Skip to main content
SpringerLink
Log in

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

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, J.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  3. Bernardi, C., Girault, V.: A local regularisation operation for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35, 1893–1916 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  6. Boussinesq, J.: Théorie analytique de la chaleur, Volume 2 of Lecture Notes in Mathematics. Gauthier-Villars, Paris (1903)

  7. 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)

  8. Clément, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9, 77–84 (1975)

    MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    MathSciNet  MATH  Google Scholar 

  11. 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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Girault, V., Raviart, P.-A.: Finite element methods for Navier–Stokes equations. In: Theory and Algorithms, SCM 5, Springer-Verlag, Berlin (1986)

  13. Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–266 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MATH  Google Scholar 

  15. Nečas, J.: Les Méthodes directes en théorie des équations elliptiques. Masson, Paris (1967)

    MATH  Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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)

  19. 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)

    Article  MATH  Google Scholar 

  20. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Vincent, D.: Le Théorème du point fixe de Brouwer. https://www-fourier.ujf-grenoble.fr/~carriere/TerDunias.pdf (2002–2003)

Download references

Author information

Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris VI, 4, Place Jussieu, 75252, Paris Cedex 05, France

    Christine Bernardi, Séréna Dib, Vivette Girault, Frédéric Hecht & François Murat

  2. Laboratoire de Mathématiques et Applications (LMA), Unité de recherche “Mathématiques et Modélisation” (MM), Faculté des Sciences, Université Saint-Joseph, B.P 11-514 Riad El Solh, Beirut, 1107 2050, Lebanon

    Séréna Dib & Toni Sayah

Authors
  1. Christine Bernardi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Séréna Dib
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vivette Girault
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Frédéric Hecht
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. François Murat
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Toni Sayah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Séréna Dib.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bernardi, C., Dib, S., Girault, V. et al. Finite element methods for Darcy’s problem coupled with the heat equation. Numer. Math. 139, 315–348 (2018). https://doi.org/10.1007/s00211-017-0938-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-017-0938-y

Mathematics Subject Classification

Search

Navigation