zbMATH — the first resource for mathematics

Mortar finite element discretization of a model coupling Darcy and Stokes equations. (English) Zbl 1138.76044
Summary: As a first draft of a model for a river flowing on a homogeneous porous ground, we consider a system where 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.

76M10 Finite element methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
76D07 Stokes and related (Oseen, etc.) flows
86A05 Hydrology, hydrography, oceanography
Full Text: DOI EuDML
[1] Y. Achdou and C. Bernardi, Un schéma de volumes ou éléments finis adaptatif pour les équations de Darcy à perméabilité variable. C. R. Acad. Sci. Paris Sér. I333 (2001) 693-698. Zbl0996.65123 · Zbl 0996.65123
[2] 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. Zbl1050.76035 · Zbl 1050.76035
[3] R.A. Adams, Sobolev Spaces. Academic Press (1975).
[4] S. Antontsev, A.V. Kazhikhov and V.N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids [English transl.], Studies in Mathematics and its Applications22. North-Holland (1990). Zbl0696.76001 · Zbl 0696.76001
[5] F. Ben Belgacem, The Mortar finite element method with Lagrangian multiplier. Numer. Math.84 (1999) 173-197. · Zbl 0944.65114
[6] C. Bernardi and G. Raugel, Analysis of some finite elements for the Stokes problem. Math. Comput.44 (1985) 71-79. · Zbl 0563.65075
[7] C. Bernardi, Y. Maday and A.T. Patera, 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. · Zbl 0797.65094
[8] C. Bernardi, B. Métivet and R. Verfürth, Analyse numérique d’indicateurs d’erreur, in Maillage et adaptation, Chap. 8, P.-L. George Ed., Hermès (2001) 251-278.
[9] C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Collection Mathématiques et Applications45. Springer-Verlag (2004).
[10] C. Bernardi, F. Hecht and O. Pironneau, Coupling Darcy and Stokes equations for porous media with cracks. ESAIM: M2AN39 (2005) 7-35. · Zbl 1079.76041
[11] C. Bernardi, Y. Maday and F. Rapetti, Basics and some applications of the mortar element method. GAMM - Gesellschaft für Angewandte Mathematik und Mechanik28 (2005) 97-123. · Zbl 1177.65178
[12] C. Bernardi, F. Hecht and Z. Mghazli, Mortar finite element discretization for the flow in a non homogeneous porous medium. Comput. Methods Appl. Mech. Engrg.196 (2007) 1554-1573. · Zbl 1173.76331
[13] J. Boland and R. Nicolaides, Stability of finite elements under divergence constraints. SIAM J. Numer. Anal.20 (1983) 722-731. · Zbl 0521.76027
[14] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal.33 (1996) 2431-2444. · Zbl 0866.65071
[15] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics15. Springer-Verlag (1991). · Zbl 0788.73002
[16] E. Burman and P. Hansbo, A unified stabilized method for Stokes’ and Darcy’s equations. J. Comput. Applied Math.198 (2007) 35-51. · Zbl 1101.76032
[17] D.-G. Calugaru, 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] C. Carstensen and J. Hu, A unifying theory of a posteriori error control for nonconforming finite element methods. Numer. Math.107 (2007) 473-502. Zbl1127.65083 · Zbl 1127.65083
[19] M. Dauge, Neumann and mixed problems on curvilinear polyhedra. Integr. Equat. Oper. Th.15 (1992) 227-261. Zbl0767.46026 · Zbl 0767.46026
[20] G. de Marsily, Quantitative Hydrology. Groundwater Hydrology for Engineers. Academic Press, New York (1986).
[21] M. Discacciati, E. Miglio and A. Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math.43 (2002) 57-74. · Zbl 1023.76048
[22] M. Fortin, Old and new elements for incompressible flows. Internat. J. Numer. Methods Fluids1 (1981) 347-364. · Zbl 0467.76030
[23] J. Galvis and M. Sarkis, Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. (Submitted). · Zbl 1170.76024
[24] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer Series in Computational Mathematics5. Springer-Verlag (1986). · Zbl 0585.65077
[25] V. Girault, R. Glowinski and H. López, A domain decomposition and mixed method for a linear parabolic boundary value problem. IMA J. Numer. Anal.24 (2004) 491-520. · Zbl 1062.65101
[26] P. Grisvard, Elliptic Problems in Nonsmooth Domains . Pitman (1985). Zbl0695.35060 · Zbl 0695.35060
[27] F. Hecht and O. Pironneau, FreeFem++, see www.freefem.org.
[28] W.J. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow. SIAM J. Numer. Anal.40 (2002) 2195-2218. Zbl1037.76014 · Zbl 1037.76014
[29] R. Lewandowski, Analyse mathématique et océanographie, Collection Recherches en Mathématiques Appliquées. Masson (1997).
[30] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod (1968). · Zbl 0165.10801
[31] J.-C. Nédélec, Mixed finite elements in \Bbb R 3 . Numer. Math.35 (1980) 315-341. · Zbl 0419.65069
[32] M. Orlt and A.-M. Sändig, 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. Zbl0810.00016 · Zbl 0810.00016
[33] K.R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Models Methods Appl. Sci.17 (2007) 215-252. · Zbl 1123.76066
[34] P.-A. Raviart and J.-M. Thomas, 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. · Zbl 0362.65089
[35] J.M. Urquiza, D. N’Dri, A. Garon and M.C. Delfour, Coupling Stokes and Darcy equations. Applied Numer. Math. (2007) (in press). · Zbl 1125.76047
[36] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques . Wiley & Teubner (1996). · Zbl 0853.65108
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.