Finite element discretization of Darcy's equations with pressure dependent porosity
Girault, Vivette ; Murat, François ; Salgado, Abner
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010), p. 1155-1191 / Harvested from Numdam
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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.

Publié le : 2010-01-01
DOI : https://doi.org/10.1051/m2an/2010019
Classification:  76S05,  65N30 
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     author = {Girault, Vivette and Murat, Fran\c cois and Salgado, Abner},
     title = {Finite element discretization of Darcy's equations with pressure dependent porosity},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {44},
     year = {2010},
     pages = {1155-1191},
     doi = {10.1051/m2an/2010019},
     mrnumber = {2769053},
     zbl = {pre05835017},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2010__44_6_1155_0}
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Girault, Vivette; Murat, François; Salgado, Abner. Finite element discretization of Darcy's equations with pressure dependent porosity. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) pp. 1155-1191. doi : 10.1051/m2an/2010019. http://gdmltest.u-ga.fr/item/M2AN_2010__44_6_1155_0/

[1] R.A. Adams, Sobolev spaces. Academic Press (1975). | Zbl 1098.46001

[2] G. Allaire, Homogeneization of the Navier-Stokes equations with slip boundary conditions. Comm. Pure Appl. Math. 44 (1991) 605-641. | Zbl 0738.35059

[3] 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). | Zbl pre05817268

[4] I. Babuška, The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179-192. | Zbl 0258.65108

[5] W. Bangerth, R. Hartman and G. Kanschat, deal.II - a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33 (2007) 24.

[6] J. Berg and J. Löfström, Interpolation spaces: An introduction, Comprehensive Studies in Mathematics 223. Springer-Verlag (1976). | Zbl 0344.46071

[7] 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).

[8] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, Texts in applied mathematics 15. Third edition, Springer-Verlag (2008). | Zbl 1135.65042

[9] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Anal. Numér. R2 (1974) 129-151. | Numdam | Zbl 0338.90047

[10] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics. Springer-Verlag, New York (1991). | Zbl 0788.73002

[11] F. Brezzi, J. Rappaz and P.-A. Raviart, Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions. Numer. Math. 36 (1980) 1-25. | Zbl 0488.65021

[12] 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. | Zbl 0875.65086

[13] D. Cioranescu, P. Donato and H.I. Ene, Homogeneization of the Stokes problem with non-homogeneous boundary conditions. Math. Appl. Sci. 19 (1996) 857-881. | Zbl 0869.35012

[14] H. Darcy, Les fontaines publiques de la ville de Dijon. Victor Dalmont, Paris, France (1856).

[15] J. Douglas and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem. Math. Comp. 29 (1975) 689-696. | Zbl 0306.65072

[16] H.I. Ene and E. Sanchez-Palencia, Équations et phénomènes de surface pour l'écoulement dans un modèle de milieu poreux. J. Mécanique 14 (1975) 73-108. | Zbl 0304.76037

[17] A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences 159. Springer-Verlag, New York, USA (2004). | Zbl 1059.65103

[18] G.B. Folland, Real analysis, modern techniques and their applications. Second edition, Wiley Interscience (1999). | Zbl 0549.28001

[19] P. Forchheimer, Wasserbewegung durch Boden. Z. Ver. Deutsh. Ing. 45 (1901) 1782-1788.

[20] 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). | Zbl 0585.65077

[21] V. Girault and M.F. Wheeler, Numerical discretization of a Darcy-Forchheimer model. Numer. Math. 110 (2008) 161-198. | Zbl 1143.76035

[22] V. Girault, R. Nochetto and L.R. Scott, Maximum-norm stability of the finite-element Stokes projection. J. Math. Pure. Appl. 84 (2005) 279-330. | Zbl 1210.76051 | Zbl pre02164965

[23] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics 24. Pitman, Boston, USA (1985). | Zbl 0695.35060

[24] 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).

[25] A.Ya. Helemskii, Lectures and exercises on functional analysis, Translations of Mathematical Monographs 233. American Mathematical Society, USA (2006). | Zbl 1123.46001

[26] L.V. Kantorovich and G.P. Akilov, Functional analysis. Third edition, Nauka (1984) [in Russian]. | Zbl 0555.46001

[27] D. Kim and E.J. Park, Primal mixed finite-element approximation of elliptic equations with gradient nonlinearities. Comput. Math. Appl. 51 (2006) 793-804. | Zbl 1134.65367

[28] J.L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, I. Dunod, Paris, France (1968). | Zbl 0165.10801

[29] E.J. Park, Mixed finite element methods for nonlinear second order elliptic problems. SIAM J. Numer. Anal. 32 (1995) 865-885. | Zbl 0834.65108

[30] S.E. Pastukhova, Substantiation of the Darcy Law for a porous medium with condition of partial adhesion. Sbornik Math. 189 (1998) 1871-1888. | Zbl 0932.35162

[31] K.R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solids. M3AS 17 (2007) 215-252. | Zbl 1123.76066

[32] 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. | Zbl 0875.65090

[33] J. Schöberl and W. Zulehner, Symmetric indefinite preconditioners for saddle point problems with applications to pde-constrained optimization problems. SIAM J. Matrix Anal. Appl. 29 (2007) 752-773. | Zbl 1154.65029

[34] E. Skjetne and J.L. Auriault, Homogeneization of wall-slip gas flow through porous media. Transp. Porous Media 36 (1999) 293-306.

[35] L. Tartar, An introduction to Sobolev spaces and interpolation spaces, Lecture Notes of the Unione Matematica Italiana 3. Springer-Verlag, Berlin-Heidelberg (2007). | Zbl 1126.46001

[36] W. Zulehner, Analysis of iterative methods for saddle point problems: a unified approach. Math. Comp. 71 (2001) 479-505. | Zbl 0996.65038