Deriving Robust Unfitted Finite Element Methods from Augmented Lagrangian Formulations

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 121)

Abstract

In this paper we will discuss different coupling methods suitable for use in the framework of the recently introduced CutFEM paradigm, cf. Burman et al. (Int. J. Numer. Methods Eng. 104(7):472–501, 2015). In particular we will consider mortaring using Lagrange multipliers on the one hand and Nitsche’s method on the other. For simplicity we will first discuss these methods in the setting of uncut meshes, and end with some comments on the extension to CutFEM. We will, for comparison, discuss some different types of problems such as high contrast problems and problems with stiff coupling or adhesive contact. We will review some of the existing methods for these problems and propose some alternative methods resulting from crossovers from the Lagrange multiplier framework to Nitsche’s method and vice versa.

Notes

Acknowledgements

The contribution of the first author was supported in part by the EPSRC grants EP/J002313/2 and EP/P01576X/1, the contribution of the second author was supported in part by the Swedish Foundation for Strategic Research Grant No. AM13-0029 and the Swedish Research Council Grant No. 2011-4992.

References

  1. 1.
    Alart, P., Curnier, A.: A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Eng. 92(3), 353–375 (1991)Google Scholar
  2. 2.
    Barbosa, H.J.C., Hughes, T.J.R.: The finite element method with Lagrange multipliers on the boundary: circumventing the Babuška-Brezzi condition. Comput. Methods Appl. Mech. Eng. 85(1), 109–128 (1991)Google Scholar
  3. 3.
    Barrenechea, G.R., Chouly, F.: A local projection stabilized method for fictitious domains. Appl. Math. Lett. 25(12), 2071–2076 (2012)Google Scholar
  4. 4.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)Google Scholar
  5. 5.
    Burman, E.: Projection stabilization of Lagrange multipliers for the imposition of constraints on interfaces and boundaries. Numer. Methods Partial Differ. Equ. 30(2), 567–592 (2014)Google Scholar
  6. 6.
    Burman, E., Hansbo, P.: A unified stabilized method for Stokes’ and Darcy’s equations. J. Comput. Appl. Math. 198(1), 35–51 (2007)Google Scholar
  7. 7.
    Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method. Comput. Methods Appl. Mech. Eng. 199(41–44), 2680–2686 (2010)Google Scholar
  8. 8.
    Burman, E., Hansbo, P.: Interior-penalty-stabilized Lagrange multiplier methods for the finite-element solution of elliptic interface problems. IMA J. Numer. Anal. 30(3), 870–885 (2010)Google Scholar
  9. 9.
    Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62(4), 328–341 (2012)Google Scholar
  10. 10.
    Burman, E., Zunino, P.: A domain decomposition method based on weighted interior penalties for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 44(4), 1612–1638 (2006)Google Scholar
  11. 11.
    Burman, E., Zunino, P.: Numerical approximation of large contrast problems with the unfitted Nitsche method. In: Frontiers in Numerical Analysis—Durham 2010. Lecture Notes in Computational Science and Engineering, vol. 85, pp. 227–282. Springer, Heidelberg (2012)Google Scholar
  12. 12.
    Burman, E., Claus, S., Hansbo, P., Larson, M.G., Massing, A.: CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Methods Eng. 104(7), 472–501 (2015)Google Scholar
  13. 13.
    Burman, E., Hansbo, P., Larson, M.G.: Augmented Lagrangian finite element methods for contact problems. ArXiv e-prints (2016)Google Scholar
  14. 14.
    Burman, E., Guzmán, J., Sánchez, M.A., Sarkis, M. Robust flux error estimation of an unfitted Nitsche method for high-contrast interface problems. IMA J. Numer. Anal. drx017.  https://doi.org/10.1093/imanum/drx017
  15. 15.
    Chouly, F., Hild, P.: A Nitsche-based method for unilateral contact problems: numerical analysis. SIAM J. Numer. Anal. 51(2), 1295–1307 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Chouly, F., Hild, P., Renard, Y.: Symmetric and non-symmetric variants of Nitsche’s method for contact problems in elasticity: theory and numerical experiments. Math. Comput. 84(293), 1089–1112 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Chouly, F., Mathieu, F., Hild, P., Mlika, R., Pousin, J., Renard, Y.: An overview of recent results on Nitsche’s method for contact problems. In: Geometrically Unfitted FEM, Theory and Applications. Proceedings from the 2016 UCL Workshop. Springer, Berlin (2018)Google Scholar
  18. 18.
    Fernández, M.A., Landajuela, M.: Splitting schemes for incompressible fluid/thin-walled structure interaction with unfitted meshes. C. R. Math. Acad. Sci. Paris 353(7), 647–652 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fortin, M., Glowinski, R.: Augmented Lagrangian Methods. Studies in Mathematics and its Applications, vol. 15. North-Holland, Amsterdam (1983). Applications to the numerical solution of boundary value problems, Translated from the French by B. Hunt and D. C. Spicer.Google Scholar
  20. 20.
    Glowinski, R., Le Tallec, P.: Numerical solution of problems in incompressible finite elasticity by augmented Lagrangian methods. I. Two-dimensional and axisymmetric problems. SIAM J. Appl. Math. 42(2), 400–429 (1982)MATHGoogle Scholar
  21. 21.
    Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191(47–48), 5537–5552 (2002)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hansbo, A., Hansbo, P.: A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Eng. 193(33–35), 3523–3540 (2004)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Juntunen, M.: On the connection between the stabilized Lagrange multiplier and Nitsche’s methods. Numer. Math. 131(3), 453–471 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Juntunen, M., Stenberg, R.: Nitsche’s method for general boundary conditions. Math. Comput. 78(267), 1353–1374 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lozinski, A., Fournié, M.: Stability and optimal convergence of unfitted extended finite element methods with lagrange multipliers for the stokes equations. In: Geometrically Unfitted FEM, Theory and Applications. Proceedings from the 2016 UCL Workshop. Springer, Berlin (2018)Google Scholar
  26. 26.
    Nitsche, J.A.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Univ. Hamburg 36, 9–15 (1971)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Stenberg, R.: On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63(1–3), 139–148 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK
  2. 2.Department of Mechanical EngineeringJönköping UniversityJönköpingSweden

Personalised recommendations