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[J]. Electronic Research Archive, 2020, 28(3): 1191-1205. doi: 10.3934/era.2020065
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.
| [1] |
Numerical solution to a mixed Navier-Stokes/Darcy model by the two-grid approach. SIMA J. Numer. Anal. (2009) 47: 3325-3338. 
|
| [2] |
Mortar element method for the time dependent coupling of Stokes and Darcy flows. J. Sci. Comput. (2019) 80: 1310-1329.
|
| [3] |
An efficient and modular grad-div stabilization. Comput. Methods Appl. Mech. Engrg. (2018) 355: 327-346.
|
| [4] |
Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Engrg. (1988) 69: 89-129.
|
| [5] |
DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition. SIAM J. Numer. Anal. (2009) 47: 2052-2089.
|
| [6] |
A modified two-grid decoupling method for the mixed Navier-Stokes/Darcy model. Comput. Math.Appl. (2016) 72: 1142-1152.
|
| [7] |
Partitioned time stepping method for fully evolutionary Navier-Stokes/ Darcy flow with BJS interface conditions. Adv. Appl. Math. Mech. (2019) 11: 381-405.
|
| [8] |
Decoupled characteristic stabilized finite element method for time-dependent Navier-Stokes/Darcy model. Numer. Meth. Part D. E. (2019) 35: 267-294.
|
| [9] | A decoupling method with different subdomain time steps for the non-stationary Navier-Stokes/Darcy model. J. Comput. Math. (2017) 35: 319-345. |
| [10] |
On the convergence rate of grad-giv stabilized Taylor-Hood to Scott-Vogelius solutions of incompressible flow problems. J. Math. Anal. Appl. (2011) 381: 612-626.
|
| [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
|
| [12] |
Decoupled schemes for a non-stationary mixed Stokes-Darcy model. Math. Comput. (2010) 79: 707-731.
|
| [13] |
Numerical analysis of two grad-div stabilization methods for the time-dependent Stokes/Darcy model. Comput. Math. Appl. (2020) 79: 817-832.
|
| [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
|
| [15] |
Two-grid finite element methods for the steady Navier-Stokes/Darcy model. East Asian J. Appl. Math. (2016) 6: 60-79.
|
| [16] |
A decoupling two-grid algorithm for the mixed Stokes-Darcy model with the Beavers-Joseph interface condition. Numer. Methods Partial Differential Equations (2014) 30: 1066-1082.
|
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Jiangshan Wang, Lingxiong Meng, Hongen Jia. Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model[J]. Electronic Research Archive, 2020, 28(3): 1191-1205. doi: 10.3934/era.2020065
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