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Automatic simplification of Darcy’s equations with pressure dependent permeability

Published online by Cambridge University Press:  07 October 2013

Etienne Ahusborde ,
Mejdi Azaïez ,
Faker Ben Belgacem  and
Christine Bernardi
Etienne Ahusborde
Affiliation:
Laboratoire de Mathématiques et de leurs Applications (U.M.R. 5142 CNRS), Bâtiment IPRA, Université de Pau et des Pays de l’Adour, avenue de l’Université, B.P. 1155, 64013 Pau Cedex, France.. etienne.ahusborde@univ-pau.fr
Mejdi Azaïez
Affiliation:
Université de Bordeaux, I2M (UMR C.N.R.S. 5295), 33405 Talence, France.; azaiez@ipb.fr
Faker Ben Belgacem
Affiliation:
LMAC (E.A. 2222), Université de Technologie de Compiègne, B.P. 20529, 60205 Compiègne Cedex, France, and Université de Bordeaux, I2M (UMR C.N.R.S. 5295), 33405 Talence, France.; faker.ben-belgacem@utc.fr
Christine Bernardi
Affiliation:
Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France.; bernardi@ann.jussieu.fr
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Abstract

We consider the flow of a viscous incompressible fluid in a rigid homogeneous porous medium provided with mixed boundary conditions. Since the boundary pressure can present high variations, the permeability of the medium also depends on the pressure, so that the model is nonlinear. A posteriori estimates allow us to omit this dependence where the pressure does not vary too much. We perform the numerical analysis of a spectral element discretization of the simplified model. Finally we propose a strategy which leads to an automatic identification of the part of the domain where the simplified model can be used without increasing significantly the error.

Keywords

Darcy’s equationsspectral elementsa posteriori analysis
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013 , pp. 1797 - 1820
Copyright
© EDP Sciences, SMAI 2013

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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. Google Scholar
Azaïez, M., Ben Belgacem, F. and Bernardi, C., The mortar spectral element method in domains of operators, Part I: The divergence operator and Darcy’s equations. IMA J. Numer. Anal. 26 (2006) 131–154. Google Scholar
Azaïez, M., Ben Belgacem, F., Bernardi, C. and Chorfi, N., Spectral discretization of Darcy’s equations with pressure dependent porosity. Appl. Math. Comput. 217 (2010) 1838–1856. Google Scholar
Azaïez, M., Ben Belgacem, F., Grundmann, M. and Khallouf, H., Staggered grids hybrid-dual spectral element method for second-order elliptic problems, Application to high-order time splitting methods for Navier–Stokes equations. Comput. Methods Appl. Mech. Engrg. 166 (1998) 183–199. Google Scholar
Bernardi, C., Indicateurs d’erreur en h − N version des éléments spectraux. Modél. Math. et Anal. Numér. 30 (1996) 1–38. Google Scholar
Bernardi, C., Blouza, A., Chorfi, N. and Kharrat, N., A penalty algorithm for the spectral element discretization of the Stokes problem. Math. Model. Numer. Anal. 45 (2011) 201–216. Google Scholar
Bernardi, C., Chacón Rebollo, T., Hecht, F. and Lewandowski, R., Automatic insertion of a turbulence model in the finite element discretization of the Navier–Stokes equations. Math. Models Methods Appl. Sci. 19 (2009) 1139–1183. Google Scholar
C. Bernardi, F. Coquel and P.-A. Raviart, Automatic coupling and finite element discretization of the Navier–Stokes and heat equations, Internal Report R10001, Labotatoire Jacques-Louis Lions, Paris (2010).
C. Bernardi, M. Dauge and Y. Maday, Polynomials in Sobolev Spaces and Application to the Mortar Spectral Element Method, in preparation.
C. Bernardi and Y. Maday, Spectral Methods, in the Handbook of Numerical Analysis V, edited by P.G. Ciarlet and J.-L. Lions. North-Holland (1997) 209–485.
C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Collection Mathématiques et Applications vol. 45. Springer-Verlag (2004).
Braack, M. and Ern, A., A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 1 (2003) 221–238. Google Scholar
Brezis, H. and Mironescu, P., Gagliardo–Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ. 1 (2001), 387–404. Google Scholar
Brezzi, F., Rappaz, J. and Raviart, P.-A., Finite dimensional approximation of nonlinear problems, Part I: Branches of nonsingular solutions. Numer. Math. 36 (1980) 1–25. Google Scholar
Chacón Rebollo, T., Del Pino, S. and Yakoubi, D., An iterative procedure to solve a coupled two-fluids turbulence model. Math. Model. Numer. Anal. 44 (2010) 693–713. Google Scholar
Chaillou, A.L. and Suri, M., Computable error estimators for the approximation of nonlinear problems by linearized models. Comput. Methods Appl. Mech. Engrg. 196 (2006) 210–224. Google Scholar
M. Daadaa, Discrétisation spectrale et par éléments spectraux des équations de Darcy, Ph.D. Thesis, Université Pierre et Marie Curie, Paris (2009).
Dauge, M., Neumann and mixed problems on curvilinear polyhedra. Integr. Equ. Oper. Th. 15 (1992) 227–261. Google Scholar
El Alaoui, L., Ern, A. and Vohralík, M., Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems. Comput. Methods Appl. Mech. Engrg. 200 (2011) 2782–2795. Google Scholar
V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer–Verlag (1986).
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. I. Dunod, Paris (1968).
Meyers, N.G., An L p-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa 17 (1963) 189–206. Google Scholar
Pousin, J. and Rappaz, J., Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69 (1994) 213–231. Google Scholar
Rajagopal, K.R., On a hierarchy of approximate models for flows of incompressible fluids through porous solid. Math. Models Methods Appl. Sci. 17 (2007) 215–252. Google Scholar
Talenti, G., Best constant in Sobolev inequality. Ann. Math. Pura ed Appl. Serie IV 110 (1976) 353–372. Google Scholar
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley ans Teubner (1996).