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Embedded Systems and Artificial Intelligence pp 233–242Cite as

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

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Part of the book series: Advances in Intelligent Systems and Computing ((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.

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References

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Elakkad, A., Elkhalfi, A.: Analysis of estimators for stokes problem using a mixed approximation. Bol. Soc. Paran. Mat. (3s.) (2018)

    Google Scholar 

  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. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Method, Computational Mathematics. Springer, New York (1991)

    Book  Google Scholar 

  7. Raviart, P.A., Thomas, J.: Introduction l’analyse numérique des à équations aux dérivées partielles. Masson, Paris (1983)

    MATH  Google Scholar 

  8. Ainsworth, M., Oden, J.: A posteriori error estimates for Stokes’ and Oseen’s equations. SIAM J. Numer. Anal 34, 228–245 (1997)

    Article  MathSciNet  Google Scholar 

  9. Verfurth, R.: A posteriori error estimators for the Stokes equations. Numer. Math 55, 309–325 (1989)

    Article  MathSciNet  Google Scholar 

  10. Ern, A.: Aide-mémoire Eléments Finis. Dunod, Paris (2005)

    Google Scholar 

  11. Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations. Springer, Berlin, Heiderlberg, New York (1981)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  13. Creuse, E., Kunert, G., Nicaise, S.: A posteriori error estimation for the Stokes problem: anisotropic and isotropic discretizations. MMAS 14, 1297–1341 (2004)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  15. Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

  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. Clement, P.: Approximation by finite element functions using local regularization. RAIRO. Anal. Numer. 2, 77–84 (1975)

    MathSciNet  MATH  Google Scholar 

  18. Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44, 283–301 (1985)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratory of Mathematics and Applications, ENS - Hassan II University, Casablanca, Morocco

    Omar El Moutea & Hassan El Amri

  2. Centre Régional des Métiers d’Education et de Formation (CRMEF), Fes, Morocco

    Hassan El Amri & Abdeslam Elakkad

Authors
  1. Omar El Moutea
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  2. Hassan El Amri
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  3. Abdeslam Elakkad
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Corresponding author

Correspondence to Omar El Moutea .

Editor information

Editors and Affiliations

  1. Department of Electronics and Communication Engineering, Shri Ramswaroop Memorial Group of Professional Colleges (SRMGPC), Lucknow, Uttar Pradesh, India

    Vikrant Bhateja

  2. School of Computer Engineering, Kalinga Institute of Industrial Technology (KIIT), Bhubaneswar, Odisha, India

    Suresh Chandra Satapathy

  3. Department of Computer Sciences, Faculty of Sciences Dhar Mahraz, Sidi Mohammed Ben Abbdallah University, Fez, Morocco

    Hassan Satori

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El Moutea, O., El Amri, H., Elakkad, A. (2020). Resolution of Brinkman Equations with a New Boundary Condition by Using Mixed Finite Element Method. In: Bhateja, V., Satapathy, S., Satori, H. (eds) Embedded Systems and Artificial Intelligence. Advances in Intelligent Systems and Computing, vol 1076. Springer, Singapore. https://doi.org/10.1007/978-981-15-0947-6_23

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