Skip to main content
SpringerLink
Log in

Grad-div Stabilization for the Evolutionary Oseen Problem with Inf-sup Stable Finite Elements

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and Crank–Nicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical 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

Similar content being viewed by others

References

  1. Ayuso, B., García-Archilla, B., Novo, J.: The postprocessed mixed finite-element method for the Navier–Stokes equations. SIAM J. Numer. Anal. 43(3), 1091–1111 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Börm, S., Le Borne, S.: \(\cal H\)-LU factorization in preconditioners for augmented Lagrangian and grad-div stabilized saddle point systems. Int. J. Numer. Methods Fluids 68(1), 83–98 (2012)

    Article  MATH  Google Scholar 

  3. Bowers, A.L., Le Borne, S., Rebholz, L.G.: Error analysis and iterative solvers for Navier–Stokes projection methods with standard and sparse grad-div stabilization. Comput. Methods Appl. Mech. Eng. 275, 1–19 (2014)

    Article  MATH  Google Scholar 

  4. Braack, M., Burman, E., John, V., Lube, G.: Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Eng. 196(4–6), 853–866 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Case, M.A., Ervin, V.J., Linke, A., Rebholz, L.G.: A connection between Scott–Vogelius and grad-div stabilized Taylor–Hood FE approximations of the Navier–Stokes equations. SIAM J. Numer. Anal. 49(4), 1461–1481 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ciarlet, Philippe G.: The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, Vol. 4

  7. Codina, R.: Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput. Methods Appl. Mech. Eng. 191(39–40), 4295–4321 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Codina, R.: Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales. Appl. Numer. Math. 58(3), 264–283 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Codina, R., Principe, J., Guasch, O., Badia, S.: Time dependent subscales in the stabilized finite element approximation of incompressible flow problems. Comput. Methods Appl. Mech. Eng. 196(21–24), 2413–2430 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dallmann, H., Arndt, D., Lube, G.: Local projection stabilization for the Oseen problem. Preprint 09, Institut für Numerische und Angewandte Mathematik, University of Göttingen, (2014)

  11. de Frutos, J., García-Archilla, B., John, V., Novo, J.: Grad-div stabilization for the time-dependent Navier–Stokes equations with inf-sup stable slements. In preparation, (2015)

  12. de Frutos, J., García-Archilla, B., Novo, J.: Local error estimates for the SUPG method applied to evolutionary convection–reaction–diffusion equations. J. Sci. Comput. (2015). doi:10.1007/s10915-015-0035-2

  13. Gelhard, T., Lube, G., Olshanskii, M.A., Starcke, J.-H.: Stabilized finite element schemes with LBB-stable elements for incompressible flows. J. Comput. Appl. Math. 177, 243–267 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Girault, V., Raviart, P.A.: Finite element methods for Navier–Stokes equations, volume 5 of springer series in computational mathematics. Springer, Berlin (1986)

    Google Scholar 

  15. Heister, T., Rapin, G.: Efficient augmented Lagrangian-type preconditioning for the Oseen problem using grad-div stabilization. Int. J. Numer. Methods Fluids 71(1), 118–134 (2013)

    Article  MathSciNet  Google Scholar 

  16. Jenkins, E.W., John, V., Linke, A., Rebholz, L.G.: On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math. 40(2), 491–516 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. John, V., Kindl, A.: Numerical studies of finite element variational multiscale methods for turbulent flow simulations. Comput. Methods Appl. Mech. Eng. 199(1316), 841–852 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. John, V., Matthies, G.: MooNMD–a program package based on mapped finite element methods. Comput. Vis. Sci. 6(2–3), 163–169 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lube, G., Rapin, G.: Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math. Models Methods Appl. Sci. 16, 949–966 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Matthies, G., Lube, G., Röhe, L.: Some remarks on residual-based stabilisation of inf-sup stable discretisations of the generalised Oseen problem. Comput. Methods Appl. Math. 9, 368–390 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Olshanskii, M., Lube, G., Heister, T., Löwe, J.: Grad-div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 198(49–52), 3975–3988 (2009)

    Article  MATH  Google Scholar 

  22. Olshanskii, M.A.: A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a stabilization issue and iterative methods. Comput. Methods Appl. Mech. Eng. 191(47–48), 5515–5536 (2002)

    Article  MathSciNet  Google Scholar 

  23. Olshanskii, M.A., Reusken, A.: Grad-div stabilization for Stokes equations. Math. Comput. 73(248), 1699–1718 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Martin, R., Lutz, T.: Robust Numerical Methods for Singularly Perturbed Differential Equations, Volume 24 of Springer Series in Computational Mathematics. Convection–Diffusion–Reaction and Flow Problems, 2nd edn. Springer, Berlin (2008)

    Google Scholar 

  25. Scott, L.R., Zhang, Shangyou: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tobiska, L., Verfürth, R.: Analysis of a streamline diffusion finite element method for the Stokes and Navier–Stokes equations. SIAM J. Numer. Anal. 33, 107–127 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Investigación en Matemáticas (IMUVA), Universidad de Valladolid, Valladolid, Spain

    Javier de Frutos

  2. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain

    Bosco García-Archilla

  3. Weierstrass Institute for Applied Analysis and Stochastics, Leibniz Institute in Forschungsverbund Berlin e.V. (WIAS), Mohrenstr. 39, 10117, Berlin, Germany

    Volker John

  4. Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 6, 14195, Berlin, Germany

    Volker John

  5. Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain

    Julia Novo

Authors
  1. Javier de Frutos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bosco García-Archilla
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Volker John
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Julia Novo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julia Novo.

Additional information

The research was supported by Spanish MICINN under Grant Nos. MTM2013-42538-P, MTM2012-31821, MTM2013-42538-P, MTM2013-42538-P and European Cooperation in Science and Technology through COST Action IS1104.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Frutos, J., García-Archilla, B., John, V. et al. Grad-div Stabilization for the Evolutionary Oseen Problem with Inf-sup Stable Finite Elements. J Sci Comput 66, 991–1024 (2016). https://doi.org/10.1007/s10915-015-0052-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0052-1

Keywords

Search

Navigation