Calcolo

, Volume 47, Issue 3, pp 129–147 | Cite as

Analysis of finite element methods for the Brinkman problem

Article

Abstract

The parameter dependent Brinkman problem, covering a field of problems from the Darcy equations to the Stokes problem, is studied. A mathematical framework is introduced for analyzing the problem. Using this uniform a priori and a posteriori estimates for two families of finite element methods are proved. Nitsche’s method for imposing boundary conditions is discussed.

Brinkman equation Stokes equation Darcy equation Nitsche’s method Mixed finite element methods Stabilized methods 

Mathematics Subject Classification (2000)

65N30 

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References

  1. 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) CrossRefMathSciNetGoogle Scholar
  2. 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) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Arbogast, T., Lehr, H.L.: Homogenization of a Darcy-Stokes system modeling vuggy porous media. Comput. Geosci. 10(3), 291–302 (2006) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo 21(4), 337–344 (1985) CrossRefMathSciNetGoogle Scholar
  5. 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) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 2nd edn. Texts in Applied Mathematics, vol. 15. Springer, New York (2002) MATHGoogle Scholar
  7. 7.
    Clément, P.: Approximation of finite element functions using local regularization. RAIRO Anal. Numer. 9, 77–84 (1975) Google Scholar
  8. 8.
    Franca, L.P., Stenberg, R.: Error analysis of some Galerkin least-squares methods for the elasticity equations. SIAM J. Numer. Anal. 28(6), 1680–1699 (1991) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hannukainen, A., Juntunen, M., Stenberg, R.: Computations with finite element methods for the Brinkman problem, Helsinki University of Technology, Institute of Mathematics Research Report, A569 (2009) Google Scholar
  10. 10.
    Hansbo, P., Juntunen, M.: Weakly imposed Dirichlet boundary conditions for the Brinkman model of porous media flow. Appl. Numer. Math. 59(6), 1274–1289 (2009) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Juntunen, M., Stenberg, R.: A residual based a posteriori estimator for the reaction-diffusion problem. C. R. Math. Acad. Sci. Paris 347, 555–558 (2009) MATHMathSciNetGoogle Scholar
  12. 12.
    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 (1981). Erratum (17), 1239 (1981) MATHGoogle Scholar
  13. 13.
    Mardal, K.A., Tai, X.-C., Winther, R.: A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal. 40(5), 1605–1631 (2002) (electronic) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Semin. Univ. Hamb. 36, 9–15 (1970/1971) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Pitkäranta, J.: Boundary subspaces for the finite element method with Lagrange multipliers. Numer. Math. 33, 273–289 (1979) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    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) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Stenberg, R.: On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63(1–3), 139–148 (1995) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Verfürth, R.: Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numer. 18, 175–182 (1984) MATHMathSciNetGoogle Scholar
  19. 19.
    Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner and Wiley, Stuttgart (1996) MATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institute of Mathematics and Systems AnalysisHelsinki University of Technology (TKK)EspooFinland

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