Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T15:34:40.260Z Has data issue: false hasContentIssue false
  • English
  • Français

Coupling Darcy and Stokes equationsfor porous media with cracks

Published online by Cambridge University Press:  15 March 2005

Christine Bernardi ,
Frédéric Hecht  and
Olivier Pironneau
Christine Bernardi
Affiliation:
Laboratoire Jacques-Louis Lions, CNRS & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. bernardi@ann.jussieu.fr; hecht@ann.jussieu.fr; pironneau@ann.jussieu.fr
Frédéric Hecht
Affiliation:
Laboratoire Jacques-Louis Lions, CNRS & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. bernardi@ann.jussieu.fr; hecht@ann.jussieu.fr; pironneau@ann.jussieu.fr
Olivier Pironneau
Affiliation:
Laboratoire Jacques-Louis Lions, CNRS & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. bernardi@ann.jussieu.fr; hecht@ann.jussieu.fr; pironneau@ann.jussieu.fr
Get access

Abstract

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 equationsfinite elementserror estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 1 , January 2005 , pp. 7 - 35
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Achdou, Y., Bernardi, C. and Coquel, F., A priori and a posteriori analysis of finite volume discretizations of Darcy's equations. Numer. Math. 96 (2003) 17–42. CrossRef
Amara, M., Capatina-Papaghiuc, D., Chacón-Vera, E. and Trujillo, D., Vorticity–velocity–pressure formulation for Navier–Stokes equations. Comput. Vis. Sci. 6 (2004) 47–52. CrossRef
Amrouche, C., Bernardi, C., Dauge, M. and Girault, V., Vector potentials in three-dimensional nonsmooth domains. Math. Meth. Appl. Sci. 21 (1998) 823–864. 3.0.CO;2-B>CrossRef
C. Bègue, C. Conca, F. Murat and O. Pironneau, 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.
Bernardi, C., Canuto, C. and Maday, Y., 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.
Bernardi, C., Canuto, C. and Maday, Y., Generalized inf-sup condition for Chebyshev spectral approximation of the Stokes problem. SIAM J. Numer. Anal. 25 (1988) 1237–1271. CrossRef
Bertoluzza, S. and Perrier, V., The mortar method in the wavelet context. ESAIM: M2AN 35 (2001) 647–673. CrossRef
Braess, D. and Verfürth, R., A posteriori error estimators for the Raviart–Thomas element. SIAM J. Numer. Anal. 33 (1996) 2431–2444. CrossRef
D.-G. Calugaru, 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).
Crouzeix, M. and Raviart, P.-A., Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7 (1973) 33–76.
Discacciati, M., Miglio, E. and Quarteroni, A., Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43 (2002) 57–74. CrossRef
M. Discacciati and A. Quarteroni, 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).
Discacciati, M. and Quarteroni, A., 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. CrossRef
Dubois, F., Vorticity–velocity–pressure formulation for the Stokes problem. Math. Meth. Appl. Sci. 25 (2002) 1091–1119. CrossRef
F. Dubois, M. Salaün and S. Salmon, First vorticity–velocity–pressure scheme for the Stokes problem, Internal Report 356, Institut Aérotechnique, Conservatoire National des Arts et Métiers, France (2002) (submitted).
P.J. Frey and P.-L. George, Maillages, applications aux éléments finis. Hermès, Paris (1999).
P.-L. George and F. Hecht, Nonisotropic grids. Handbook of Grid Generation, J.F. Thompson, B.K. Soni & N.P. Weatherhill Eds., CRC Press (1998).
V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms . Springer–Verlag (1986).
Hecht, F., Construction d'une base de fonctions P 1 non conforme à divergence nulle dans $\mathbb{R}^3$ . RAIRO Anal. Numér. 15 (1981) 119–150. CrossRef
F. Hecht and O. Pironneau, FreeFem++, see www.freefem.org.
H. Kawarada, E. Baba and H. Suito, 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).
Kawarada, H., Baba, E. and Suito, H., 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.
W.J. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow, Preprint of the University of Magdebourg, report N° 22-01 (2001).
Nedelec, J.-C., Mixed finite elements in $\mathbb{R}^3$ . Numer. Math. 35 (1980) 315–341. CrossRef
Nicolaides, R.A., Existence, uniqueness and approximation for generalized saddle point problems. SIAM J. Numer. Anal. 19 (1982) 349–357. CrossRef
Raviart, P.-A. and Thomas, J.-M., 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. CrossRef
S. Salmon, Développement numérique de la formulation tourbillon–vitesse–pression pour le problème de Stokes. Thesis, Université Pierre et Marie Curie, Paris (1999).
R. Temam, Theory and Numerical Analysis of the Navier–Stokes Equations . North-Holland (1977).
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques . Wiley & Teubner (1996).