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Finite element discretization of Darcy's equations with pressure dependent porosity

Published online by Cambridge University Press:  23 February 2010

Vivette Girault
UPMC Univ. Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
François Murat
CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France.
Abner Salgado
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA.
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We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. The permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a finite element discretization of this problem and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence on the pressure is exponential, we propose a splitting scheme which involves solving two linear systems, but parts of the analysis of this method are still heuristic. Numerical tests are presented, which illustrate the introduced methods.

Research Article
© EDP Sciences, SMAI, 2010

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R.A. Adams, Sobolev spaces. Academic Press (1975).
Allaire, G., Homogeneization of the Navier-Stokes equations with slip boundary conditions. Comm. Pure Appl. Math. 44 (1991) 605641. CrossRef
M. Azaïez, F. Ben Belgacem, C. Bernardi and N. Chorfi, Spectral discretization of Darcy's equations with pressure dependent porosity. Report 2009-10, Laboratoire Jacques-Louis Lions, France (2009).
Babuška, I., The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179192. CrossRef
Bangerth, W., Hartman, R. and Kanschat, G., deal.II – a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33 (2007) 24. CrossRef
J. Berg and J. Löfström, Interpolation spaces: An introduction, Comprehensive Studies in Mathematics 223. Springer-Verlag (1976).
D. Boffi, F. Brezzi, L. Demkowicz, R. Durán, R. Falk and M. Fortin, Mixed finite elements, compatibility conditions, and applications, Lecture Notes in Mathematics 939. Springer-Verlag, Berlin, Germany (2008).
S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, Texts in applied mathematics 15. Third edition, Springer-Verlag (2008).
Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Anal. Numér. R2 (1974) 129151.
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics. Springer-Verlag, New York (1991).
Brezzi, F., Rappaz, J. and Raviart, P.-A., Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions. Numer. Math. 36 (1980) 125. CrossRef
P.-G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis II, Finite Element Methods (Part 1), P.G. Ciarlet and J.L. Lions Eds., Amsterdam, North-Holland (1991) 17–351.
Cioranescu, D., Donato, P. and Ene, H.I., Homogeneization of the Stokes problem with non-homogeneous boundary conditions. Math. Appl. Sci. 19 (1996) 857881.
H. Darcy, Les fontaines publiques de la ville de Dijon. Victor Dalmont, Paris, France (1856).
Douglas, J. and Dupont, T., Galerkin, A method for a nonlinear Dirichlet problem. Math. Comp. 29 (1975) 689696. CrossRef
Ene, H.I. and Sanchez-Palencia, E., Équations et phénomènes de surface pour l'écoulement dans un modèle de milieu poreux. J. Mécanique 14 (1975) 73108.
A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences 159. Springer-Verlag, New York, USA (2004).
G.B. Folland, Real analysis, modern techniques and their applications. Second edition, Wiley Interscience (1999).
Forchheimer, P., Wasserbewegung durch Boden. Z. Ver. Deutsh. Ing. 45 (1901) 17821788.
V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations – Theory and algorithms, Springer Series in Computational Mathematics 5. Springer-Verlag, Berlin, Germany (1986).
Girault, V. and Wheeler, M.F., Numerical discretization of a Darcy-Forchheimer model. Numer. Math. 110 (2008) 161198. CrossRef
Girault, V., Nochetto, R. and Scott, L.R., Maximum-norm stability of the finite-element Stokes projection. J. Math. Pure. Appl. 84 (2005) 279330. CrossRef
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics 24. Pitman, Boston, USA (1985).
F. Hecht, A. Le Hyaric, O. Pironneau and K. Ohtsuka, Freefem++. Second Edition, Version 2.24-2-2. Laboratoire J.-L. Lions, UPMC, Paris, France (2008).
A.Ya. Helemskii, Lectures and exercises on functional analysis, Translations of Mathematical Monographs 233. American Mathematical Society, USA (2006).
L.V. Kantorovich and G.P. Akilov, Functional analysis. Third edition, Nauka (1984) [in Russian].
Kim, D. and Park, E.J., Primal mixed finite-element approximation of elliptic equations with gradient nonlinearities. Comput. Math. Appl. 51 (2006) 793804. CrossRef
J.L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, I. Dunod, Paris, France (1968).
Park, E.J., Mixed finite element methods for nonlinear second order elliptic problems. SIAM J. Numer. Anal. 32 (1995) 865885. CrossRef
Pastukhova, S.E., Substantiation of the Darcy Law for a porous medium with condition of partial adhesion. Sbornik Math. 189 (1998) 18711888. CrossRef
Rajagopal, K.R., On a hierarchy of approximate models for flows of incompressible fluids through porous solids. M3AS 17 (2007) 215252.
J.E. Roberts and J.-M. Thomas, Mixed and Hybrid methods in Handbook of Numerical Analysis II: Finite Element Methods (Part 1), P.G. Ciarlet and J.L. Lions Eds., Amsterdam, North-Holland (1991) 523–639.
Schöberl, J. and Zulehner, W., Symmetric indefinite preconditioners for saddle point problems with applications to pde-constrained optimization problems. SIAM J. Matrix Anal. Appl. 29 (2007) 752773. CrossRef
Skjetne, E. and Auriault, J.L., Homogeneization of wall-slip gas flow through porous media. Transp. Porous Media 36 (1999) 293306. CrossRef
L. Tartar, An introduction to Sobolev spaces and interpolation spaces, Lecture Notes of the Unione Matematica Italiana 3. Springer-Verlag, Berlin-Heidelberg (2007).
Zulehner, W., Analysis of iterative methods for saddle point problems: a unified approach. Math. Comp. 71 (2001) 479505. CrossRef