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Article

Mabrouki, Yassine ; Satouri, Jamil
Spectral discretization of Darcy equations coupled with Navier-Stokes equations by vorticity-velocity-pressure formulation. (English). Applications of Mathematics, vol. 67 (2022), issue 4, pp. 445-470
MSC: 35Q30, 65F08, 65N30, 65N55, 76S05 | MR 4444787 | Zbl 07584080 | DOI: 10.21136/AM.2022.0372-20
Keywords:
Navier-Stokes equation; Darcy equation; spectral element
Summary:
We consider a model coupling the Darcy equations in a porous medium with the Navier-Stokes equations in the cracks, for which the coupling is provided by the pressure's continuity on the interface. We discretize the coupled problem by the spectral element method combined with a nonoverlapping domain decomposition method. We prove the existence of solution for the discrete problem and establish an error estimation. We conclude with some numerical tests confirming the results of our analysis.
References:
[1] Amrouche, C., Bernardi, C., Dauge, M., Girault, V.: Vector potentials in three-dimensional non-smooth domains. Math. Methods Appl. Sci. 21 (1998), 823-864. DOI 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B | MR 1626990 | Zbl 0914.35094
[2] Aouadi, S. M., Bernardi, C., Satouri, J.: Mortar spectral element discretization of the Stokes problem in axisymmetric domains. Numer. Methods Partial Differ. Equations 30 (2014), 44-73. DOI 10.1002/num.21794 | MR 3149400 | Zbl 1299.76049
[3] Aouadi, S. M., Satouri, J.: Mortar spectral method in axisymmetric domains. ESAIM, Math. Model. Numer. Anal. 47 (2013), 33-55. DOI 10.1051/m2an/2012018 | MR 2968694 | Zbl 1277.65101
[4] Aza{ï}ez, M., Bernardi, C., Chorfi, N.: Spectral discretization of the vorticity, velocity and pressure formulation of the Navier-Stokes equations. Numer. Math. 104 (2006), 1-26. DOI 10.1007/s00211-006-0684-z | MR 2232000 | Zbl 1138.76052
[5] Badea, L., Discacciati, M., Quarteroni, A.: Numerical analysis of the Navier-Stokes/Darcy coupling. Numer. Math. 115 (2010), 195-227. DOI 10.1007/s00211-009-0279-6 | MR 2606960 | Zbl 1423.35304
[6] Beavers, S. G., Joseph, D. D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30 (1967), 197-207. DOI 10.1017/S0022112067001375
[7] Bernardi, C., Rebollo, T. Chacón, Hecht, F., Mghazli, Z.: Mortar finite element discretization of a model coupling Darcy and Stokes equations. ESAIM, Math. Model. Numer. Anal. 42 (2008), 375-410. DOI 10.1051/m2an:2008009 | MR 2423791 | Zbl 1138.76044
[8] Bernardi, C., Chorfi, N.: Spectral discretization of the vorticity, velocity and pressure formulation of the Stokes problem. SIAM J. Numer. Anal. 44 (2006), 826-850. DOI 10.1137/050622687 | MR 2218971 | Zbl 1117.65159
[9] Bernardi, C., Dauge, M., Maday, Y.: Polynomials in the Sobolev world. Available at https://hal.archives-ouvertes.fr/hal-00153795v2 (2007), 112 pages.
[10] Bernardi, C., Hecht, F., Pironneau, O.: Coupling Darcy and Stokes equations for porous media with cracks. ESAIM, Math. Model. Numer. Anal. 39 (2005), 7-35. DOI 10.1051/m2an:2005007 | MR 2136198 | Zbl 1079.76041
[11] Bernardi, C., Maday, Y.: Spectral Methods. Handbook of Numerical Analysis. Volume 5 P. G. Ciarlet, J. L. Lions North-Holland, Amsterdam (1997), 209-485. DOI 10.1016/S1570-8659(97)80003-8 | MR 1470226 | Zbl 0884.65001
[12] Brezzi, F., Rappaz, J., Raviart, P. A.: Finite dimensional approximation of nonlinear problems. 1. Branches of nonsingular solutions. Numer. Math. 36 (1980), 1-25. DOI 10.1007/BF01395985 | MR 0595803 | Zbl 0488.65021
[13] Cao, Y., Gunzburger, M., Hua, F., Wang, X.: Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition. Commun. Math. Sci. 8 (2010), 1-25. DOI 10.4310/CMS.2010.v8.n1.a2 | MR 2655899 | Zbl 1189.35244
[14] Chidyagwai, P., Rivière, B.: On the solution of the coupled Navier-Stokes and Darcy equations. Comput. Methods Appl. Mech. Eng. 198 (2009), 3806-3820. DOI 10.1016/j.cma.2009.08.012 | MR 2557499 | Zbl 1230.76023
[15] Costabel, M., Dauge, M.: Espaces fonctionnels Maxwell: Les gentils les méchants et les singularités. Available at \brokenlink{ https://www.yumpu.com/fr/document/read/7814987/espaces-{fonctionnels-maxwell-universite-de-rennes-1}} (1998), 6 pages French.
[16] Costabel, M., Dauge, M.: Computation of resonance frequencies for Maxwell equations in non-smooth domains. Topics in Computational Wave Propagation: Direct and Inverse Problems Springer, Berlin (2003), 125-161. DOI 10.1007/978-3-642-55483-4_4 | MR 2032869 | Zbl 1116.78002
[17] Discacciati, M., Quarteroni, A.: Navier-Stokes/Darcy coupling: Modeling, analysis, and numerical approximation. Rev. Mat. Complut. 22 (2009), 315-426. DOI 10.5209/rev_REMA.2009.v22.n2.16263 | MR 2553940 | Zbl 1172.76050
[18] Dubois, F.: Vorticity-velocity-pressure formulation for the Stokes problem. Math. Methods Appl. Sci. 25 (2002), 1091-1119. DOI 10.1002/mma.328 | MR 1924283 | Zbl 1099.76049
[19] Dubois, F., Salaün, M., Salmon, S.: Vorticity-velocity-pressure and stream function-vorticity formulations for the Stokes problem. J. Math. Pures Appl., IX. Sér. 82 (2003), 1395-1451. DOI 10.1016/j.matpur.2003.09.002 | MR 2020806 | Zbl 1070.76014
[20] Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics 5. Springer, Berlin (1986). DOI 10.1007/978-3-642-61623-5 | MR 0851383 | Zbl 0585.65077
[21] Girault, V., Rivière, B.: 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 | MR 2519594 | Zbl 1406.76082
[22] Logvinov, O. A., Malashin, A. A.: Generalized Navier-Stokes-Darcy model. Eur. J. Mech., B, Fluids 63 (2017), 100-105. DOI 10.1016/j.euromechflu.2017.01.019 | MR 3623140 | Zbl 1408.76145
[23] Mabrouki, Y., Aouadi, S. M., Satouri, J.: Spectral discretization of Darcy equations coupled with Stokes equations by vorticity-velocity-pressure formulation. Numer. Methods Partial Differ. Equations 33 (2017), 1628-1651. DOI 10.1002/num.22157 | MR 3683526 | Zbl 1394.65156
[24] Mabrouki, Y., Satouri, J.: Analysis of a Navier-Stokes-Darcy coupling problem. Adv. Pure Appl. Math. 7 (2016), 177-188. DOI 10.1515/apam-2016-0017 | MR 3518354 | Zbl 1342.76031
[25] Mikelic, A., Jäger, W.: On the interface boundary condition of Beavers, Joseph and Saffman. SIAM J. Appl. Math. 60 (2000), 1111-1127. DOI 10.1137/S003613999833678X | MR 1760028 | Zbl 0969.76088
[26] Saffman, P. G.: On the boundary condition at the interface of a porous medium. Stud. Appl. Math. 50 (1971), 93-101. DOI 10.1002/sapm197150293 | Zbl 0271.76080
[27] Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl., IV. Ser. 110 (1976), 353-372. DOI 10.1007/BF02418013 | MR 0463908 | Zbl 0353.46018
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