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Computations with finite element methods for the Brinkman problem

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Abstract

Various finite element families for the Brinkman flow (or Stokes–Darcy flow) are tested numerically. Particularly, the effect of small permeability is studied. The tested finite elements are the MINI element, the Taylor–Hood element, and the stabilized equal order methods. The numerical tests include both a priori analysis and adaptive methods.

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References

  1. Allaire, G.: Homogenization of the navier-stokes equations in open sets perforated with tiny holes. I. abstract framework, a volume distribution of holes. Arch. Ration. Mech. Anal. 113(3), 209–259 (1990)

    Article  MathSciNet  Google Scholar 

  2. Allaire, G.: Homogenization of the navier-stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes. Arch. Ration. Mech. Anal. 113(3), 261–298 (1990)

    Article  MathSciNet  Google Scholar 

  3. Arbogast, T., Lehr, H.L.: Homogenization of a darcy-stokes system modeling vuggy porous media. Comput. Geosci. 10(3), 291–302 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo 21(4), 337–344 (1984). (1985)

    Article  MATH  MathSciNet  Google Scholar 

  5. Badia, S., Codina, R.: Unified stabilized finite element formulations for the Stokes and the Darcy problems. SIAM J. Numer. Anal. 47(3), 1971–2000 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bercovier, M., Pironneau, O.: Error estimates for finite element method solution of the stokes problem in the primitive variables. Numer. Math. 33(2), 211–224 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boffi, D.: Stability of higher order triangular Hood-Taylor methods for the stationary stokes equations. Math. Models Methods Appl. Sci. 4(2), 223–235 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  8. Boffi, D.: Three-dimensional finite element methods for the Stokes problem. SIAM J. Numer. Anal. 34(2), 664–670 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brezzi, F., Falk, R.S.: Stability of higher-order hood-taylor methods. SIAM J. Numer. Anal. 28(3), 581–590 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the Stokes equations. In: Efficient Solutions of Elliptic Systems (Kiel, 1984). Notes Numer. Fluid Mech., vol. 10, pp. 11–19. Vieweg, Braunschweig (1984)

    Google Scholar 

  11. Falk, R.S.: A fortin operator for two-dimensional taylor-hood elements. M2AN Math. Model. Numer. Anal. 42(3), 411–424 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Franca, L.P., Hughes, T.J.R., Stenberg, R.: Stabilized finite element methods. In: Gunzburger, M., Nicolaides, R.A. (eds.) Incompressible Computational Fluid Dynamics, chapter 4, pp. 87–107. Cambridge University Press (1993)

  13. Franca, L.P., Stenberg, R.: Error analysis of galerkin least squares methods for the elasticity equations. SIAM J. Numer. Anal. 28(6), 1680–1697 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hansbo, P., Juntunen, M.: Weakly imposed Dirichlet boundary conditions for the brinkman model of porous media flow. Appl. Numer. Math. 59, 1274–1289 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hughes, T.J.R., Franca, L.P.: A new finite element formulation for computational fluid dynamics. VII. The stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces. Comput. Methods Appl. Mech. Eng. 65(1), 85–96 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  16. Juntunen, M., Stenberg, R.: Analysis of finite element methods for the Brinkman problem. Calcolo, (2009). doi:10.1007/s10092-009-0017-6

    Google Scholar 

  17. Juntunen, M., Stenberg, R.: Nitsche’s method for general boundary conditions. Math. Comp. 78(267), 1353–1374 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Juntunen, M., Stenberg, R.: A residual based a posteriori estimator for the reaction-diffusion problem. C. R. Acad. Sci. Paris, Ser. I 347, 555–558 (2009)

    MATH  MathSciNet  Google Scholar 

  19. 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):1239 (1981)

    MATH  Google Scholar 

  20. Mardal, K., Tai, X., Winther, R.: A robust finite element method for darcy-stokes flow. SIAM J. Numer. Anal. 40(5), 1605–1631 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mardal,K., Winther, R.: Uniform preconditioners for the time dependent Stokes problem. Numer. Math. 98, 305–327 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mardal, K., Winther, R.: Uniform preconditioners for the time dependent stokes problem. Numer. Math. 103, 171–172 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nitsche, J.A.: Über ein variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36, 9–15 (1970/71)

    Article  MATH  MathSciNet  Google Scholar 

  24. 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  MATH  MathSciNet  Google Scholar 

  25. Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér. 19(1), 111–143 (1985)

    MATH  MathSciNet  Google Scholar 

  26. Stenberg, R.: On some three-dimensional finite elements for incompressible media. Comput. Methods Appl. Mech. Eng. 63(3), 261–269 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  27. Stenberg, R.: Error analysis of some finite element methods for the Stokes problem. Math. Comp. 54, 495–508 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Systems Analysis, Helsinki University of Technology, P.O. Box 1100, 02015 TKK, Helsinki, Finland

    Antti Hannukainen, Mika Juntunen & Rolf Stenberg

Authors
  1. Antti Hannukainen
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  2. Mika Juntunen
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  3. Rolf Stenberg
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Correspondence to Mika Juntunen.

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Hannukainen, A., Juntunen, M. & Stenberg, R. Computations with finite element methods for the Brinkman problem. Comput Geosci 15, 155–166 (2011). https://doi.org/10.1007/s10596-010-9204-4

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