Time-dependent coupling of Navier-Stokes and Darcy flows
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 2, pp. 539-554.

A weak solution of the coupling of time-dependent incompressible Navier-Stokes equations with Darcy equations is defined. The interface conditions include the Beavers-Joseph-Saffman condition. Existence and uniqueness of the weak solution are obtained by a constructive approach. The analysis is valid for weak regularity interfaces.

DOI : 10.1051/m2an/2012034
Classification : 35Q30, 76N10
Mots-clés : multiphysics, weak solution, interface conditions, Beavers-Joseph-Saffman 
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     author = {Cesmelioglu, Aycil and Girault, Vivette and Rivi\`ere, B\'eatrice},
     title = {Time-dependent coupling of {Navier-Stokes} and {Darcy} flows},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {539--554},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {2},
     year = {2013},
     doi = {10.1051/m2an/2012034},
     mrnumber = {3021697},
     zbl = {1267.76096},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2012034/}
}
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Cesmelioglu, Aycil; Girault, Vivette; Rivière, Béatrice. Time-dependent coupling of Navier-Stokes and Darcy flows. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 2, pp. 539-554. doi : 10.1051/m2an/2012034. http://www.numdam.org/articles/10.1051/m2an/2012034/

[1] R. Adams, Sobolev Spaces. Academic Press, New-York (1975). | MR | Zbl

[2] T. Arbogast and D. Brunson, A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Comput. Geosci. 11 (2007) 207-218. | MR | Zbl

[3] T. Arbogast and H. Lehr, Homogenization of a Darcy-Stokes system modeling vuggy porous media. Comput. Geosci. 10 (2006) 291-302. | MR | Zbl

[4] J. Aubin, Un théorème de compacité. CRAS Paris Sér. I 256 (1963) 5042-5044. | Zbl

[5] L. Badea, M. Discacciati and A. Quarteroni, Mathematical analysis of the Navier-Stokes/Darcy coupling. Numer. Math. 1152 (2010) 195-227. | MR

[6] G. Beavers and D. Joseph, Boundary conditions at a naturally impermeable wall. J. Fluid. Mech. 30 (1967) 197-207.

[7] E. Burman and P. Hansbo, A unified stabilized method for Stokes and Darcy's equations. J. Computat. Appl. Math. 198 (2007) 35-51. | MR | Zbl

[8] Y. Cao, M. Gunzburger, F. Hua and X. Wang, Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition. Commun. Math. Sci. 8 (2010) 1-25. | MR | Zbl

[9] A. Çeşmelioğlu and B. Rivière, Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow. J. Numer. Math. 16 (2008) 249-280. | MR | Zbl

[10] A. Çeşmelioğlu and B. Rivière, Primal discontinuous Galerkin methods for time-dependent coupled surface and subsurface flow. J. Sci. Comput. 40 (2009) 115-140. | Zbl

[11] P. Chidyagwai and B. Rivière, On the solution of the coupled Navier-Stokes and Darcy equations. Comput. Methods Appl. Mech. Eng. 198 (2009) 3806-3820. | MR | Zbl

[12] P. Chidyagwai and B. Rivière, Numerical modelling of coupled surface and subsurface flow systems. Adv. Water Resour. 33 (2010) 92-105.

[13] E.A. Coddington and N. Levinson, Theory of differential equations. McGraw-Hill, New York (1955). | Zbl

[14] M. Discacciati, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows. Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne, Switzerland (2004).

[15] M. Discacciati and A. Quarteroni, Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations. in Numerical Analysis and Advanced Applications ENUMATH 2001. Springer, Milan (2003) 3-20. | MR | Zbl

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

[17] 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. | MR | Zbl

[18] 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. | MR

[19] P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, Boston, MA. Monogr. Stud. Math. 24 (1985). | MR | Zbl

[20] N. Hanspal, A. Waghode, V. Nassehi and R. Wakeman, Numerical analysis of coupled Stokes/Darcy flows in industrial filtrations. Transp. Porous Media 64 (2006) 1573-1634.

[21] J. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982) 275-311. | MR | Zbl

[22] W. Jäger and A. Mikelić, On the interface boundary condition of Beavers, Joseph and Saffman. SIAM J. Appl. Math. 60 (2000) 1111-1127. | MR | Zbl

[23] G. Kanschat and B. Rivière, A strongly conservative finite element method for the coupling of Stokes and Darcy flow. J. Computat. Phys. 229 (2010) 5933-5943. | MR

[24] W. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40 (2003) 2195-2218. | MR | Zbl

[25] J.-L. Lions, Equations différentielles opérationnelles et problèmes aux limites. Springer-Verlag, Berlin, Heidelberg, New York (1961). | Zbl

[26] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. I. Springer-Verlag, New York (1972). | MR | Zbl

[27] K.A. Mardal, X.-C. Tai and R. Winther, A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal. 40 (2002) 1605-1631 (electronic). | MR | Zbl

[28] M. Mu and J. Xu, A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 45 (2007) 1801-1813. | MR | Zbl

[29] J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). | Zbl

[30] B. Rivière, Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 22 (2005) 479-500. | MR | Zbl

[31] B. Rivière and I. Yotov, Locally conservative coupling of Stokes and Darcy flow. SIAM J. Numer. Anal. 42 (2005) 1959-1977. | MR | Zbl

[32] P. Saffman, On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50 (1971) 292-315. | Zbl

[33] J. Simon, Compact sets in the space Lp(0,T;B). Ann. Math. Pures Appl. 146 (1990) 1093-1117. | MR | Zbl

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

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