Skip to main content
SpringerLink
Log in

A stabilized Nitsche overlapping mesh method for the Stokes problem

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We develop a Nitsche-based formulation for a general class of stabilized finite element methods for the Stokes problem posed on a pair of overlapping, non-matching meshes. By extending the least-squares stabilization to the overlap region, we prove that the method is stable, consistent, and optimally convergent. To avoid an ill-conditioned linear algebra system, the scheme is augmented by a least-squares term measuring the discontinuity of the solution in the overlap region of the two meshes. As a consequence, we may prove an estimate for the condition number of the resulting stiffness matrix that is independent of the location of the interface. Finally, we present numerical examples in three spatial dimensions illustrating and confirming the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Alnæs, M.S., Logg, A., Mardal, K.-M., Skavhaug, O., Langtangen, H.P.: Unified framework for finite element assembly. Int. J. Comput. Sci. Eng. 4(4), 231–244 (2009)

    Article  Google Scholar 

  2. Alnæs, M.S., Logg, A., Ølgaard, K.B., Rognes, M.E., Wells, G.N.: Unified form language: a domain-specific language for weak formulations of partial differential equations. ACM Trans. Math. Softw. (2013, to appear). http://arxiv.org/abs/1211.4047

  3. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Num. Anal. 39, 1749–1779 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barth, T., Bochev, P., Gunzburger, M., Shadid, J.: A taxonomy of consistently stabilized finite element methods for the Stokes problem. SIAM J. Num. Anal. 25(5), 1585 (2004)

    MATH  MathSciNet  Google Scholar 

  5. Becker, R., Burman, E., Hansbo, P.: A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Eng. 198(41–44), 3352–3360 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. In: Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, Berlin (2008)

  7. Day, D., Bochev, P.: Analysis and computation of a least-squares method for consistent mesh tying. J. Comput. Appl. Math. 218(1), 21–33 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Douglas, J., Wang, J.: An absolutely stabilized finite element method for the Stokes problem. Math. Comp 52(186), 495–508 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ern, A., Guermond, J.L.: Evaluation of the condition number in linear systems arising in finite element approximations. ESAIM, Math. Model. Num. Anal. 40(1), 29–48 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Franca, L.P., Hughes, T.J.R., Stenberg, R.: Incompressible Computational Fluid Dynamics. In: Gunzburger, M.D., Nicolaides, R.A. (eds.) Stabilized finite element methods for the Stokes problem. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  11. Girault, V., Rivière, B., Wheeler, M.F.: A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier–Stokes problems. Math. Comp. 74(249), 53–84 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hansbo, A., Hansbo, P., Mats G., Larson: A finite element method on composite grids based on Nitsche’s method. ESAIM, Math. Model. Num. Anal. 37(3), 495–514 (2003)

    Article  MATH  Google Scholar 

  14. Hansbo, P.: Nitsche’s method for interface problems in computational mechanics. GAMM-Mitt 28(2), 183–206 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hansbo, P., Hermansson, J.: Nitsche’s method for coupling non-matching meshes in fluid-structure vibration problems. Comput. Mech. 32(1–2), 134–139 (2003)

    Article  MATH  Google Scholar 

  16. Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59(1), 85–99 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kirby, R.C., Logg, A.: A compiler for variational forms. ACM Trans. Math. Softw. 32(3), 417–444 (2006)

    Article  MathSciNet  Google Scholar 

  18. Logg, A.: Automating the finite element method. Arch. Comput. Methods Eng. 14(2), 93–138 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Logg, A., Mardal, K.-A., Wells, G.N., et al.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  20. Logg, A., Wells, G. N. DOLFIN.: Automated finite element computing. ACM Trans. Math. Softw. 37(2), (2010)

  21. Logg, A., Ølgaard, K.B., Rognes, M.E., Wells, G.N.: FFC: the FEniCS Form Compiler, chapter 11. Springer, Berlin (2012)

    Google Scholar 

  22. Massing, A., Larson, M.G., Logg, A.: Efficient implementation of finite element methods on non-matching and overlapping meshes in 3D. SIAM J. Sci. Comput. 35(1), C23–C47 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  23. Massing, A., Larson, M.G., Logg, A., Rognes, M.E.: A stabilized Nitsche fictitious domain method for the Stokes problem. J. Sci. Comp. (2013, submitted)

  24. Nitsche, J.: 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(1), 9–15 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  25. Quarteroni, A.: Numerical models for differential problems. In: Modeling, Simulation and Applications. Springer, Berlin (2009)

  26. Scott, R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54(190), 483–493 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  27. Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  28. Verfürth, R.: A posteriori error estimation and adaptive mesh-refinement techniques. J. Comp. Appl. Math. 50(1), 67–83 (1994)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

This work is supported by an Outstanding Young Investigator grant from the Research Council of Norway, NFR 180450. This work is also supported by a Center of Excellence grant from the Research Council of Norway to the Center for Biomedical Computing at Simula Research Laboratory.

Author information

Authors and Affiliations

  1. Simula Research Laboratory, Oslo, Norway

    André Massing & Marie E. Rognes

  2. Department of Mathematics, Umeå University, Umeå, Sweden

    Mats G. Larson

  3. Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden

    Anders Logg

Authors
  1. André Massing
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mats G. Larson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anders Logg
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Marie E. Rognes
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to André Massing.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Massing, A., Larson, M.G., Logg, A. et al. A stabilized Nitsche overlapping mesh method for the Stokes problem. Numer. Math. 128, 73–101 (2014). https://doi.org/10.1007/s00211-013-0603-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-013-0603-z

Mathematics Subject Classification (2000)

Search

Navigation