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September  2020, 28(3): 1191-1205. doi: 10.3934/era.2020065

Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model

Taiyuan University of Technology, Taiyuan 030024, China

* Corresponding author: jiahongen@aliyun.com

Received  March 2020 Revised  June 2020 Published  July 2020

Fund Project: The first author is supported by the Provincial Natural Science Foundation of Shanxi grant 201901D111123, Key Research and Development (R&D) Projects of Shanxi Province grant 201903D121038

In this paper, we construct a modular grad-div stabilization method for the Navier-Stokes/Darcy model, which is based on the first order Backward Euler scheme. This method does not enlarge the accuracy of numerical solution, but also can improve mass conservation and relax the influence of parameters. Herein, we give stability analysis and error estimations. Finally, by some numerical experiment, the scheme our proposed is shown to be valid.

Citation: Jiangshan Wang, Lingxiong Meng, Hongen Jia. Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model. Electronic Research Archive, 2020, 28 (3) : 1191-1205. doi: 10.3934/era.2020065
References:
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M. CaiM. Mu and J. Xu, Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach, SIMA J. Numer. Anal., 47 (2009), 3325-3338.  doi: 10.1137/080721868.  Google Scholar

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H. E. JiaH. Y. Jia and Y. Q. Huang, A modified two-grid decoupling method for the mixed Navier-Stokes/Darcy model, Comput. Math.Appl., 72 (2016), 1142-1152.  doi: 10.1016/j.camwa.2016.06.033.  Google Scholar

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[10]

A. LinkeL. G. Rebholz and N. E. Wilson, On the convergence rate of grad-giv stabilized Taylor-Hood to Scott-Vogelius solutions of incompressible flow problems, J. Math. Anal. Appl., 381 (2011), 612-626.  doi: 10.1016/j.jmaa.2011.03.019.  Google Scholar

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M. Mu and X. H. Zhu, Decoupled schemes for a non-stationary mixed Stokes-Darcy model, Math. Comput., 79 (2010), 707-731.  doi: 10.1090/S0025-5718-09-02302-3.  Google Scholar

[13]

Y. QinY. HouP. Huang and Y. Wang, Numerical analysis of two grad-div stabilization methods for the time-dependent Stokes/Darcy model, Comput. Math. Appl., 79 (2020), 817-832.  doi: 10.1016/j.camwa.2019.07.032.  Google Scholar

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show all references

References:
[1]

M. CaiM. Mu and J. Xu, Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach, SIMA J. Numer. Anal., 47 (2009), 3325-3338.  doi: 10.1137/080721868.  Google Scholar

[2]

Y. ChenX. Zhao and Y. Huang, Mortar element method for the time dependent coupling of Stokes and Darcy flows, J. Sci. Comput., 80 (2019), 1310-1329.  doi: 10.1007/s10915-019-00977-4.  Google Scholar

[3]

J. A. FiordilinoW. Layton and Y. Rong, An efficient and modular grad-div stabilization, Comput. Methods Appl. Mech. Engrg., 355 (2018), 327-346.  doi: 10.1016/j.cma.2018.02.023.  Google Scholar

[4]

L. P. Franca and T. J. R. Hughes, Two classes of mixed finite element methods, Comput. Methods Appl. Mech. Engrg., 69 (1988), 89-129.  doi: 10.1016/0045-7825(88)90168-5.  Google Scholar

[5]

V. Girault and B. Rivière, DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition, SIAM J. Numer. Anal., 47 (2009), 2052-2089.  doi: 10.1137/070686081.  Google Scholar

[6]

H. E. JiaH. Y. Jia and Y. Q. Huang, A modified two-grid decoupling method for the mixed Navier-Stokes/Darcy model, Comput. Math.Appl., 72 (2016), 1142-1152.  doi: 10.1016/j.camwa.2016.06.033.  Google Scholar

[7]

H. E. JiaY. S. Zhang and J. P. Yu, Partitioned time stepping method for fully evolutionary Navier-Stokes/ Darcy flow with BJS interface conditions, Adv. Appl. Math. Mech., 11 (2019), 381-405.  doi: 10.4208/aamm.oa-2018-0102.  Google Scholar

[8]

X. F. JiaJ. C. Li and H. E. Jia, Decoupled characteristic stabilized finite element method for time-dependent Navier-Stokes/Darcy model, Numer. Meth. Part D. E., 35 (2019), 267-294.  doi: 10.1002/num.22300.  Google Scholar

[9]

H. Y. JiaP. L. ShiK. T. Li and H. E. Jia, A decoupling method with different subdomain time steps for the non-stationary Navier-Stokes/Darcy model, J. Comput. Math., 35 (2017), 319-345.   Google Scholar

[10]

A. LinkeL. G. Rebholz and N. E. Wilson, On the convergence rate of grad-giv stabilized Taylor-Hood to Scott-Vogelius solutions of incompressible flow problems, J. Math. Anal. Appl., 381 (2011), 612-626.  doi: 10.1016/j.jmaa.2011.03.019.  Google Scholar

[11]

X. Lu and P. 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

[12]

M. Mu and X. H. Zhu, Decoupled schemes for a non-stationary mixed Stokes-Darcy model, Math. Comput., 79 (2010), 707-731.  doi: 10.1090/S0025-5718-09-02302-3.  Google Scholar

[13]

Y. QinY. HouP. Huang and Y. Wang, Numerical analysis of two grad-div stabilization methods for the time-dependent Stokes/Darcy model, Comput. Math. Appl., 79 (2020), 817-832.  doi: 10.1016/j.camwa.2019.07.032.  Google Scholar

[14]

Y. Zeng and P. Huang, A grad-div stabilized projection finite element method for a double-diffusive natural convection model, Numer. Heat Tr. B-Fund., (2020). doi: 10.1080/10407790.2020.1747285.  Google Scholar

[15]

J. Zhao and T. zhang, Two-grid finite element methods for the steady Navier-Stokes/Darcy model, East Asian J. Appl. Math., 6 (2016), 60-79.  doi: 10.4208/eajam.080215.111215a.  Google Scholar

[16]

L. Zuo and Y. Hou, A decoupling two-grid algorithm for the mixed Stokes-Darcy model with the Beavers-Joseph interface condition, Numer. Methods Partial Differential Equations, 30 (2014), 1066-1082.  doi: 10.1002/num.21860.  Google Scholar

Figure 1.  The global domain $ \Omega $
Table 1.  Numerical results at time T = 1 for the standard scheme
$\frac{1}{h}$$||e_{\bf u}||_{L^2}$${\bf u}_{L^2}rate$$|| e_{\bf u}||_f$${\bf u}_{H_f}rate$$||\nabla \cdot e_{\bf u}||_{L^2}$$div{\bf u}_{L^2}rate$
$4$0.01589060.03528820.042823
$8$0.008477820.9064080.01618721.124330.009784382.12983
$16$0.004367990.9567240.008050771.007650.002301982.08761
$32$0.002215160.9795590.004043690.9934540.0006099021.91623
$64$0.001115310.9899660.002031120.9933970.0001314012.2146
$\frac{1}{h}$$||e_\phi||_{L^2}$$\phi_{L^2}rate$$|| e_\phi||_p$$\phi_{H_p}rate$$|| e_p||_{L^2}$$ p_{L^2}rate$
$4$0.04047640.06530310.500655
$8$0.01824771.149370.01826491.838080.257160.961151
$16$0.009273250.9765680.007774671.232220.1305690.977854
$32$0.004686120.9846810.003706711.068640.06583430.987901
$64$0.002355840.9921520.001838881.011310.03304730.994307
$\frac{1}{h}$$||e_{\bf u}||_{L^2}$${\bf u}_{L^2}rate$$|| e_{\bf u}||_f$${\bf u}_{H_f}rate$$||\nabla \cdot e_{\bf u}||_{L^2}$$div{\bf u}_{L^2}rate$
$4$0.01589060.03528820.042823
$8$0.008477820.9064080.01618721.124330.009784382.12983
$16$0.004367990.9567240.008050771.007650.002301982.08761
$32$0.002215160.9795590.004043690.9934540.0006099021.91623
$64$0.001115310.9899660.002031120.9933970.0001314012.2146
$\frac{1}{h}$$||e_\phi||_{L^2}$$\phi_{L^2}rate$$|| e_\phi||_p$$\phi_{H_p}rate$$|| e_p||_{L^2}$$ p_{L^2}rate$
$4$0.04047640.06530310.500655
$8$0.01824771.149370.01826491.838080.257160.961151
$16$0.009273250.9765680.007774671.232220.1305690.977854
$32$0.004686120.9846810.003706711.068640.06583430.987901
$64$0.002355840.9921520.001838881.011310.03304730.994307
Table 2.  Numerical results at time T = 1 for the standard grad-div scheme
$\frac{1}{h}$$||e_{\bf u}||_{L^2}$${\bf u}_{L^2}rate$$||e_{\bf u}||_f$${\bf u}_{H_f}rate$$||\nabla \cdot e_{\bf u}||_{L^2}$$div{\bf u}_{L^2}rate$
$4$0.0157960.03499070.0332181
$8$0.008474770.8983130.01616291.114290.007956382.06179
$16$0.004367880.9562410.008048561.005880.001925462.04691
$32$0.002215150.9795290.004043510.9931230.0005518831.80277
$64$0.001115310.989960.00203110.9933470.0001129742.28837
$\frac{1}{h}$$|| e_\phi||_{L^2}$$\phi_{L^2}rate$$||e_\phi||_p$$\phi_{H_p}rate$$|| e_p||_{L^2}$$ p_{L^2}rate$
$4$0.0403890.06527550.512196
$8$0.0182391.146940.01826131.837750.2574430.992443
$16$0.009272650.9759730.007774411.231980.1305820.979297
$32$0.004686070.9846030.003706691.06860.06583570.988014
$64$0.002355830.9921430.001838881.01130.03304740.994333
$\frac{1}{h}$$||e_{\bf u}||_{L^2}$${\bf u}_{L^2}rate$$||e_{\bf u}||_f$${\bf u}_{H_f}rate$$||\nabla \cdot e_{\bf u}||_{L^2}$$div{\bf u}_{L^2}rate$
$4$0.0157960.03499070.0332181
$8$0.008474770.8983130.01616291.114290.007956382.06179
$16$0.004367880.9562410.008048561.005880.001925462.04691
$32$0.002215150.9795290.004043510.9931230.0005518831.80277
$64$0.001115310.989960.00203110.9933470.0001129742.28837
$\frac{1}{h}$$|| e_\phi||_{L^2}$$\phi_{L^2}rate$$||e_\phi||_p$$\phi_{H_p}rate$$|| e_p||_{L^2}$$ p_{L^2}rate$
$4$0.0403890.06527550.512196
$8$0.0182391.146940.01826131.837750.2574430.992443
$16$0.009272650.9759730.007774411.231980.1305820.979297
$32$0.004686070.9846030.003706691.06860.06583570.988014
$64$0.002355830.9921430.001838881.01130.03304740.994333
Table 3.  Numerical results at time T = 1 for the modular grad-div scheme
$\frac{1}{h}$$||e_{\bf u}||_{L^2}$${\bf u}_{L^2}rate$$|| e_{\bf u}||_f$${\bf u}_{H_f}rate$$||\nabla \cdot e_{\bf u}||_{L^2}$$div{\bf u}_{L^2}rate$
$4$0.01612270.04222950.00546336
$8$0.008494780.9244450.01869871.175310.001336662.03116
$16$0.004368880.9593130.008496081.138070.0002500082.41859
$32$0.002215260.9797870.00423941.002947.36766e-0051.7627
$64$0.001115310.9900310.002044151.052366.93936e-0063.40833
$\frac{1}{h}$$|| e_\phi||_{L^2}$$\phi_{L^2}rate$$|| e_\phi||_p$$\phi_{H_p}rate$$|| e_p||_{L^2}$$ p_{L^2}rate$
$4$0.04038030.06525230.500113
$8$0.01823351.147060.01825841.837470.2571520.959633
$16$0.009272490.9755630.007774361.231760.130570.977798
$32$0.004686080.9845750.00370671.068590.06583450.987908
$64$0.002355830.9921460.001838881.011310.03304730.994311
$\frac{1}{h}$$||e_{\bf u}||_{L^2}$${\bf u}_{L^2}rate$$|| e_{\bf u}||_f$${\bf u}_{H_f}rate$$||\nabla \cdot e_{\bf u}||_{L^2}$$div{\bf u}_{L^2}rate$
$4$0.01612270.04222950.00546336
$8$0.008494780.9244450.01869871.175310.001336662.03116
$16$0.004368880.9593130.008496081.138070.0002500082.41859
$32$0.002215260.9797870.00423941.002947.36766e-0051.7627
$64$0.001115310.9900310.002044151.052366.93936e-0063.40833
$\frac{1}{h}$$|| e_\phi||_{L^2}$$\phi_{L^2}rate$$|| e_\phi||_p$$\phi_{H_p}rate$$|| e_p||_{L^2}$$ p_{L^2}rate$
$4$0.04038030.06525230.500113
$8$0.01823351.147060.01825841.837470.2571520.959633
$16$0.009272490.9755630.007774361.231760.130570.977798
$32$0.004686080.9845750.00370671.068590.06583450.987908
$64$0.002355830.9921460.001838881.011310.03304730.994311
Table 4.  The $||\nabla \cdot e_{\bf u}||_f$ for the standard without grad-div scheme, standard scheme and grad-div scheme with vaying hydraulic conductivity tensor $\mathbf K$
${\bf K}$Non-stabilizedStandard grad-divmodular grad-div
${\bf I}$0.01691730.01080850.077557
$1e-1{\bf I}$0.02029740.01304370.0775562
$1e-2{\bf I}$0.04646250.03018790.077554
$1e-3{\bf I}$0.1242380.08249880.0775521
${\bf K}$Non-stabilizedStandard grad-divmodular grad-div
${\bf I}$0.01691730.01080850.077557
$1e-1{\bf I}$0.02029740.01304370.0775562
$1e-2{\bf I}$0.04646250.03018790.077554
$1e-3{\bf I}$0.1242380.08249880.0775521
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