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February  2021, 26(2): 815-845. doi: 10.3934/dcdsb.2020143

Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows

Xiaoli Lu 1,2, , Pengzhan Huang 1,, and  Yinnian He 2,

1. 

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
2. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, P.R. China

* Corresponding author: Pengzhan Huang

Received  July 2018 Revised  February 2020 Published  May 2020

Fund Project: This work is supported by the NSF of China (grant numbers 11861067, 11771348)

We devote the present paper to a fully discrete finite element scheme for the 2D/3D nonstationary incompressible magnetohydrodynamic-Voigt regularization model. This scheme is based on a finite element approximation for space discretization and the Crank-Nicolson-type scheme for time discretization, which is a two-step method. Moreover, we study stability and convergence of the fully discrete finite element scheme and obtain unconditional stability and error estimates of velocity and magnetic fields, respectively. Finally, several numerical experiments are investigated to confirm our theoretical findings.

Keywords: Magnetohydrodynamic-Voigt model, finite element approximation, Crank-Nicolson-type scheme, unconditional stability, convergence.
Mathematics Subject Classification: Primary: 65N30.
Citation: Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143
References:
[1]

H. Alfvén, Existence of electromagnetic-hydrodynamic waves, Nature, 150 (1942), 3763-3767.   Google Scholar

[2]

L. Barleon, V. Casal and L. Lenhart, MHD flow in liquid-metal-cooled blankets, Fusion Eng. Des., 14 (1991), 401-412.   Google Scholar

[3]

J. D. Barrow, R. Maartens and C. G. Tsagas, Cosmology with inhomogeneous magnetic fields, Phys. Rep., 449 (2007), 131-171.  doi: 10.1016/j.physrep.2007.04.006.  Google Scholar

[4]

R. Bermejo, P. Galán del Sastre and 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.  Google Scholar

[5]

P. Bodenheimer, G. P. Laughlin, M. Różyczka and H. W. Yorke, Numerical Methods in Astrophysics, Series in Astronomy and Astrophysics, Taylor and Francis, New York, 2007.  Google Scholar

[6]

M. A. Case, A. Labovsky, L. G. Rebholz and N. E. Wilson, A high physical accuracy method for incompressible magnetohydrodynamics, Int. J. Numer. Anal. Model. ser. B, 1 (2010), 217-236.   Google Scholar

[7]

D. Catania, Global existence for a regularized magnetohydrodynamic-$\alpha$ model, Ann. Univ. Ferrara., 56 (2010), 1-20.  doi: 10.1007/s11565-009-0069-1.  Google Scholar

[8] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511626333.  Google Scholar
[9]

E. Dormy and A. M. Soward, Mathematical Aspects of Natural Dynamos, , Fluid Mechanics of Astrophysics and Geophysics, Grenoble Sciences, vol. 13, Universite Joseph Fourier, Grenoble, 2007. doi: 10.1201/9781420055269.  Google Scholar

[10]

M. A. Ebrahimi, M. Holst and E. Lunasin, The Navier-Stokes-Voight model for image inpainting, IMA J. Appl. Math., 78 (2013), 869-894.  doi: 10.1093/imamat/hxr069.  Google Scholar

[11]

J. A. Font, General relativistic hydrodynamics and magnetohydrodynamics: Hyperbolic systems in relativistic astrophysics, Hyperbolic Problems: Theory, Numerics, Applications, Springer, Berlin, 2008, 3–17. doi: 10.1007/978-3-540-75712-2_1.  Google Scholar

[12] J. F. Gerbeau, C. L. Bris and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006.   Google Scholar
[13]

M. Gunzburger, A. Meir and J. Peterson, On the existence, uniquess and finite element approximation of solutions of the equations of sationary, incompressible magnetohydrodynamic, Math. Comput., 56 (1991), 523-563.  doi: 10.1090/S0025-5718-1991-1066834-0.  Google Scholar

[14]

H. Hashizume, Numerical and experimental research to solve MHD problem in liquid blanket system, Fusion Eng. Des., 81 (2006), 1431-1438.  doi: 10.1016/j.fusengdes.2005.08.086.  Google Scholar

[15]

J. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes equations, Ⅳ: Error analysis for second order time discretizations, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.  Google Scholar

[16]

W. Hillebrandt and F. Kupka, Interdisciplinary Aspects of Turbulence, , Lecture Notes in Physics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-78961-1.  Google Scholar

[17]

N. Jiang, M. Kubacki, W. Layton, M. Moraiti and H. Tran, A Crank-Nicolson Leapfrog stabilization: Unconditional stability and two applications, J. Comput. Appl. Math., 281 (2015), 263-276.  doi: 10.1016/j.cam.2014.09.026.  Google Scholar

[18]

V. K. Kalantarov, B. Levant and E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152.  doi: 10.1007/s00332-008-9029-7.  Google Scholar

[19]

B. Khouider and E. Titi, An inviscid regularization for the surface quasi-geostrophic equation, Comm. Pure Appl. Math., 61 (2008), 1331-1346.  doi: 10.1002/cpa.20218.  Google Scholar

[20]

P. Kuberry, A. Larios, L. Rebholz and N. Wilson, Numerical approximation of the Voigt regularization for incompressible Navier-Stokes and magnetohydrodynamic flows, Comput. Math. Appl., 64 (2012), 2647-2662.  doi: 10.1016/j.camwa.2012.07.010.  Google Scholar

[21]

A. Labovsky, W. Layton, C. Manica, M. Neda and L. Rebholz, The stabilized extrapolated trapezoidal finite element method for the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 198 (2009), 958-974.  doi: 10.1016/j.cma.2008.11.004.  Google Scholar

[22]

A. Larios, E. Lunasin and E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differ. Equ., 255 (2013), 2636-2654.  doi: 10.1016/j.jde.2013.07.011.  Google Scholar

[23]

A. Larios and E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 603-627.  doi: 10.3934/dcdsb.2010.14.603.  Google Scholar

[24]

A. Larios and E. S. Titi, Higher-order global regularity of an inviscid Voigt-regularization of the three-dimensional inviscid resistive magnetohydrodynamic equations, J. Math. Fluid Mech., 16 (2014), 59-76.  doi: 10.1007/s00021-013-0136-3.  Google Scholar

[25]

W. Layton and C. Trenchea, Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations, Appl. Numer. Math., 62 (2012), 112-120.  doi: 10.1016/j.apnum.2011.10.006.  Google Scholar

[26]

B. Levant, F. Ramos and E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Commun. Math. Sci., 8 (2010), 277-293.  doi: 10.4310/CMS.2010.v8.n1.a14.  Google Scholar

[27]

T. Lin, J. Gilbert, R. Kossowsky and P. College, Sea-Water Magnetohydrodynamic Propulsion for Next-Generation Undersea Vehicles, Defens Technical Information Center, 1990. Google Scholar

[28]

X. L. Lu and P. Z. Huang, Unconditional stability of fully discrete scheme for the Kelvin-Voigt model, Univ. Politeh. Buchar. Sci. Bull. Ser. A Appl. Math. Phys., 81 (2019), 137-142.   Google Scholar

[29]

X. L. Lu, L. Zhang and P. Z. Huang, A fully discrete finite element scheme for the Kelvin-Voigt model, Filomat, 33 (2019), 5813-5827.  doi: 10.2298/FIL1918813L.  Google Scholar

[30]

X. L. Lu and P. Z. Huang, A modular grad-div stabilization for the 2D/3D nonstationary incompressible magnetohydrodynamic equations, J. Sci. Comput., 82 (2020), Paper No. 3, 24 pp. doi: 10.1007/s10915-019-01114-x.  Google Scholar

[31]

R. Moreau, Magneto-hydrodynamics, Kluwer Academic Publishers, Dordrecht, 1990. Google Scholar

[32]

B. Punsly, Black Hole Gravitohydromagnetics, Astrophysics and Space Science Library, Springer-Verlag, Berlin, 2008.  Google Scholar

[33]

F. Ramos and E. S. Titi, Invariant measure for the 3D Navier-Stokes-Voigt equations and their Navier-Stokes limit, Discrete Contin. Dyn. Syst., 28 (2010), 375-403.  doi: 10.3934/dcds.2010.28.375.  Google Scholar

[34]

S. Smolentsev, R. Moreau, L. Bühler and C. Mistrangelo, MHD thermofluid issues of liquid-metal blankets: Phenomena and advances, Fusion Eng. Des., 85 (2010), 1196-1205.  doi: 10.1016/j.fusengdes.2010.02.038.  Google Scholar

[35]

P. Wang, P. Huang and J. Wu, Superconvergence of the stationary incompressible magnetohydrodynamics equations, Univ. Politeh. Buchar. Sci. Bull. Ser. A Appl. Math. Phys., 80 (2018), 281-292.   Google Scholar

[36]

L. Wang, J. Li and P. Z. Huang, An efficient two-level algorithm for the 2D/3D stationary incompressible magnetohydrodynamics based on the finite element method, Int. Commun. Heat Mass Transf., 98 (2018), 183-190.  doi: 10.1016/j.icheatmasstransfer.2018.02.019.  Google Scholar

[37]

J. Yang, Y. N. He and G. Zhang, On an efficient second order backward difference Newton scheme for MHD system, J. Math. Anal. Appl., 458 (2018), 676-714.  doi: 10.1016/j.jmaa.2017.09.024.  Google Scholar

[38]

G. D. Zhang and 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.  Google Scholar

[39]

G. D. Zhang, J. J. Yang and C. J. Bi, Second order unconditionally convergent and energy stable linearized scheme for MHD equations, Adv. Comput. Math., 44 (2018), 505-540.  doi: 10.1007/s10444-017-9552-x.  Google Scholar

show all references

References:
[1]

H. Alfvén, Existence of electromagnetic-hydrodynamic waves, Nature, 150 (1942), 3763-3767.   Google Scholar

[2]

L. Barleon, V. Casal and L. Lenhart, MHD flow in liquid-metal-cooled blankets, Fusion Eng. Des., 14 (1991), 401-412.   Google Scholar

[3]

J. D. Barrow, R. Maartens and C. G. Tsagas, Cosmology with inhomogeneous magnetic fields, Phys. Rep., 449 (2007), 131-171.  doi: 10.1016/j.physrep.2007.04.006.  Google Scholar

[4]

R. Bermejo, P. Galán del Sastre and 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.  Google Scholar

[5]

P. Bodenheimer, G. P. Laughlin, M. Różyczka and H. W. Yorke, Numerical Methods in Astrophysics, Series in Astronomy and Astrophysics, Taylor and Francis, New York, 2007.  Google Scholar

[6]

M. A. Case, A. Labovsky, L. G. Rebholz and N. E. Wilson, A high physical accuracy method for incompressible magnetohydrodynamics, Int. J. Numer. Anal. Model. ser. B, 1 (2010), 217-236.   Google Scholar

[7]

D. Catania, Global existence for a regularized magnetohydrodynamic-$\alpha$ model, Ann. Univ. Ferrara., 56 (2010), 1-20.  doi: 10.1007/s11565-009-0069-1.  Google Scholar

[8] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511626333.  Google Scholar
[9]

E. Dormy and A. M. Soward, Mathematical Aspects of Natural Dynamos, , Fluid Mechanics of Astrophysics and Geophysics, Grenoble Sciences, vol. 13, Universite Joseph Fourier, Grenoble, 2007. doi: 10.1201/9781420055269.  Google Scholar

[10]

M. A. Ebrahimi, M. Holst and E. Lunasin, The Navier-Stokes-Voight model for image inpainting, IMA J. Appl. Math., 78 (2013), 869-894.  doi: 10.1093/imamat/hxr069.  Google Scholar

[11]

J. A. Font, General relativistic hydrodynamics and magnetohydrodynamics: Hyperbolic systems in relativistic astrophysics, Hyperbolic Problems: Theory, Numerics, Applications, Springer, Berlin, 2008, 3–17. doi: 10.1007/978-3-540-75712-2_1.  Google Scholar

[12] J. F. Gerbeau, C. L. Bris and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006.   Google Scholar
[13]

M. Gunzburger, A. Meir and J. Peterson, On the existence, uniquess and finite element approximation of solutions of the equations of sationary, incompressible magnetohydrodynamic, Math. Comput., 56 (1991), 523-563.  doi: 10.1090/S0025-5718-1991-1066834-0.  Google Scholar

[14]

H. Hashizume, Numerical and experimental research to solve MHD problem in liquid blanket system, Fusion Eng. Des., 81 (2006), 1431-1438.  doi: 10.1016/j.fusengdes.2005.08.086.  Google Scholar

[15]

J. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes equations, Ⅳ: Error analysis for second order time discretizations, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.  Google Scholar

[16]

W. Hillebrandt and F. Kupka, Interdisciplinary Aspects of Turbulence, , Lecture Notes in Physics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-78961-1.  Google Scholar

[17]

N. Jiang, M. Kubacki, W. Layton, M. Moraiti and H. Tran, A Crank-Nicolson Leapfrog stabilization: Unconditional stability and two applications, J. Comput. Appl. Math., 281 (2015), 263-276.  doi: 10.1016/j.cam.2014.09.026.  Google Scholar

[18]

V. K. Kalantarov, B. Levant and E. S. Titi, Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152.  doi: 10.1007/s00332-008-9029-7.  Google Scholar

[19]

B. Khouider and E. Titi, An inviscid regularization for the surface quasi-geostrophic equation, Comm. Pure Appl. Math., 61 (2008), 1331-1346.  doi: 10.1002/cpa.20218.  Google Scholar

[20]

P. Kuberry, A. Larios, L. Rebholz and N. Wilson, Numerical approximation of the Voigt regularization for incompressible Navier-Stokes and magnetohydrodynamic flows, Comput. Math. Appl., 64 (2012), 2647-2662.  doi: 10.1016/j.camwa.2012.07.010.  Google Scholar

[21]

A. Labovsky, W. Layton, C. Manica, M. Neda and L. Rebholz, The stabilized extrapolated trapezoidal finite element method for the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 198 (2009), 958-974.  doi: 10.1016/j.cma.2008.11.004.  Google Scholar

[22]

A. Larios, E. Lunasin and E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differ. Equ., 255 (2013), 2636-2654.  doi: 10.1016/j.jde.2013.07.011.  Google Scholar

[23]

A. Larios and E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 603-627.  doi: 10.3934/dcdsb.2010.14.603.  Google Scholar

[24]

A. Larios and E. S. Titi, Higher-order global regularity of an inviscid Voigt-regularization of the three-dimensional inviscid resistive magnetohydrodynamic equations, J. Math. Fluid Mech., 16 (2014), 59-76.  doi: 10.1007/s00021-013-0136-3.  Google Scholar

[25]

W. Layton and C. Trenchea, Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations, Appl. Numer. Math., 62 (2012), 112-120.  doi: 10.1016/j.apnum.2011.10.006.  Google Scholar

[26]

B. Levant, F. Ramos and E. S. Titi, On the statistical properties of the 3D incompressible Navier-Stokes-Voigt model, Commun. Math. Sci., 8 (2010), 277-293.  doi: 10.4310/CMS.2010.v8.n1.a14.  Google Scholar

[27]

T. Lin, J. Gilbert, R. Kossowsky and P. College, Sea-Water Magnetohydrodynamic Propulsion for Next-Generation Undersea Vehicles, Defens Technical Information Center, 1990. Google Scholar

[28]

X. L. Lu and P. Z. Huang, Unconditional stability of fully discrete scheme for the Kelvin-Voigt model, Univ. Politeh. Buchar. Sci. Bull. Ser. A Appl. Math. Phys., 81 (2019), 137-142.   Google Scholar

[29]

X. L. Lu, L. Zhang and P. Z. Huang, A fully discrete finite element scheme for the Kelvin-Voigt model, Filomat, 33 (2019), 5813-5827.  doi: 10.2298/FIL1918813L.  Google Scholar

[30]

X. L. Lu and P. Z. Huang, A modular grad-div stabilization for the 2D/3D nonstationary incompressible magnetohydrodynamic equations, J. Sci. Comput., 82 (2020), Paper No. 3, 24 pp. doi: 10.1007/s10915-019-01114-x.  Google Scholar

[31]

R. Moreau, Magneto-hydrodynamics, Kluwer Academic Publishers, Dordrecht, 1990. Google Scholar

[32]

B. Punsly, Black Hole Gravitohydromagnetics, Astrophysics and Space Science Library, Springer-Verlag, Berlin, 2008.  Google Scholar

[33]

F. Ramos and E. S. Titi, Invariant measure for the 3D Navier-Stokes-Voigt equations and their Navier-Stokes limit, Discrete Contin. Dyn. Syst., 28 (2010), 375-403.  doi: 10.3934/dcds.2010.28.375.  Google Scholar

[34]

S. Smolentsev, R. Moreau, L. Bühler and C. Mistrangelo, MHD thermofluid issues of liquid-metal blankets: Phenomena and advances, Fusion Eng. Des., 85 (2010), 1196-1205.  doi: 10.1016/j.fusengdes.2010.02.038.  Google Scholar

[35]

P. Wang, P. Huang and J. Wu, Superconvergence of the stationary incompressible magnetohydrodynamics equations, Univ. Politeh. Buchar. Sci. Bull. Ser. A Appl. Math. Phys., 80 (2018), 281-292.   Google Scholar

[36]

L. Wang, J. Li and P. Z. Huang, An efficient two-level algorithm for the 2D/3D stationary incompressible magnetohydrodynamics based on the finite element method, Int. Commun. Heat Mass Transf., 98 (2018), 183-190.  doi: 10.1016/j.icheatmasstransfer.2018.02.019.  Google Scholar

[37]

J. Yang, Y. N. He and G. Zhang, On an efficient second order backward difference Newton scheme for MHD system, J. Math. Anal. Appl., 458 (2018), 676-714.  doi: 10.1016/j.jmaa.2017.09.024.  Google Scholar

[38]

G. D. Zhang and 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.  Google Scholar

[39]

G. D. Zhang, J. J. Yang and C. J. Bi, Second order unconditionally convergent and energy stable linearized scheme for MHD equations, Adv. Comput. Math., 44 (2018), 505-540.  doi: 10.1007/s10444-017-9552-x.  Google Scholar

Figure 1.  $ H_{a} = 0.5 $, $ Re = Re_{m} = 0.1 $ (left: velocity; right: magnetic field)
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Figure 2.  $ H_{a}=5 $, $ Re=Re_{m}=1 $ (left: velocity; right: magnetic field)
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Figure 3.  $ H_{a}=50 $, $ Re=Re_{m}=10 $ (left: velocity; right: magnetic field)
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Figure 4.  $ H_{a} = 150 $, $ Re = Re_{m} = 30 $ (left: velocity; right: magnetic field)
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Table 1.  $ \|\mathbf{u}_{h}^{n}\|_{0} $ of the considered scheme for the 2D problem
$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2^{2} $ 0.34080 0.34067 0.34014 0.32677
$ 2^{3} $ 0.35282 0.35269 0.35215 0.33852
$ 2^{4} $ 0.35376 0.35363 0.35308 0.33944
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$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2^{2} $ 0.34080 0.34067 0.34014 0.32677
$ 2^{3} $ 0.35282 0.35269 0.35215 0.33852
$ 2^{4} $ 0.35376 0.35363 0.35308 0.33944
Table 2.  $ \|\nabla\mathbf{u}_{h}^{n}\|_{0} $ of the considered scheme for the 2D problem
$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2^{2} $ 2.49610 2.49517 2.49125 2.39290
$ 2^{3} $ 2.56163 2.56067 2.55668 2.45729
$ 2^{4} $ 2.56670 2.56574 2.56174 2.46228
Table Options

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$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2^{2} $ 2.49610 2.49517 2.49125 2.39290
$ 2^{3} $ 2.56163 2.56067 2.55668 2.45729
$ 2^{4} $ 2.56670 2.56574 2.56174 2.46228
Table 3.  $ \|\mathbf{B}_{h}^{n}\|_{0} $ of the considered scheme for the 2D problem
$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2^{2} $ 0.25873 0.25863 0.25824 0.25554
$ 2^{3} $ 0.26001 0.25991 0.25951 0.25682
$ 2^{4} $ 0.26001 0.25999 0.25960 0.25690
Table Options

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$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2^{2} $ 0.25873 0.25863 0.25824 0.25554
$ 2^{3} $ 0.26001 0.25991 0.25951 0.25682
$ 2^{4} $ 0.26001 0.25999 0.25960 0.25690
Table 4.  $ \|\nabla\mathbf{B}_{h}^{n}\|_{0} $ of the considered scheme for the 2D problem
$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2^{2} $ 1.15137 1.15093 1.14917 1.13716
$ 2^{3} $ 1.15530 1.15486 1.15310 1.14113
$ 2^{4} $ 1.15556 1.15512 1.15336 1.14140
Table Options

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$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2^{2} $ 1.15137 1.15093 1.14917 1.13716
$ 2^{3} $ 1.15530 1.15486 1.15310 1.14113
$ 2^{4} $ 1.15556 1.15512 1.15336 1.14140
Table 5.  $ \|\mathbf{u}_{h}^{n}\|_{0} $ of the considered scheme for the 3D problem
$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2 $ 0.04996 0.04994 0.04998 0.05332
$ 4 $ 0.09893 0.09890 0.09840 0.07804
Table Options

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$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2 $ 0.04996 0.04994 0.04998 0.05332
$ 4 $ 0.09893 0.09890 0.09840 0.07804
Table 6.  $ \|\nabla\mathbf{u}_{h}^{n}\|_{0} $ of the considered scheme for the 3D problem
$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2 $ 0.59884 0.59861 0.60165 0.71084
$ 4 $ 0.97397 0.97361 0.96881 0.78582
Table Options

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$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2 $ 0.59884 0.59861 0.60165 0.71084
$ 4 $ 0.97397 0.97361 0.96881 0.78582
Table 7.  $ \|\mathbf{B}_{h}^{n}\|_{0} $ of the considered scheme for the 3D problem
$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2 $ 0.21840 0.21832 0.21787 0.17749
$ 4 $ 0.26288 0.26278 0.26223 0.21420
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$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2 $ 0.21840 0.21832 0.21787 0.17749
$ 4 $ 0.26288 0.26278 0.26223 0.21420
Table 8.  $ \|\nabla\mathbf{B}_{h}^{n}\|_{0} $ of the considered scheme for the 3D problem
$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2 $ 1.59528 1.59468 1.59254 1.41689
$ 4 $ 1.84022 1.83953 1.83450 1.54426
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$ \frac{1}{h} $ $ \frac{1}{\tau} $
$ 2^{6} $ $ 2^{5} $ $ 2^{4} $ $ 2^{3} $
$ 2 $ 1.59528 1.59468 1.59254 1.41689
$ 4 $ 1.84022 1.83953 1.83450 1.54426
Table 9.  Error and convergence rates for the considered scheme with $ {\tau} = \mathbb{O}(h) $ for the 2D problem
$ h $ $ \|E(\mathbf{u})\| $ Rate $ \|E(\mathbf{B})\| $ Rate $ \|E(p)\| $ Rate
1/10 0.062879 - 0.009524 - 0.005226 -
1/20 0.015703 2.001 0.002346 2.021 0.001348 1.955
1/40 0.003903 2.008 0.000581 2.014 0.000303 2.153
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$ h $ $ \|E(\mathbf{u})\| $ Rate $ \|E(\mathbf{B})\| $ Rate $ \|E(p)\| $ Rate
1/10 0.062879 - 0.009524 - 0.005226 -
1/20 0.015703 2.001 0.002346 2.021 0.001348 1.955
1/40 0.003903 2.008 0.000581 2.014 0.000303 2.153
Table 10.  Numerical convergence rates for velocity in $ H^{1} $-norm with variation in $ \kappa_{1} $ and $ \kappa_{2} $
$ h $ Rate Rate Rate Rate Rate
$ \kappa_{1}=\kappa_{2} $=1E-2 $ \kappa_{1}=\kappa_{2} $=1E-4 $ \kappa_{1}=\kappa_{2} $=1E-8 $ \kappa_{1} $=1E-2, $ \kappa_{2} $=1E-8 $ \kappa_{1} $=1E-8, $ \kappa_{2} $=1E-2
1/10 - - - - -
1/20 2.001 2.008 2.008 2.001 2.008
1/40 2.008 2.010 2.011 2.008 2.011
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$ h $ Rate Rate Rate Rate Rate
$ \kappa_{1}=\kappa_{2} $=1E-2 $ \kappa_{1}=\kappa_{2} $=1E-4 $ \kappa_{1}=\kappa_{2} $=1E-8 $ \kappa_{1} $=1E-2, $ \kappa_{2} $=1E-8 $ \kappa_{1} $=1E-8, $ \kappa_{2} $=1E-2
1/10 - - - - -
1/20 2.001 2.008 2.008 2.001 2.008
1/40 2.008 2.010 2.011 2.008 2.011
Table 11.  Numerical convergence rates for magnetic in $ H^{1} $-norm with variation in $ \kappa_{1} $ and $ \kappa_{2} $
$ h $ Rate Rate Rate Rate Rate
$ \kappa_{1}=\kappa_{2} $=1E-2 $ \kappa_{1}=\kappa_{2} $=1E-4 $ \kappa_{1}=\kappa_{2} $=1E-8 $ \kappa_{1} $=1E-2, $ \kappa_{2} $=1E-8 $ \kappa_{1} $=1E-8, $ \kappa_{2} $=1E-2
1/10 - - - - -
1/20 2.021 2.027 2.026 2.026 2.021
1/40 2.014 2.016 2.016 2.016 2.013
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$ h $ Rate Rate Rate Rate Rate
$ \kappa_{1}=\kappa_{2} $=1E-2 $ \kappa_{1}=\kappa_{2} $=1E-4 $ \kappa_{1}=\kappa_{2} $=1E-8 $ \kappa_{1} $=1E-2, $ \kappa_{2} $=1E-8 $ \kappa_{1} $=1E-8, $ \kappa_{2} $=1E-2
1/10 - - - - -
1/20 2.021 2.027 2.026 2.026 2.021
1/40 2.014 2.016 2.016 2.016 2.013
Table 12.  Error and convergence rates for the considered scheme with $ {\tau} = \mathbb{O}(h) $ for the 3D problem
$ h $ $ \|E(\mathbf{u})\| $ Rate $ \|E(\mathbf{B})\| $ Rate $ \|E(p)\| $ Rate
1/2 0.690006 - 0.325897 - 0.256334 -
1/4 0.273947 1.333 0.135307 1.268 0.077825 1.720
1/6 0.163001 1.280 0.086166 1.113 0.040745 1.596
1/8 0.117710 1.132 0.066413 0.905 0.026852 1.449
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$ h $ $ \|E(\mathbf{u})\| $ Rate $ \|E(\mathbf{B})\| $ Rate $ \|E(p)\| $ Rate
1/2 0.690006 - 0.325897 - 0.256334 -
1/4 0.273947 1.333 0.135307 1.268 0.077825 1.720
1/6 0.163001 1.280 0.086166 1.113 0.040745 1.596
1/8 0.117710 1.132 0.066413 0.905 0.026852 1.449
Table 13.  Numerical convergence rates for velocity in $ H^{1} $-norm with variation in $ \kappa_{1} $ and $ \kappa_{2} $
$ h $ Rate Rate Rate Rate Rate
$ \kappa_{1}=\kappa_{2} $=1E-2 $ \kappa_{1}=\kappa_{2} $=1E-4 $ \kappa_{1}=\kappa_{2} $=1E-8 $ \kappa_{1} $=1E-2, $ \kappa_{2} $=1E-8 $ \kappa_{1} $=1E-8, $ \kappa_{2} $=1E-2
1/2 - - - - -
1/4 1.333 1.058 1.049 1.333 1.049
1/6 1.280 1.038 1.016 1.280 1.016
1/8 1.132 1.013 0.974 1.132 0.974
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$ h $ Rate Rate Rate Rate Rate
$ \kappa_{1}=\kappa_{2} $=1E-2 $ \kappa_{1}=\kappa_{2} $=1E-4 $ \kappa_{1}=\kappa_{2} $=1E-8 $ \kappa_{1} $=1E-2, $ \kappa_{2} $=1E-8 $ \kappa_{1} $=1E-8, $ \kappa_{2} $=1E-2
1/2 - - - - -
1/4 1.333 1.058 1.049 1.333 1.049
1/6 1.280 1.038 1.016 1.280 1.016
1/8 1.132 1.013 0.974 1.132 0.974
Table 14.  Numerical convergence rates for magnetic in $ H^{1} $-norm with variation in $ \kappa_{1} $ and $ \kappa_{2} $
$ h $ Rate Rate Rate Rate Rate
$ \kappa_{1}=\kappa_{2} $=1E-2 $ \kappa_{1}=\kappa_{2} $=1E-4 $ \kappa_{1}=\kappa_{2} $=1E-8 $ \kappa_{1} $=1E-2, $ \kappa_{2} $=1E-8 $ \kappa_{1} $=1E-8, $ \kappa_{2} $=1E-2
1/2 - - - - -
1/4 1.268 1.055 1.048 1.048 1.268
1/6 1.113 0.960 0.948 0.948 1.113
1/8 0.905 0.889 0.864 0.864 0.905
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$ h $ Rate Rate Rate Rate Rate
$ \kappa_{1}=\kappa_{2} $=1E-2 $ \kappa_{1}=\kappa_{2} $=1E-4 $ \kappa_{1}=\kappa_{2} $=1E-8 $ \kappa_{1} $=1E-2, $ \kappa_{2} $=1E-8 $ \kappa_{1} $=1E-8, $ \kappa_{2} $=1E-2
1/2 - - - - -
1/4 1.268 1.055 1.048 1.048 1.268
1/6 1.113 0.960 0.948 0.948 1.113
1/8 0.905 0.889 0.864 0.864 0.905
Table 15.  Errors for the different methods of 3D Hartmann flow at T = 10
Methods $ {\tau}=h $ $ \|\mathbf{u}(T)-\mathbf{u}_{h}^{N}\|_{0,2} $ $ \|\mathbf{B}(T)-\mathbf{B}_{h}^{N}\|_{0,2} $
Algorithm 3.1 1/4 8.20E-02 3.47E-02
Zhang's algorithm [39] 1/4 9.49E-02 7.22E-02
Linearized Crank-Nicolson [39] 1/4 9.50E-02 7.22E-02
[5pt] Algorithm 3.1 1/8 2.44E-02 1.27E-02
Zhang's algorithm [39] 1/8 3.58E-02 3.24E-02
Linearized Crank-Nicolson [39] 1/8 3.58E-02 3.24E-02
[5pt] Algorithm 3.1 1/16 1.09E-02 9.38E-03
Zhang's algorithm [39] 1/16 1.15E-02 1.08E-02
Linearized Crank-Nicolson [39] 1/16 1.15E-02 1.08E-02
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Methods $ {\tau}=h $ $ \|\mathbf{u}(T)-\mathbf{u}_{h}^{N}\|_{0,2} $ $ \|\mathbf{B}(T)-\mathbf{B}_{h}^{N}\|_{0,2} $
Algorithm 3.1 1/4 8.20E-02 3.47E-02
Zhang's algorithm [39] 1/4 9.49E-02 7.22E-02
Linearized Crank-Nicolson [39] 1/4 9.50E-02 7.22E-02
[5pt] Algorithm 3.1 1/8 2.44E-02 1.27E-02
Zhang's algorithm [39] 1/8 3.58E-02 3.24E-02
Linearized Crank-Nicolson [39] 1/8 3.58E-02 3.24E-02
[5pt] Algorithm 3.1 1/16 1.09E-02 9.38E-03
Zhang's algorithm [39] 1/16 1.15E-02 1.08E-02
Linearized Crank-Nicolson [39] 1/16 1.15E-02 1.08E-02
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