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.
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: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {7--35},
year = {2005},
publisher = {EDP Sciences},
volume = {39},
number = {1},
doi = {10.1051/m2an:2005007},
mrnumber = {2136198},
zbl = {1079.76041},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an:2005007/}
}
TY - JOUR AU - Bernardi, Christine AU - Hecht, Frédéric AU - Pironneau, Olivier TI - Coupling Darcy and Stokes equations for porous media with cracks JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2005 SP - 7 EP - 35 VL - 39 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an:2005007/ DO - 10.1051/m2an:2005007 LA - en ID - M2AN_2005__39_1_7_0 ER -
%0 Journal Article %A Bernardi, Christine %A Hecht, Frédéric %A Pironneau, Olivier %T Coupling Darcy and Stokes equations for porous media with cracks %J ESAIM: Modélisation mathématique et analyse numérique %D 2005 %P 7-35 %V 39 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an:2005007/ %R 10.1051/m2an:2005007 %G en %F M2AN_2005__39_1_7_0
Bernardi, Christine; Hecht, Frédéric; Pironneau, Olivier. Coupling Darcy and Stokes equations for porous media with cracks. ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 1, pp. 7-35. doi: 10.1051/m2an:2005007
[1] , and, A priori and a posteriori analysis of finite volume discretizations of Darcy's equations. Numer. Math. 96 (2003) 17-42. | Zbl
[2] ,, and, Vorticity-velocity-pressure formulation for Navier-Stokes equations. Comput. Vis. Sci. 6 (2004) 47-52. | Zbl
[3] ,, and, Vector potentials in three-dimensional nonsmooth domains. Math. Meth. Appl. Sci. 21 (1998) 823-864. | Zbl
[4] ,, and, Les équations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression. Nonlinear Partial Differ. Equ. Appl., Collège de France Seminar IX (1988) 179-264. | Zbl
[5] , and, Un problème variationnel abstrait. Application d'une méthode de collocation pour les équations de Stokes. C.R. Acad. Sci. Paris série I 303 (1986) 971-974. | Zbl
[6] , and, Generalized inf-sup condition for Chebyshev spectral approximation of the Stokes problem. SIAM J. Numer. Anal. 25 (1988) 1237-1271. | Zbl
[7] and, The mortar method in the wavelet context. ESAIM: M2AN 35 (2001) 647-673. | Zbl | Numdam | EuDML
[8] and, A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal. 33 (1996) 2431-2444. | Zbl
[9] , Modélisation et simulation numérique du transport de radon dans un milieu poreux fissuré ou fracturé. Problème direct et problèmes inverses comme outils d'aide à la prédiction sismique, Thesis, Université de Franche-Comté, Besançon (2002).
[10] and, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7 (1973) 33-76. | Zbl | Numdam
[11] , and, Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43 (2002) 57-74. | Zbl
[12] and, Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations, in Proc. of ENUMATH, F. Brezzi Ed., Springer-Verlag (to appear). | MR
[13] and, Convergence analysis of a subdomain iterative method for the finite element approximation of the coupling of Stokes and Darcy equations. Comput. Vis. Sci. 6 (2004) 93-104.
[14] , Vorticity-velocity-pressure formulation for the Stokes problem. Math. Meth. Appl. Sci. 25 (2002) 1091-1119. | Zbl
[15] , and, First vorticity-velocity-pressure scheme for the Stokes problem, Internal Report 356, Institut Aérotechnique, Conservatoire National des Arts et Métiers, France (2002) (submitted).
[16] and, Maillages, applications aux éléments finis. Hermès, Paris (1999).
[17] and, Nonisotropic grids. Handbook of Grid Generation, J.F. Thompson, B.K. Soni & N.P. Weatherhill Eds., CRC Press (1998).
[18] and, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986). | Zbl
[19] , Construction d’une base de fonctions non conforme à divergence nulle dans . RAIRO Anal. Numér. 15 (1981) 119-150. | Zbl | Numdam
[20] and, FreeFem++, see www.freefem.org.
[21] , and, Effects of spilled oil on coastal ecosystems, in the Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering 2000, CD-ROM proceedings (2001).
[22] , and, Effects of wave breaking action on flows in tidal-flats, in Computational Fluid Dynamics for the 21st Century, M. Hafez, K. Morinishi and J. Périaux Eds., Springer. Notes on Numerical Fluid Mechanics 78 (2001) 275-289.
[23] , and, Coupling fluid flow with porous media flow 22-01 (2001). | Zbl | MR
[24] , Mixed finite elements in . Numer. Math. 35 (1980) 315-341. | Zbl
[25] , Existence, uniqueness and approximation for generalized saddle point problems. SIAM J. Numer. Anal. 19 (1982) 349-357. | Zbl
[26] and, A mixed finite element method for second order elliptic problems, Mathematical Aspects of Finite Element Methods. Springer, Berlin. Lect. Notes Math. 606 (1977) 292-315. | Zbl
[27] , Développement numérique de la formulation tourbillon-vitesse-pression pour le problème de Stokes. Thesis, Université Pierre et Marie Curie, Paris (1999).
[28] , Theory and Numerical Analysis of the Navier-Stokes Equations. North-Holland (1977). | Zbl
[29] , A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner (1996). | Zbl
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