As a first draft of a model for a river flowing on a homogeneous porous ground, we consider a system where the Darcy and Stokes equations are coupled via appropriate matching conditions on the interface. We propose a discretization of this problem which combines the mortar method with standard finite elements, in order to handle separately the flow inside and outside the porous medium. We prove a priori and a posteriori error estimates for the resulting discrete problem. Some numerical experiments confirm the interest of the discretization.
Mots-clés : Mortar method, finite elements, Darcy equations, Stokes equations
@article{M2AN_2008__42_3_375_0,
author = {Bernardi, Christine and Rebollo, Tom\'as Chac\'on and Hecht, Fr\'ed\'eric and Mghazli, Zoubida},
title = {Mortar finite element discretization of a model coupling {Darcy} and {Stokes} equations},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {375--410},
publisher = {EDP-Sciences},
volume = {42},
number = {3},
year = {2008},
doi = {10.1051/m2an:2008009},
mrnumber = {2423791},
zbl = {1138.76044},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2008009/}
}
TY - JOUR AU - Bernardi, Christine AU - Rebollo, Tomás Chacón AU - Hecht, Frédéric AU - Mghazli, Zoubida TI - Mortar finite element discretization of a model coupling Darcy and Stokes equations JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 375 EP - 410 VL - 42 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2008009/ DO - 10.1051/m2an:2008009 LA - en ID - M2AN_2008__42_3_375_0 ER -
%0 Journal Article %A Bernardi, Christine %A Rebollo, Tomás Chacón %A Hecht, Frédéric %A Mghazli, Zoubida %T Mortar finite element discretization of a model coupling Darcy and Stokes equations %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 375-410 %V 42 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2008009/ %R 10.1051/m2an:2008009 %G en %F M2AN_2008__42_3_375_0
Bernardi, Christine; Rebollo, Tomás Chacón; Hecht, Frédéric; Mghazli, Zoubida. Mortar finite element discretization of a model coupling Darcy and Stokes equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 3, pp. 375-410. doi : 10.1051/m2an:2008009. http://www.numdam.org/articles/10.1051/m2an:2008009/
[1] and , Un schéma de volumes ou éléments finis adaptatif pour les équations de Darcy à perméabilité variable. C. R. Acad. Sci. Paris Sér. I 333 (2001) 693-698. | MR | Zbl
[2] , and , A priori and a posteriori analysis of finite volume discretizations of Darcy's equations. Numer. Math. 96 (2003) 17-42. | MR | Zbl
[3] , Sobolev Spaces. Academic Press (1975). | MR | Zbl
[4] , and , Boundary Value Problems in Mechanics of Nonhomogeneous Fluids [English transl.], Studies in Mathematics and its Applications 22. North-Holland (1990). | MR | Zbl
[5] , The Mortar finite element method with Lagrangian multiplier. Numer. Math. 84 (1999) 173-197. | MR | Zbl
[6] and , Analysis of some finite elements for the Stokes problem. Math. Comput. 44 (1985) 71-79. | MR | Zbl
[7] , and , A new nonconforming approach to domain decomposition: the mortar element method, in Collège de France Seminar XI, H. Brezis and J.-L. Lions Eds., Pitman (1994) 13-51. | MR | Zbl
[8] , and , Analyse numérique d'indicateurs d'erreur, in Maillage et adaptation, Chap. 8, P.-L. George Ed., Hermès (2001) 251-278.
[9] , and , Discrétisations variationnelles de problèmes aux limites elliptiques, Collection Mathématiques et Applications 45. Springer-Verlag (2004). | MR | Zbl
[10] , and , Coupling Darcy and Stokes equations for porous media with cracks. ESAIM: M2AN 39 (2005) 7-35. | Numdam | MR | Zbl
[11] , and , Basics and some applications of the mortar element method. GAMM - Gesellschaft für Angewandte Mathematik und Mechanik 28 (2005) 97-123. | MR
[12] , and , Mortar finite element discretization for the flow in a non homogeneous porous medium. Comput. Methods Appl. Mech. Engrg. 196 (2007) 1554-1573. | MR
[13] and , Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722-731. | MR | Zbl
[14] and , A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal. 33 (1996) 2431-2444. | MR | Zbl
[15] and , Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15. Springer-Verlag (1991). | MR | Zbl
[16] and , A unified stabilized method for Stokes' and Darcy's equations. J. Comput. Applied Math. 198 (2007) 35-51. | MR | Zbl
[17] , Modélisation et simulation numérique du transport de radon dans un milieu poreux fissuré ou fracturé. Problème direct et problèmes inverses comme outils d'aide à la prédiction sismique. Ph.D. thesis, Université de Franche-Comté, Besançon, France (2002).
[18] and , A unifying theory of a posteriori error control for nonconforming finite element methods. Numer. Math. 107 (2007) 473-502. | MR | Zbl
[19] , Neumann and mixed problems on curvilinear polyhedra. Integr. Equat. Oper. Th. 15 (1992) 227-261. | MR | Zbl
[20] , Quantitative Hydrology. Groundwater Hydrology for Engineers. Academic Press, New York (1986).
[21] , and , Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43 (2002) 57-74. | MR | Zbl
[22] , Old and new elements for incompressible flows. Internat. J. Numer. Methods Fluids 1 (1981) 347-364. | MR | Zbl
[23] and , Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. (Submitted).
[24] and , Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer Series in Computational Mathematics 5. Springer-Verlag (1986). | MR | Zbl
[25] , and , A domain decomposition and mixed method for a linear parabolic boundary value problem. IMA J. Numer. Anal. 24 (2004) 491-520. | MR | Zbl
[26] , Elliptic Problems in Nonsmooth Domains. Pitman (1985). | MR | Zbl
[27] and , FreeFem++, see www.freefem.org.
[28] , and , Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40 (2002) 2195-2218. | MR | Zbl
[29] , Analyse mathématique et océanographie, Collection Recherches en Mathématiques Appliquées. Masson (1997).
[30] and , Problèmes aux limites non homogènes et applications, Vol. 1. Dunod (1968). | MR | Zbl
[31] , Mixed finite elements in . Numer. Math. 35 (1980) 315-341. | MR | Zbl
[32] and , Regularity of viscous Navier-Stokes flows in nonsmooth domains, in Proc. Conf. Boundary Value Problems and Integral Equations in Nonsmooth Domains, M. Costabel, M. Dauge and S. Nicaise Eds., Lect. Notes Pure Appl. Math. 167, Dekker (1995) 185-201. | MR | Zbl
[33] , On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Models Methods Appl. Sci. 17 (2007) 215-252. | MR | Zbl
[34] and , A mixed finite element method for second order elliptic problems, in Mathematical Aspects of Finite Element Methods, Lect. Notes Math. 606, Springer (1977) 292-315. | MR | Zbl
[35] , , and , Coupling Stokes and Darcy equations. Applied Numer. Math. (2007) (in press). | MR | Zbl
[36] , A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner (1996). | Zbl
Cité par Sources :
