Numerische Mathematik

, Volume 139, Issue 2, pp 315–348 | Cite as

Finite element methods for Darcy’s problem coupled with the heat equation

  • Christine Bernardi
  • Séréna DibEmail author
  • Vivette Girault
  • Frédéric Hecht
  • François Murat
  • Toni Sayah


In this article, we study theoretically and numerically the heat equation coupled with Darcy’s law by a nonlinear viscosity depending on the temperature. We establish existence of a solution by using a Galerkin method and we prove uniqueness. We propose and analyze two numerical schemes based on finite element methods. An optimal a priori error estimate is then derived for each numerical scheme. Numerical experiments are presented that confirm the theoretical accuracy of the discretization.

Mathematics Subject Classification

35K05 25B45 74S05 


  1. 1.
    Abboud, H., Girault, V., Sayah, T.: A second order accuracy in time for a full discretized time-dependent Navier–Stockes equations by a two-grid scheme. Numer. Math. 114, 189–231 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adams, J.A.: Sobolev Spaces. Academic Press, New York (1975)zbMATHGoogle Scholar
  3. 3.
    Bernardi, C., Girault, V.: A local regularisation operation for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35, 1893–1916 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bernardi, C., Métivet, B., Pernaud-Thomas, B.: Couplage des équations de Navier–Stokes et de la chaleur: le modèle et son approximation par éléments finis. RAIRO Modél. Math. Anal. Numér. 29(7), 871–921 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bernardi, C., Maarouf, S., Yakoubi, D.: Spectral discretization of Darcy’s equations coupled with the heat equation. IMA J. Numer. Anal. 36(3), 1193–1216 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Boussinesq, J.: Théorie analytique de la chaleur, Volume 2 of Lecture Notes in Mathematics. Gauthier-Villars, Paris (1903)Google Scholar
  7. 8.
    Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Handbook of Numerical Analysis, Finite Element Methods (Part I), II, pp. 17–343. North-Holland (1991)Google Scholar
  8. 9.
    Clément, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9, 77–84 (1975)MathSciNetzbMATHGoogle Scholar
  9. 10.
    Deteix, J., Jendoubi, A., Yakoubi, D.: A coupled prediction scheme for solving the Navier–Stokes and convection–diffusion equations. SIAM J. Numer. Anal. 52(5), 2415–2439 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 11.
    Gaultier, M., Lezaun, M.: Équations de Navier–Stokes couplées à des équations de la chaleur: résolution par une méthode de point fixe en dimension infinie. Ann. Sci. Math. Québec 13(1), 1–17 (1989)MathSciNetzbMATHGoogle Scholar
  11. 12.
    Girault, V., Lions, J.-L.: Two-grid finite-element schemes for the transient Navier–Stokes problem. M2AN Math. Model. Numer. Anal. 35(5), 945–980 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 13.
    Girault, V., Raviart, P.-A.: Finite element methods for Navier–Stokes equations. In: Theory and Algorithms, SCM 5, Springer-Verlag, Berlin (1986)Google Scholar
  13. 14.
    Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–266 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 15.
    Hooman, K., Gurgenci, H.: Effects of temperature-dependent viscosity variation on entropy generation, heat and fluid flow through a porous-saturated duct of rectangular cross-section. ISSN Appl. Math. Mech. 28, 69–78 (2007)CrossRefzbMATHGoogle Scholar
  15. 16.
    Nečas, J.: Les Méthodes directes en théorie des équations elliptiques. Masson, Paris (1967)zbMATHGoogle Scholar
  16. 17.
    Oyarzua, R., Schötzau, D., Quin, T.: An exactly divergence-free finite element method for a generalized Boussinesq problem. IMA J. Numer. Anal. 34(3), 1104–1135 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 18.
    Rashad, A.M.: Effects of radiation and variable viscosity on unsteady MHD flow of a rotating fluid from stretching surface in porous medium. J. Egypt. Math. Soc. 22, 134–142 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 19.
    Roberts, J.E., Thomas, J.-M.: Mixed and hybrid methods. In: Handbook of Numerical Analysis, Finite Element Methods (Part I), II, pp. 523–637. North-Holland (1991)Google Scholar
  19. 20.
    Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Annales de l’institut Fourier (Grenoble) 15, 189–258 (1985)CrossRefzbMATHGoogle Scholar
  20. 21.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 22.
    Vincent, D.: Le Théorème du point fixe de Brouwer. (2002–2003)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Christine Bernardi
    • 1
  • Séréna Dib
    • 1
    • 2
    Email author
  • Vivette Girault
    • 1
  • Frédéric Hecht
    • 1
  • François Murat
    • 1
  • Toni Sayah
    • 2
  1. 1.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie, Paris VIParis Cedex 05France
  2. 2.Laboratoire de Mathématiques et Applications (LMA), Unité de recherche “Mathématiques et Modélisation” (MM), Faculté des SciencesUniversité Saint-JosephBeirutLebanon

Personalised recommendations