Coupling Darcy and Stokes equations for porous media with cracks
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 1, p. 7-35

In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive a priori and a posteriori error estimates. We present some numerical experiments that are in good agreement with the analysis.

DOI : https://doi.org/10.1051/m2an:2005007
Classification:  65N30,  65N50,  76D07,  76S05
Keywords: Darcy and Stokes equations, finite elements, error estimates
@article{M2AN_2005__39_1_7_0,
     author = {Bernardi, Christine and Hecht, Fr\'ed\'eric and Pironneau, Olivier},
     title = {Coupling Darcy and Stokes equations for porous media with cracks},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {1},
     year = {2005},
     pages = {7-35},
     doi = {10.1051/m2an:2005007},
     zbl = {1079.76041},
     mrnumber = {2136198},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2005__39_1_7_0}
}
Bernardi, Christine; Hecht, Frédéric; Pironneau, Olivier. Coupling Darcy and Stokes equations for porous media with cracks. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 1, pp. 7-35. doi : 10.1051/m2an:2005007. http://www.numdam.org/item/M2AN_2005__39_1_7_0/

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