A second order in time incremental pressure correction finite element method for the Navier-Stokes/Darcy problem

Wang, Yunxia; Li, Shishun; Si, Zhiyong

doi:10.1051/m2an/2017049

Wang, Yunxia ¹ ; Li, Shishun ² ; Si, Zhiyong ²

¹ School of Materials Science and Engineering, Henan Polytechnic University, 454003, Jiaozuo, P.R. China
² School of Mathematics and Information Science, Henan Polytechnic University, 454003, Jiaozuo, P.R. China

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1477-1500.

Résumé

In this paper, we give a second order in time incremental pressure correction finite element method for the Navier-Stokes/Darcy problem. In this method, the Navier-Stokes/Darcy problem is solved in three steps: a convection-diffusion step, a projection correction (incremental pressure correction) step and a Darcy step. In this way, the Navier-Stokes/Darcy equation is solved in a fractional step way, which is a decoupled method. In order to decouple the equation, we use the numerical solutions at the last time level to give the interface conditions. The stability analysis shows that the second order in time incremental pressure correction finite element method is unconditionally stable. The optimal error estimate is also given. Finally, we present some numerical results to show the efficiency of the method.

Reçu le : 2016-04-25
Accepté le : 2017-09-25

MR Zbl

DOI : 10.1051/m2an/2017049

Classification : 76D05, 35Q30, 65M60, 65N30
Mots-clés : Navier-Stokes/Darcy equations, projection method, second order in time, incremental pressure correction method, stability analysis, optimal error analysis

Affiliations des auteurs :

Wang, Yunxia ¹ ; Li, Shishun ² ; Si, Zhiyong ²

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     title = {A second order in time incremental pressure correction finite element method for the {Navier-Stokes/Darcy} problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1477--1500},
     publisher = {EDP-Sciences},
     volume = {52},
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Wang, Yunxia; Li, Shishun; Si, Zhiyong. A second order in time incremental pressure correction finite element method for the Navier-Stokes/Darcy problem. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 4, pp. 1477-1500. doi : 10.1051/m2an/2017049. http://archive.numdam.org/articles/10.1051/m2an/2017049/

Bibliographie
Cité par

[1] I. Babuska and G. Gatica, A residual-based a posteriori error estimator for the Stokes-Darcy coupled problem. SIAM J. Numer. Anal. 48 (2010) 498–523. | DOI | MR | Zbl

[2] S. Badia and R. Codina, Unified stabilized finite element formulations for the Stokes and the Darcy problems. SIAM J. Numer. Anal. 47 (2009) 1971–2000. | DOI | MR | Zbl

[3] L. Badea, M. Discacciati and A. Quarteroni, Numerical analysis of the Navier-Stokes/Darcy coupling. Numer. Math. 115 (2010) 195–227. | DOI | MR | Zbl

[4] G. Beavers and D.D. Joseph, Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30 (1967) 197–207. | DOI

[5] M. Cai, M. Mu and J. Xu, Numerical solution to a mixed Navier–Stokes/Darcy model by the two-grid approach. SIAM J. Numer. Anal. 47 (2009) 3325–3338. | DOI | MR | Zbl

[6] Y. Cao, M. Gunzburger, X. M. He and X. Wang, Robin-Robin domain decomposition methods for the steady–state Stokes– Darcy system with the Beavers–Joseph interface condition. Numer. Math. 117 (2011) 601–629. | DOI | MR | Zbl

[7] A. Çeçmelioğlu and B. Rivière, Existence of a weak solution for the fully coupled Navier-Stokes/Darcy-transport problem. J. Differ. Equ. 252 (2012) 4138–4175. | DOI | MR | Zbl

[8] P. Chidyagwai and B. Rivière, On the solution of the coupled Navier-Stokes and Darcy equations. Comput. Method Appl. M. 198 (2009) 3806–3820. | DOI | MR | Zbl

[9] A. J. Chorin, Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745–762. | DOI | MR | Zbl

[10] W. Chen, M. Gunzburger, F. Hua and X. Wang, A parallel Robin-Robin domain decomposition method for the Stokes-Darcy system. SIAM J. Numer. Anal. 49 (2011) 1064–1084. | DOI | MR | Zbl

[11] A. J. Chorin, Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968) 745–762. | DOI | MR | Zbl

[12] M. Cui and N. Yan, A posteriori error estimate for the Stokes-Darcy system. Math. Method Appl. Sci. 34 (2011) 1050–1064. | DOI | MR | Zbl

[13] M. Discacciati and A. Quarteroni, Navier-Stokes/Darcy coupling: Modeling, analysis, and numerical approximation. Rev. Mat. Comp. 22 (2009) 315–426. | MR | Zbl

[14] M. Discacciati, A. Quarteroni and A. Valli, Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM J. Numer. Anal. 45 (2007) 1246–1268. | DOI | MR | Zbl

[15] W.E. and J. Liu, Projection method I: Convergence and numerical boundary layers. SIAM J. Numer. Anal. 32 (1995) 1017– 1057. | MR | Zbl

[16] W.E. and J. Liu, Gauge method for viscous incompressible flows. Commun. Math. Sci. 1 (2003) 317–332. | DOI | MR | Zbl

[17] G. N. Gatica, R. Oyarzua and F. J. Sayas, Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem. Math. Comput. 80 (2011) 1911–1948. | DOI | MR | Zbl

[18] V. Girault and B. Riviere, DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition. SIAM J. Numer. Anal. 47 (2009) 2052–2089. | DOI | MR | Zbl

[19] J. Guermond, Some practical implementations of projection methods for Navier Stokes equations. ESAIM: M2AN 30 (1996) 637–C667. | DOI | Numdam | MR | Zbl

[20] J. Guermond, Un résultat de convergence d’ordre deux en temps pour l’approximation des équations de Navier–Stokes par une technique deprojection incrementale. ESAIM: M2AN 33 (1999) 169–189. | DOI | Numdam | MR | Zbl

[21] J. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows. Comput. Method Appl. Math. 195 (2006) 6011–6045. | MR | Zbl

[22] J. Guermond and L. Quartapelle, On the approximation of the unsteady Navier-Stokes equations by finite element projection methods. Numer. Math. 80 (1998) 207–238. | DOI | MR | Zbl

[23] J. Guermond and P. Minev, Analysis of a projection/characteristic scheme for incompressible flow. Comm. Numer. Methods Eng. 19 (2003) 535–550. | DOI | MR | Zbl

[24] J. Guermond, P. Minev and J. Shen, Error analysis of pressure-correction schemes for the Navier-Stokes equations with open boundary conditions. SIAM J. Numer. Anal. 43 (2005) 239–258. | DOI | MR | Zbl

[25] J. Guermond and L. Quartapelle, On stability and convergence of projection methods based on pressure Poisson equation. Int. J. Numer. Methods Fluids. 26 (1998) 1039–1053. | DOI | MR | Zbl

[26] J. Guermond and L. Quartapelle, On the approximation of the unsteady Navier Stokes equations by finite element projection methods. Numer. Math. 80 (1998) 207–238. | DOI | MR | Zbl

[27] J. Guermond and J. Shen, Velocity-correction projection methods for incompressible flows. SIAM J. Numer. Anal. 41 (2003) 112–134. | DOI | MR | Zbl

[28] J. Guermond and J. Shen, A new class of truly consistent splitting schemes for incompressible flows. J. Comput . Phys. 192 (2003) 262–276. | DOI | MR | Zbl

[29] J. Guermond and J. Shen, On the error estimates of rotational pressure-correction projection methods. Math. Comput. 73 (2004) 1719–1737. | DOI | MR | Zbl

[30] F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. | MR | Zbl

[31] W. Layton, Fluid-porous interface conditions with the “inertia term” ½ǁu_fluidǁ² are not Galilean invariant. Tech. Rep. TR-MATH 09-30, University of Pittsburgh (2009).

[32] W. Layton, H. Tran and X. Xiong, Long time stability of four methods for splitting the evolutionary Stokes-Darcy problems into Stokes and Darcy subproblems. J. Comput. Appl. Math. 236 (2012) 3198–3217. | DOI | MR | Zbl

[33] M. Moraiti, On the quasistatic approximation in the Stokes-Darcy model of groundwater-surface water flows. J. Math. Anal. Appl. 394 (2012) 796–808. | DOI | MR | Zbl

[34] M. Mu and X. H. Zhu, Decoupled schemes for a non-stationary mixed Stokes-Darcy model. Math. Comput. 79 (2010) 707–731. | DOI | MR | Zbl

[35] R. Nochetto and J. Pyo, Error estimates for semi-discrete gauge methods for the Navier-Stokes equations. Math. Comput. 74 (2005) 521–542. | DOI | MR | Zbl

[36] P. Saffman, On the boundary at the surface of a porous medium. Studies Appl. Math. 50 (1971) 93–101. | DOI | Zbl

[37] L. Shan, H. Zheng and W. Layton, A decoupling method with different subdomain time steps for the nonstationary stokes–darcy model. Numer. Methods Partial Diff. Equ. 29 (2013) 549–583. | DOI | MR | Zbl

[38] J. Shen, On error estimates of the projection methods for the Navier-Stokes equations: First-order schemes. SIAM J. Numer. Anal. 29 (1992) 57–77. | DOI | MR | Zbl

[39] J. Shen, On error estimates of projection methods for the Navier-Stokes equations: second-order schemes. Math. Comput. 65 (1996) 1039–1065. | DOI | MR | Zbl

[40] J. Shen, A new pseudo-compressibility method for the Navier-Stokes equations. Appl. Numer. Math. 21 (1996) 71–90. | DOI | MR | Zbl

[41] J. Shen and X. Yang, Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. DCDS-B. 8 (2007) 663–676. | DOI | MR | Zbl

[42] Z. Si, Y. Wang and S. Li, Decoupled modified characteristics finite element method for the time dependent Navier-Stokes/Darcy problem. Math. Method Appl. Sci. 37 (2014) 1392–1404. | DOI | MR | Zbl

[43] H. Sun, Y. He and X. Feng, On error estimates of the pressure-correction projection methods for the time-dependent Navier-Stokes equations. Int. J. Numer. Anal. Model 8 (2011) 70–85. | MR | Zbl

[44] R. Temam, Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II. Arch. Rat. Mech. Anal. 33 (1969) 377–385. | DOI | MR | Zbl

[45] D. Vassilev and I. Yotov, Coupling Stokes-Darcy flow with transport. SIAM J. Sci. Comput. 31 (2009) 3661–3684. | DOI | MR | Zbl

[46] T. Zhang, D. Pedro and J. Yuan, A large time stepping viscosity-splitting finite element method for the viscoelastic flow problem. Adv. Comput. Math. 41 (2015) 149–190. | DOI | MR | Zbl

[47] L. Zuo and Y. Hou, Numerical analysis for the mixed Navier-Stokes and Darcy Problem with the Beavers-Joseph interface condition. Numer. Methods Partial Diff. Equ. 31 (2015) 1009–1030. | DOI | MR | Zbl

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