appl. Mech. Eng. 346, 982–1001 (2019)

    MathSciNet  MATH  Google Scholar 

  43. Zhang, G.D., He, X.M., Yang, X.: A decoupled, linear and unconditionally energy stable scheme with finite element discretizations for magneto-hydrodynamic equations. J. Sci. Comput. 81(3), 1678–1711 (2019)

    MathSciNet  MATH  Google Scholar 

  44. Zhang, K., Su, H., Feng, X.: Second order unconditional linear energy stable, rotational velocity correction method for unsteady incompressible magneto-hydrodynamic equations. Comput. Fluids 236, 105300 (2022)

    MathSciNet  MATH  Google Scholar 

  45. Yang, J., Mao, S.: Second order fully decoupled and unconditionally energy-stable finite element algorithm for the incompressible MHD equations. Appl. Math. Lett. 121, 107467 (2021)

    MathSciNet  MATH  Google Scholar 

  46. Zhang, G.D., He, X.M., Yang, X.: A fully decoupled linearized finite element method with second-order temporal accuracy and unconditional energy stability for incompressible MHD equations. J. Comput. Phys. 448, 110752 (2022)

    MathSciNet  MATH  Google Scholar 

  47. Yang, X., Huang, P., He, Y.: A Voigt-regularization of the thermally coupled inviscid, resistive magnetohydrodynamic. Int. J. Numer. Anal. Model. 21(4), 476–503 (2024)

    MathSciNet  MATH  Google Scholar 

  48. Yang, X., Zhang, G., He, X.M.: Convergence analysis of an unconditionally energy stable projection scheme for magneto-hydrodynamic equations. Appl. Numer. Math. 136, 235–256 (2019)

    MathSciNet  MATH  Google Scholar 

  49. Zhang, G., He, X.M., Yang, X.: Fully decoupled, linear and unconditionally energy stable time discretization scheme for solving the magneto-hydrodynamic equations. J. Comput. Appl. Math. 369, 112636 (2020)

    MathSciNet  MATH  Google Scholar 

  50. Ding, Q., He, X.M., Long, X., Mao, S.: Error analysis of a fully discrete projection method for magnetohydrodynamic system. Numer. Methods Partial Differ. Equ. 39(2), 1449–1477 (2023)

    MathSciNet  MATH  Google Scholar 

  51. Chen, R., Zhang, H.: Second-order energy stable schemes for the new model of the Cahn-Hilliard-MHD equations. Adv. Comput. Math. 46(6), 79 (2020)

    MathSciNet  MATH  Google Scholar 

  52. Qiu, H.: Error analysis of fully discrete scheme for the Cahn-Hilliard-magneto-hydrodynamics problem. J. Sci. Comput. 95(1), 16 (2023)

    MathSciNet  MATH  Google Scholar 

  53. Chen, C., Zhang, T.: Unconditional stability and optimal error estimates of first order semi-implicit stabilized finite element method for two phase magnetohydrodynamic diffuse interface model. Appl. Math. Comput. 429, 127238 (2022)

    MathSciNet  MATH  Google Scholar 

  54. Zhao, J., Chen, R., Su, H.: An energy-stable finite element method for incompressible magnetohydrodynamic-Cahn-Hilliard coupled model. Adv. Appl. Math. Mech. 13(4), 761–790 (2021)

    MathSciNet  MATH  Google Scholar 

  55. Su, H., Zhang, G.D.: Highly efficient and energy stable schemes for the 2D/3D diffuse interface model of two-phase magnetohydrodynamics. J. Sci. Comput. 90(1), 63 (2022)

    MathSciNet  MATH  Google Scholar 

  56. Su, H., Zhang, G.D.: Energy stable schemes with second order temporal accuracy and decoupled structure for diffuse interface model of two-phase magnetohydrodynamics. Commun. Nonlinear Sci. Numer. Simul. 119, 107126 (2023)

    MathSciNet  MATH  Google Scholar 

  57. Yang, X.: On a novel fully decoupled, second-order accurate energy stable numerical scheme for a binary fluid-surfactant phase-field model. SIAM J. Sci. Comput. 43(2), B479–B507 (2021)

    MathSciNet  MATH  Google Scholar 

  58. Yang, X.: A novel fully-decoupled, second-order and energy stable numerical scheme of the conserved Allen-Cahn type flow-coupled binary surfactant model. Comput. Methods Appl. Mech. Eng. 373, 113502 (2021)

    MathSciNet  MATH  Google Scholar 

  59. Yang, X.: Numerical approximations of the Navier-Stokes equation coupled with volume-conserved multi-phase-field vesicles system: fully-decoupled, linear, unconditionally energy stable and second-order time-accurate numerical scheme. Comput. Methods Appl. Mech. Eng. 375, 113600 (2021)

    MathSciNet  MATH  Google Scholar 

  60. Yang, X.: A novel fully-decoupled, second-order time-accurate, unconditionally energy stable scheme for a flow-coupled volume-conserved phase-field elastic bending energy model. J. Comput. Phys. 432, 110015 (2021)

    MathSciNet  MATH  Google Scholar 

  61. Yang, X.: A novel fully decoupled scheme with second-order time accuracy and unconditional energy stability for the Navier-Stokes equations coupled with mass-conserved Allen-Cahn phase-field model of two-phase incompressible flow. Internat. J. Numer. Methods Engrg. 122(5), 1283–1306 (2021)

    MathSciNet  MATH  Google Scholar 

  62. Yang, X., He, X.M.: A fully-discrete decoupled finite element method for the conserved Allen-Cahn type phase-field model of three-phase fluid flow system. Comput. Methods Appl. Mech. Eng. 389, 114376 (2022)

    MathSciNet  MATH  Google Scholar 

  63. Zhang, G., He, X.M., Yang, X.: A unified framework of the SAV-ZEC method for a mass-conserved Allen-Cahn type two-phase ferrofluid flow model. SIAM J. Sci. Comput. 46(2), B77–B106 (2024)

    MathSciNet  MATH  Google Scholar 

  64. Zhang, G.D., He, X.M., Yang, X.: Reformulated weak formulation and efficient fully discrete finite element method for a two-phase ferrohydrodynamics Shliomis model. SIAM J. Sci. Comput. 45(3), B253–B282 (2023)

  65. Shen, J., Zheng, N.: Efficient and unconditional energy stable schemes for the micropolar Navier-Stokes equations. CSIAM Trans. Appl. Math. 3(1), 57–81 (2022)

    MathSciNet  MATH  Google Scholar 

  66. Lin, L., Yang, Z., Dong, S.: Numerical approximation of incompressible Navier-Stokes equations based on an auxiliary energy variable. J. Comput. Phys. 388, 1–22 (2019)

    MathSciNet  MATH  Google Scholar 

  67. Wang, W.: Novel pressure-correction schemes based on scalar auxiliary variable method for the MHD equations. Appl. Math. Comput. 437, 127550 (2023)

    MathSciNet  MATH  Google Scholar 

  68. Gao, Y., He, X.M., Nie, Y.: Second-order, fully decoupled, linearized, and unconditionally stable SAV schemes for Cahn-Hilliard-Darcy system. Numer. Methods Partial Differ. Equ. 38(6), 1658–1683 (2022)

    MATH  Google Scholar 

  69. Gao, Y., Li, R., He, X.M., Lin, Y.: A fully decoupled numerical method for Cahn-Hilliard-Navier-Stokes-Darcy equations based on auxiliary variable approaches. J. Comput. Appl. Math. 436, 115363 (2024)

    MathSciNet  MATH  Google Scholar 

  70. Zhang, T., Yuan, J.: Unconditional stability and optimal error estimates of Euler implicit/explicit-SAV scheme for the Navier-Stokes equations. J. Sci. Comput. 90(1), 1 (2022)

    MathSciNet  MATH  Google Scholar 

  71. Li, X., Shen, J.: Error estimate of a consistent splitting GSAV scheme for the Navier-Stokes equations. Appl. Numer. Math. 188, 62–74 (2023)

    MathSciNet  MATH  Google Scholar 

  72. Zhang, T., Yang, J.: Decoupled and linearized scalar auxiliary variable finite element method for the time-dependent incompressible magnetohydrodynamic equations: unconditional stability and convergence analysis. Numer. Methods Partial Differ. Equ. 38(5), 1499–1525 (2022)

    MathSciNet  MATH  Google Scholar 

  73. Li, X., Wang, W., Shen, J.: Stability and error analysis of IMEX SAV schemes for the magneto-hydrodynamic equations. SIAM J. Numer. Anal. 60(3), 1026–1054 (2022)

    MathSciNet  MATH  Google Scholar 

  74. Li, X., Shen, J.: On fully decoupled MSAV schemes for the Cahn-Hilliard-Navier-Stokes model of two-phase incompressible flows. Math. Models Methods Appl. Sci. 32(3), 457–495 (2022)

    MathSciNet  MATH  Google Scholar 

  75. Fei, J., Xie, S., Chen, C.: A scalar auxiliary variable (SAV) and operator splitting compact finite difference method for peritectic phase field model. Int. J. Numer. Anal. Model. 19(1), 85–100 (2022)

    MathSciNet  MATH  Google Scholar 

  76. Li, X., Shen, J.: Error analysis of the SAV-MAC scheme for the Navier-Stokes equations. SIAM J. Numer. Anal. 58(5), 2465–2491 (2020)

    MathSciNet  MATH  Google Scholar 

  77. Li, X., Shen, J., Liu, Z.: New SAV-pressure correction methods for the Navier-Stokes equations: stability and error analysis. Math. Comput. 91(333), 141–167 (2022)

    MathSciNet  MATH  Google Scholar 

  78. Nochetto, R.H., Salgado, A.J., Tomas, I.: A diffuse interface model for two-phase ferrofluid flows. Comput. Methods Appl. Mech. Eng. 309, 497–531 (2016)

    MathSciNet  MATH  Google Scholar 

  79. Wu, J., Yang, J., Tan, Z.: Unconditionally energy-stable time-marching methods for the multi-phase conservative Allen-Cahn fluid models based on a modified SAV approach. Comput. Methods Appl. Mech. Eng. 398, 115291 (2022)

    MathSciNet  MATH  Google Scholar 

  80. Zhang, G., He, X.M., Yang, X.: Decoupled, linear, and unconditionally energy stable fully-discrete finite element numerical scheme for a two-phase ferrohydrodynamics model. SIAM J. Sci. Comput. 43(1), B167–B193 (2021)

    MathSciNet  MATH  Google Scholar 

  81. Klawonn, A.: An optimal preconditioner for a class of saddle point problems with a penalty term. SIAM J. Sci. Comput. 19(2), 540–552 (1998)

    MathSciNet  MATH  Google Scholar 

  82. Pavarino, L.F.: Indefinite overlapping Schwarz methods for time-dependent Stokes problems. Comput. Methods Appl. Mech. Eng. 187(1–2), 35–51 (2000)

    MathSciNet  MATH  Google Scholar 

  83. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Numerical Mathematics and Scientific Computation, 2nd edn. Oxford University Press, Oxford (2014)

    MATH  Google Scholar 

  84. Guermond, J.L., Shen, J.: Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41(1), 112–134 (2003)

    MathSciNet  MATH  Google Scholar 

  85. Chen, X.: Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differ. Equ. 19(7–8), 1371–1395 (1994)

    MathSciNet  MATH  Google Scholar 

  86. Feng, X., Prohl, A.: Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99(1), 47–84 (2004)

    MathSciNet  MATH  Google Scholar 

  87. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, vol. 15. Springer, New York (2008). https://doi.org/10.1007/978-0-387-75934-0

    Book  MATH  Google Scholar 

  88. Huang, Y., Qiu, W., Sun, W.: New analysis of mixed finite element methods for incompressible magnetohydrodynamics. J. Sci. Comput. 95(3), 72 (2023)

    MathSciNet  MATH  Google Scholar 

  89. Gao, H., Qiu, W., Sun, W.: New analysis of mixed FEMs for dynamical incompressible magnetohydrodynamics. Numer. Math. 153(2–3), 327–358 (2023)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  91. Cai, W., Sun, W., Wang, J., Yang, Z.: Optimal \(L^2\) error estimates of unconditionally stable finite element schemes for the Cahn-Hilliard-Navier-Stokes system. SIAM J. Numer. Anal. 61(3), 1218–1245 (2023)

    MathSciNet  MATH  Google Scholar 

Download references

Funding

The first author is supported by Natural Science Foundation of Henan (202300410489), Key Scientific Research Projects of Higher Education Institutions in Henan Province(24A110013,24ZX008) and Scientific Research Team Plan of Zhengzhou University of Aeronautics(23ZHTD01003), the second author is supported by National Natural Science Foundation of China (No. 12271514) and National Key Research and Development Program of China 2023YFC3705701.

Author information

Authors and Affiliations

  1. School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou, 450046, P.R. China

    Jinjin Yang

  2. LSEC, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100190, P.R. China

    Shipeng Mao

  3. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, 65409, USA

    Xiaoming He

Authors
  1. Jinjin Yang
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Shipeng Mao
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Xiaoming He
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Xiaoming He.

Ethics declarations

Conflict of interest

The authors have no competing interests to declare that are relevant to the content of this article.

Code availability

Custom code.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, J., Mao, S. & He, X. Unconditionally Optimal Convergent Zero-Energy-Contribution Scheme for Two Phase MHD Model. J Sci Comput 102, 55 (2025). https://doi.org/10.1007/s10915-024-02773-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-024-02773-1

Keywords

Mathematics Subject Classification