Free access
Issue
ESAIM: M2AN
Volume 42, Number 3, May-June 2008
Page(s) 375 - 410
DOI http://dx.doi.org/10.1051/m2an:2008009
Published online 03 April 2008
  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. I 333 (2001) 693–698.
  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. [CrossRef] [MathSciNet]
  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 Applications 22. North-Holland (1990).
  5. F. Ben Belgacem, The Mortar finite element method with Lagrangian multiplier. Numer. Math. 84 (1999) 173–197. [CrossRef] [MathSciNet]
  6. C. Bernardi and G. Raugel, Analysis of some finite elements for the Stokes problem. Math. Comput. 44 (1985) 71–79. [CrossRef]
  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.
  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 Applications 45. Springer-Verlag (2004).
  10. C. Bernardi, F. Hecht and O. Pironneau, Coupling Darcy and Stokes equations for porous media with cracks. ESAIM: M2AN 39 (2005) 7–35. [CrossRef] [EDP Sciences]
  11. C. Bernardi, Y. Maday and F. Rapetti, Basics and some applications of the mortar element method. GAMM – Gesellschaft für Angewandte Mathematik und Mechanik 28 (2005) 97–123.
  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. [CrossRef] [MathSciNet]
  13. J. Boland and R. Nicolaides, Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722–731. [CrossRef] [MathSciNet]
  14. D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal. 33 (1996) 2431–2444. [CrossRef] [MathSciNet]
  15. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics 15. Springer-Verlag (1991).
  16. E. Burman and P. Hansbo, A unified stabilized method for Stokes' and Darcy's equations. J. Comput. Applied Math. 198 (2007) 35–51. [CrossRef] [MathSciNet]
  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. [CrossRef] [MathSciNet]
  19. M. Dauge, Neumann and mixed problems on curvilinear polyhedra. Integr. Equat. Oper. Th. 15 (1992) 227–261. [CrossRef]
  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. [CrossRef] [MathSciNet]
  22. M. Fortin, Old and new elements for incompressible flows. Internat. J. Numer. Methods Fluids 1 (1981) 347–364. [CrossRef] [MathSciNet]
  23. J. Galvis and M. Sarkis, Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations. (Submitted).
  24. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms, Springer Series in Computational Mathematics 5. Springer-Verlag (1986).
  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. [CrossRef] [MathSciNet]
  26. P. Grisvard, Elliptic Problems in Nonsmooth Domains . Pitman (1985).
  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. [CrossRef] [MathSciNet]
  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).
  31. J.-C. Nédélec, Mixed finite elements in Formula . Numer. Math. 35 (1980) 315–341. [CrossRef] [MathSciNet]
  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.
  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. [CrossRef] [MathSciNet]
  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.
  35. J.M. Urquiza, D. N'Dri, A. Garon and M.C. Delfour, Coupling Stokes and Darcy equations. Applied Numer. Math. (2007) (in press).
  36. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques . Wiley & Teubner (1996).

Recommended for you