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

Xiaoli Lu; Pengzhan Huang; Yinnian He

doi:10.3934/dcdsb.2020143

Article Contents

2021, Volume 26, Issue 2: 815-845. Doi: 10.3934/dcdsb.2020143

This issue Previous Article Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations Next Article Chaos control in a special pendulum system for ultra-subharmonic resonance

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

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
^* Corresponding author: Pengzhan Huang

Received: June 30, 2018

Revised: January 31, 2020

Early access: May 2020

Published: February 2021

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

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Abstract

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:

Mathematics Subject Classification: Primary: 65N30.

Citation:

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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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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

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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

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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

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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

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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

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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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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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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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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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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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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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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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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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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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