Automatic simplification of Darcy's equations with pressure dependent permeability
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 6, pp. 1797-1820.

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.

DOI: 10.1051/m2an/2013089
Classification: 76S05, 65N35
Keywords: Darcy's equations, spectral elements, a posteriori analysis
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     author = {Ahusborde, Etienne and Aza{\"\i}ez, Mejdi and Ben Belgacem, Faker and Bernardi, Christine},
     title = {Automatic simplification of {Darcy's} equations with pressure dependent permeability},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1797--1820},
     publisher = {EDP-Sciences},
     volume = {47},
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     doi = {10.1051/m2an/2013089},
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     zbl = {1311.76128},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2013089/}
}
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Ahusborde, Etienne; Azaïez, Mejdi; Ben Belgacem, Faker; Bernardi, Christine. Automatic simplification of Darcy's equations with pressure dependent permeability. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 6, pp. 1797-1820. doi : 10.1051/m2an/2013089. http://www.numdam.org/articles/10.1051/m2an/2013089/

[1] Y. Achdou, C. Bernardi and F. Coquel, A priori and a posteriori analysis of finite volume discretizations of Darcy's equations. Numer. Math. 96 (2003) 17-42. | MR | Zbl

[2] M. Azaïez, F. Ben Belgacem and C. Bernardi, 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. | MR | Zbl

[3] M. Azaïez, F. Ben Belgacem, C. Bernardi and N. Chorfi, Spectral discretization of Darcy's equations with pressure dependent porosity. Appl. Math. Comput. 217 (2010) 1838-1856. | MR

[4] M. Azaïez, F. Ben Belgacem, M. Grundmann and H. Khallouf, 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. | MR | Zbl

[5] C. Bernardi, Indicateurs d'erreur en h − N version des éléments spectraux. Modél. Math. et Anal. Numér. 30 (1996) 1-38. | Numdam | MR | Zbl

[6] C. Bernardi, A. Blouza, N. Chorfi and N. Kharrat, A penalty algorithm for the spectral element discretization of the Stokes problem. Math. Model. Numer. Anal. 45 (2011) 201-216. | Numdam | MR | Zbl

[7] C. Bernardi, T. Chacón Rebollo, F. Hecht and R. Lewandowski, 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. | MR | Zbl

[8] 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).

[9] C. Bernardi, M. Dauge and Y. Maday, Polynomials in Sobolev Spaces and Application to the Mortar Spectral Element Method, in preparation.

[10] 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. | MR | Zbl

[11] C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Collection Mathématiques et Applications vol. 45. Springer-Verlag (2004). | MR | Zbl

[12] M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 1 (2003) 221-238. | MR | Zbl

[13] H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ. 1 (2001), 387-404. | MR | Zbl

[14] 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. | MR | Zbl

[15] T. Chacón Rebollo, S. Del Pino and D. Yakoubi, An iterative procedure to solve a coupled two-fluids turbulence model. Math. Model. Numer. Anal. 44 (2010) 693-713. | Numdam | MR | Zbl

[16] A.L. Chaillou and M. Suri, Computable error estimators for the approximation of nonlinear problems by linearized models. Comput. Methods Appl. Mech. Engrg. 196 (2006) 210-224. | MR | Zbl

[17] M. Daadaa, Discrétisation spectrale et par éléments spectraux des équations de Darcy, Ph.D. Thesis, Université Pierre et Marie Curie, Paris (2009).

[18] M. Dauge, Neumann and mixed problems on curvilinear polyhedra. Integr. Equ. Oper. Th. 15 (1992) 227-261. | MR | Zbl

[19] L. El Alaoui, A. Ern and M. Vohralík, 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. | MR | Zbl

[20] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag (1986). | MR | Zbl

[21] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. I. Dunod, Paris (1968). | Zbl

[22] N.G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Sup. Pisa 17 (1963) 189-206. | Numdam | MR | Zbl

[23] J. Pousin and J. Rappaz, Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numer. Math. 69 (1994) 213-231. | MR | Zbl

[24] K.R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solid. Math. Models Methods Appl. Sci. 17 (2007) 215-252. | MR | Zbl

[25] G. Talenti, Best constant in Sobolev inequality. Ann. Math. Pura ed Appl. Serie IV 110 (1976) 353-372. | MR | Zbl

[26] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley ans Teubner (1996). | Zbl

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