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Resolution of Brinkman Equations with a New Boundary Condition by Using Mixed Finite Element Method

  • Omar El MouteaEmail author
  • Hassan El Amri
  • Abdeslam Elakkad
Conference paper
  • 110 Downloads
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1076)

Abstract

This paper considers numerical methods for solving Brinkman equations with a new boundary condition summing Dirichlet and Neumann conditions. We develop here a robust stabilized mixed finite element method (MFEM), and two types of a posteriori error indicator are introduced to give global error estimates; there are equivalent to the true error. We present numerical simulations.

Keywords

Brinkman equations \( C_{a, \mu ^{*}} \) boundary Mixed finite element methods Residual error estimator Numerical simulations 

References

  1. 1.
    Wu, D.H., Currie, I.G.: Analysis of a posteriori error indicator in viscous flows. Int. J. Num. Meth. Heat Fluid Flow 12, 306–327 (2002)CrossRefGoogle Scholar
  2. 2.
    Rajagopal, K.R.: On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math. Models Methods Appl. Sci. 17(2), 215–252 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Lévy, T.: Loi de Darcy ou loi de Brinkman? C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre. 292(12), 871–874, Erratum (17):12–39 (1981)Google Scholar
  4. 4.
    Elakkad, A., Elkhalfi, A.: Analysis of estimators for stokes problem using a mixed approximation. Bol. Soc. Paran. Mat. (3s.) (2018)Google Scholar
  5. 5.
    El-Mekkaoui, J., Elkhalfi, A., Elakkad, A.: Resolution of Stokes equations with the Ca;b boundary condition using mixed finite element method. WSEAS Trans. Math. (2015)Google Scholar
  6. 6.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Method, Computational Mathematics. Springer, New York (1991)CrossRefGoogle Scholar
  7. 7.
    Raviart, P.A., Thomas, J.: Introduction l’analyse numérique des à équations aux dérivées partielles. Masson, Paris (1983)zbMATHGoogle Scholar
  8. 8.
    Ainsworth, M., Oden, J.: A posteriori error estimates for Stokes’ and Oseen’s equations. SIAM J. Numer. Anal 34, 228–245 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Verfurth, R.: A posteriori error estimators for the Stokes equations. Numer. Math 55, 309–325 (1989)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ern, A.: Aide-mémoire Eléments Finis. Dunod, Paris (2005)Google Scholar
  11. 11.
    Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations. Springer, Berlin, Heiderlberg, New York (1981)zbMATHGoogle Scholar
  12. 12.
    Kay, D., Silvester, D.: A posteriori error estimation for stabilized mixed approximations of the Stokes equations. SIAM J. Sci. Comput. 21, 1321–1336 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Creuse, E., Kunert, G., Nicaise, S.: A posteriori error estimation for the Stokes problem: anisotropic and isotropic discretizations. MMAS 14, 1297–1341 (2004)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 27–34 (1948)zbMATHGoogle Scholar
  15. 15.
    Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford (2005)zbMATHGoogle Scholar
  16. 16.
    Roberts, J., Thomas, J.M.: Mixed and Hybrid Methods, Handbook of Numerical Analysis II, Finite Element Methods 1. P. Ciarlet and J. Lions, Amsterdam (1989)Google Scholar
  17. 17.
    Clement, P.: Approximation by finite element functions using local regularization. RAIRO. Anal. Numer. 2, 77–84 (1975)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44, 283–301 (1985)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Omar El Moutea
    • 1
    Email author
  • Hassan El Amri
    • 1
    • 2
  • Abdeslam Elakkad
    • 2
  1. 1.Laboratory of Mathematics and ApplicationsENS - Hassan II UniversityCasablancaMorocco
  2. 2.Centre Régional des Métiers d’Education et de Formation (CRMEF)FesMorocco

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