Article Contents
Article Contents

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

• * Corresponding author: Pengzhan Huang

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.

Mathematics Subject Classification: Primary: 65N30.

 Citation:

• Figure 1.  $H_{a} = 0.5$, $Re = Re_{m} = 0.1$ (left: velocity; right: magnetic field)

Figure 2.  $H_{a}=5$, $Re=Re_{m}=1$ (left: velocity; right: magnetic field)

Figure 3.  $H_{a}=50$, $Re=Re_{m}=10$ (left: velocity; right: magnetic field)

Figure 4.  $H_{a} = 150$, $Re = Re_{m} = 30$ (left: velocity; right: magnetic field)

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

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 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 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 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 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 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

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

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

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

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

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

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

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

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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