Stability and Optimal Convergence of Unfitted Extended Finite Element Methods with Lagrange Multipliers for the Stokes Equations

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

Abstract

We study a fictitious domain approach with Lagrange multipliers to discretize Stokes equations on a mesh that does not fit the boundaries. A mixed finite element method is used for fluid flow. Several stabilization terms are added to improve the approximation of the normal trace of the stress tensor and to avoid the inf-sup conditions between the spaces of the velocity and the Lagrange multipliers. We generalize first an approach based on eXtended Finite Element Method due to Haslinger-Renard (SIAM J Numer Anal 47(2):1474–1499, 2009) involving a Barbosa-Hughes stabilization and a robust reconstruction on the badly cut elements. Secondly, we adapt the approach due to Burman-Hansbo (Comput Methods Appl Mech Eng 199(41–44):2680–2686, 2010) involving a stabilization only on the Lagrange multiplier. Multiple choices for the finite elements for velocity, pressure and multiplier are considered. Additional stabilization on pressure (Brezzi-Pitkäranta, Interior Penalty) is added, if needed. We prove the stability and the optimal convergence of several variants of these methods under appropriate assumptions. Finally, we perform numerical tests to illustrate the capabilities of the methods.

Notes

Acknowledgements

We wish to thank Prof. Erik Burman for giving us the occasion to participate in the “Unfitted FEM” workshop and to contribute to this volume. We are indebted to Prof. Yves Renard—the main developer of GetFEM++ library, used for all our numerical experiments—for adapting this library for our needs and for useful advice.

References

  1. 1.
    Amestoy, P.R., Duff, I.S., L’Excellent, J.Y., Koster, J.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001). http://dx.doi.org/10.1137/S0895479899358194
  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). https://dx.doi.org/10.1016/0045-7825(91)90125-P
  3. 3.
    Boffi, D., Brezzi, F., Fortin, M.: Finite elements for the Stokes problem. In: Boffi, D., Gastaldi, L. (eds.) Mixed Finite Elements, Compatibility Conditions, and Applications. Lecture Notes in Mathematics. Lectures Given at the C.I.M.E. Summer School held in Cetraro, June 26–July 1, 2006, vol. 1939. Springer/Fondazione C.I.M.E., Berlin/Florence (2008). http://dx.doi.org/10.1007/978-3-540-78319-0
  4. 4.
    Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the Stokes equations. In: Efficient Solutions of Elliptic Systems (Kiel, 1984). Notes on Numerical Fluid Mechanics, vol. 10, pp. 11–19. Friedrich Vieweg, Braunschweig (1984)Google Scholar
  5. 5.
    Burman, E.: Ghost penalty. C.R. Math. 348(21), 1217–1220 (2010)Google Scholar
  6. 6.
    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). http://dx.doi.org/10.1016/j.cma.2010.05.011
  7. 7.
    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).  https://doi.org/10.1093/imanum/drn081
  8. 8.
    Burman, E., Hansbo, P.: Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem. ESAIM Math. Model. Numer. Anal. 48(3), 859–874 (2014). http://dx.doi.org/10.1051/m2an/2013123
  9. 9.
    Court, S., Fournié, M.: A fictitious domain finite element method for simultations of fluid-structure interactions: the Navier-Stokes equations coupled with a moving solid. J. Fluids Struct. 55, 398–408 (2015). http://dx.doi.org/10.1016/j.jfluidstructs.2015.03.013
  10. 10.
    Court, S., Fournié, M., Lozinski, A.: A fictitious domain approach for the Stokes problem based on the extended finite element method. Int. J. Numer. Methods Fluids 74(2), 73–99 (2014).  http://dx.doi.org/10.1002/fld.3839
  11. 11.
    Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004). http://dx.doi.org/10.1007/978-1-4757-4355-5
  12. 12.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986). http://dx.doi.org/10.1007/978-3-642-61623-5
  13. 13.
    Guzmán, J., Olshanskii, M.A.: Inf-sup stability of geometrically unfitted Stokes finite elements. ArXiv e-prints (2016)Google Scholar
  14. 14.
    Haslinger, J., Renard, Y.: A new fictitious domain approach inspired by the extended finite element method. SIAM J. Numer. Anal. 47(2), 1474–1499 (2009). http://dx.doi.org/10.1137/070704435 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kirchhart, M., Gross, S., Reusken, A.: Analysis of an XFEM discretization for Stokes interface problems. SIAM J. Sci. Comput. 38(2), A1019–A1043 (2016). http://dx.doi.org/10.1137/15M1011779 MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité de Toulouse, UPSToulouse Cedex 9France
  2. 2.CNRS, Institut de Mathématiques, UMR 5219Toulouse Cedex 9France
  3. 3.Laboratoire de Mathématiques de BesançonUMR CNRS 6623, Univ. Bourgogne Franche-ComtéBesançonFrance

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