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

Vivette Girault; François Murat; Abner Salgado

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 6, page 1155-1191
- ISSN: 0764-583X

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topGirault, Vivette, Murat, François, and Salgado, Abner. "Finite element discretization of Darcy's equations with pressure dependent porosity." ESAIM: Mathematical Modelling and Numerical Analysis 44.6 (2010): 1155-1191. <http://eudml.org/doc/250842>.

@article{Girault2010,

abstract = {
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.
},

author = {Girault, Vivette, Murat, François, Salgado, Abner},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Porous media flows; Darcy equations; finite elements; exponential porosity; bounded porosity; convergence; splitting scheme},

language = {eng},

month = {10},

number = {6},

pages = {1155-1191},

publisher = {EDP Sciences},

title = {Finite element discretization of Darcy's equations with pressure dependent porosity},

url = {http://eudml.org/doc/250842},

volume = {44},

year = {2010},

}

TY - JOUR

AU - Girault, Vivette

AU - Murat, François

AU - Salgado, Abner

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/10//

PB - EDP Sciences

VL - 44

IS - 6

SP - 1155

EP - 1191

AB -
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.

LA - eng

KW - Porous media flows; Darcy equations; finite elements; exponential porosity; bounded porosity; convergence; splitting scheme

UR - http://eudml.org/doc/250842

ER -

## References

top- R.A. Adams, Sobolev spaces. Academic Press (1975).
- G. Allaire, Homogeneization of the Navier-Stokes equations with slip boundary conditions. Comm. Pure Appl. Math.44 (1991) 605–641.
- 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).
- I. Babuška, The finite element method with Lagrangian multipliers. Numer. Math.20 (1973) 179–192.
- W. Bangerth, R. Hartman and G. Kanschat, deal.II – a general-purpose object-oriented finite element library. ACM Trans. Math. Softw.33 (2007) 24.
- J. Berg and J. Löfström, Interpolation spaces: An introduction, Comprehensive Studies in Mathematics223. 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 Mathematics939. Springer-Verlag, Berlin, Germany (2008).
- S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, Texts in applied mathematics15. Third edition, Springer-Verlag (2008).
- F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Anal. Numér.R2 (1974) 129–151.
- F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics. Springer-Verlag, New York (1991).
- 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.
- P.-G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical AnalysisII, Finite Element Methods (Part 1), P.G. Ciarlet and J.L. Lions Eds., Amsterdam, North-Holland (1991) 17–351.
- 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.
- H. Darcy, Les fontaines publiques de la ville de Dijon. Victor Dalmont, Paris, France (1856).
- J. Douglas and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem. Math. Comp.29 (1975) 689–696.
- H.I. Ene and E. Sanchez-Palencia, Équations et phénomènes de surface pour l'écoulement dans un modèle de milieu poreux. J. Mécanique14 (1975) 73–108.
- A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences159. Springer-Verlag, New York, USA (2004).
- G.B. Folland, Real analysis, modern techniques and their applications. Second edition, Wiley Interscience (1999).
- P. Forchheimer, Wasserbewegung durch Boden. Z. Ver. Deutsh. Ing.45 (1901) 1782–1788.
- V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations – Theory and algorithms, Springer Series in Computational Mathematics5. Springer-Verlag, Berlin, Germany (1986).
- V. Girault and M.F. Wheeler, Numerical discretization of a Darcy-Forchheimer model. Numer. Math.110 (2008) 161–198.
- 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.
- P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics24. 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 Monographs233. American Mathematical Society, USA (2006).
- L.V. Kantorovich and G.P. Akilov, Functional analysis. Third edition, Nauka (1984) [in Russian].
- D. Kim and E.J. Park, Primal mixed finite-element approximation of elliptic equations with gradient nonlinearities. Comput. Math. Appl.51 (2006) 793–804.
- J.L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, I. Dunod, Paris, France (1968).
- E.J. Park, Mixed finite element methods for nonlinear second order elliptic problems. SIAM J. Numer. Anal.32 (1995) 865–885.
- S.E. Pastukhova, Substantiation of the Darcy Law for a porous medium with condition of partial adhesion. Sbornik Math.189 (1998) 1871–1888.
- K.R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solids. M3AS17 (2007) 215–252.
- J.E. Roberts and J.-M. Thomas, Mixed and Hybrid methods in Handbook of Numerical AnalysisII: Finite Element Methods (Part 1), P.G. Ciarlet and J.L. Lions Eds., Amsterdam, North-Holland (1991) 523–639.
- 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.
- E. Skjetne and J.L. Auriault, Homogeneization of wall-slip gas flow through porous media. Transp. Porous Media36 (1999) 293–306.
- L. Tartar, An introduction to Sobolev spaces and interpolation spaces, Lecture Notes of the Unione Matematica Italiana3. Springer-Verlag, Berlin-Heidelberg (2007).
- W. Zulehner, Analysis of iterative methods for saddle point problems: a unified approach. Math. Comp.71 (2001) 479–505.

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