Numerische Mathematik

, Volume 126, Issue 4, pp 635–677 | Cite as

Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow

  • Gabriel N. Gatica
  • Luis F. Gatica
  • Antonio Márquez
Article

Abstract

In this paper we introduce and analyze a new mixed finite element method for the two-dimensional Brinkman model of porous media flow with mixed boundary conditions. We use a dual-mixed formulation in which the main unknown is given by the pseudostress. The original velocity and pressure unknowns are easily recovered through a simple postprocessing. In addition, since the Neumann boundary condition becomes essential, we impose it in a weak sense, which yields the introduction of the trace of the fluid velocity over the Neumann boundary as the associated Lagrange multiplier. We apply the Babuška–Brezzi theory to establish sufficient conditions for the well-posedness of the resulting continuous and discrete formulations. In particular, a feasible choice of finite element subspaces is given by Raviart–Thomas elements of order \(k \ge 0\) for the pseudostress, and continuous piecewise polynomials of degree \(k + 1\) for the Lagrange multiplier. We also derive a reliable and efficient residual-based a posteriori error estimator for this problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Raviart–Thomas and Clément interpolation operators are the main tools for proving the reliability. Then, Helmholtz’s decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, several numerical results illustrating the performance and the robustness of the method, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are provided.

Mathematics Subject Classification (2000)

65N30 65N12 65N15 65J15 74B20 

References

  1. 1.
    Agmon, S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton (1965)MATHGoogle Scholar
  2. 2.
    Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Arnold, D.N., Brezzi, F., Douglas, J.: PEERS: a new mixed finite element method for plane elasticity. Jpn. J. Appl. Math. 1, 347–367 (1984)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Arnold, D.N., Douglas, J., Gupta, ChP: A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45(1), 1–22 (1984)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Babuška, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the finite element method. In: Aziz, A.K. (ed.) The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Academic Press, New York (1972)Google Scholar
  6. 6.
    Babuška, I., Gatica, G.N.: On the mixed finite element method with Lagrange multipliers. Numer. Methods Partial Differ. Equ. 19(2), 192–210 (2003)CrossRefMATHGoogle Scholar
  7. 7.
    Barrios, T.P., Gatica, G.N., González, M., Heuer, N.: A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity. ESAIM Math. Model. Numer. Anal. 40(5), 843–869 (2006)CrossRefMATHGoogle Scholar
  8. 8.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)Google Scholar
  9. 9.
    Burman, E., Hansbo, P.: A unified stabilized method for Stokes’ and Darcy’s equations. J. Comput. Appl. Math. 198(1), 35–51 (2007)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Cai, Z., Lee, B., Wang, P.: Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. SIAM J. Numer. Anal. 42(2), 843–859 (2004)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Cai, Z., Starke, G.: Least-squares methods for linear elasticity. SIAM J. Numer. Anal. 42(2), 826–842 (2004)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Cai, Z., Tong, Ch., Vassilevski, P.S., Wang, Ch.: Mixed finite element methods for incompressible flow: stationary Stokes equations. Numer Methods Partial Differ. Equ. 26(4), 957–978 (2009)MathSciNetGoogle Scholar
  13. 13.
    Carstensen, C.: An a posteriori error estimate for a first-kind integral equation. Math. Comput. 66(217), 139–155 (1997)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Carstensen, C.: A posteriori error estimate for the mixed finite element method. Math. Comput. 66(218), 465–478 (1997)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Carstensen, C., Dolzmann, G.: A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81(2), 187–209 (1998)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATHGoogle Scholar
  17. 17.
    Clément, P.: Approximation by finite element functions using local regularisation. RAIRO Model. Math. et Anal. Numer. 9, 77–84 (1975)MATHGoogle Scholar
  18. 18.
    Ervin, V.J., Howell, J.S., Stanculescu, I.: A dual-mixed approximation method for a three-field model of a nonlinear generalized Stokes problem. Comput. Methods Appl. Mech. Eng. 197(33–40), 2886–2900 (2008)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Figueroa, L., Gatica, G.N., Heuer, N.: A priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows. Comput. Methods Appl. Mech. Eng. 198(2), 280–291 (2008)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Figueroa, L.E., Gatica, G.N., Márquez, A.: Augmented mixed finite element methods for the stationary Stokes Equations. SIAM J. Sci. Comput. 31(2), 1082–1119 (2008)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Gatica, G.N.: A note on the efficiency of residual-based a-posteriori error estimators for some mixed finite element methods. Electron. Trans. Numer. Anal. 17, 218–233 (2004)MATHMathSciNetGoogle Scholar
  22. 22.
    Gatica, G.N.: Analysis of a new augmented mixed finite element method for linear elasticity allowing \(\mathbb{RT}_0-\mathbb{P}_1-\mathbb{P}_0\) approximations. ESAIM Math. Model. Numer. Anal. 40(1), 1–28 (2006)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Gatica, G.N., González, M., Meddahi, S.: A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. I: a priori error analysis. Comput. Methods Appl. Mech. Eng. 193(9–11), 881–892 (2004)CrossRefMATHGoogle Scholar
  24. 24.
    Gatica, G.N., Hsiao, G., Meddahi, S.: A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem. Numer. Math. 114(1), 63–106 (2009)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Gatica, G.N., Márquez, A., Meddahi, S.: An augmented mixed finite element method for 3D linear elasticity problems. J. Comput. Appl. Math. 231(2), 526–540 (2009)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Gatica, G.N., Márquez, A., Sánchez, M.: Analysis of a velocity–pressure–pseudostress formulation for the stationary Stokes equations. Comput. Methods Appl. Mech. Eng. 199(17–20), 1064–1079 (2010)CrossRefMATHGoogle Scholar
  27. 27.
    Gatica, G.N., Márquez, A., Sánchez, M.: A priori and a posteriori error analyses of a velocity–pseudostress formulation for a class of quasi-Newtonian Stokes flows. Comput. Methods Appl. Mech. Eng. 200(17–20), 1619–1636 (2011)CrossRefMATHGoogle Scholar
  28. 28.
    Gatica, G.N., Oyarzúa, R., Sayas, F.J.: Analysis of fully-mixed finite element methods for the Stokes–Darcy coupled problem. Math. Comput. 80(276), 1911–1948 (2011)CrossRefMATHGoogle Scholar
  29. 29.
    Girault, V., Raviart, P.A.: Finite element methods for Navier–Stokes equations. In: Theory and Algorithms. Springer, Berlin (1986)Google Scholar
  30. 30.
    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)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Howell, J.S.: Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity gradients. J. Comput. Appl. Math. 231(2), 780–792 (2009)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Juntunen, M., Stenberg, R.: Analysis of finite element methods for the Brinkman problem. Calcolo 47(3), 129–147 (2010)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Lions, J.-L., Magenes, E.: Problèmes aux Limites non Homogènes et Applications I. Dunod, Paris (1968)Google Scholar
  35. 35.
    Mc Lean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, London (2000)Google Scholar
  36. 36.
    Márquez, A., Meddahi, S., Sayas, F.-J.: Strong coupling of finite element methods for the Stokes–Darcy problem. arXiv:1203.4717v1[math.NA]Google Scholar
  37. 37.
    Nečas, J.: Les Méthodes Directes en Théorie des Équations Elliptiques. Masson, Paris (1967)MATHGoogle Scholar
  38. 38.
    Prössdorf, S., Silbermann, B.: Numerical Analysis for Integral and Related Operator Equations. Birkhäuser, Basel (1991)Google Scholar
  39. 39.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Heidelberg (1996)Google Scholar
  40. 40.
    Roberts, J.E., Thomas, J.M.: Mixed and hybrid methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis. Finite Element Methods (Part 1), vol. II. Nort-Holland, Amsterdam (1991)Google Scholar
  41. 41.
    Verfürth, R.: A posteriori error estimation and adaptive-mesh-refinement techniques. J. Comput. Appl. Math. 50(1–3), 67–83 (1994)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive-Mesh-Refinement Techniques. Wiley, Chichester (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Gabriel N. Gatica
    • 1
  • Luis F. Gatica
    • 2
  • Antonio Márquez
    • 3
  1. 1.CI²MA and Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile
  2. 2.CI²MA, Universidad de Concepción, and Facultad de IngenieríaUniversidad Católica de la Santísima ConcepciónConcepciónChile
  3. 3.Departamento de Construcción e Ingeniería de FabricaciónUniversidad de OviedoOviedoSpain

Personalised recommendations