On the parameter choice in grad-div stabilization for the Stokes equations

Abstract

Standard error analysis for grad-div stabilization of inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be \(\mathcal O(1)\). This paper revisits this choice for the Stokes equations on the basis of minimizing the \(H^{1}(\Omega )\) error of the velocity and the \(L^{2}(\Omega )\) error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. In particular, the approximation property of the pointwise divergence-free subspace plays a key role. With such an optimal approximation property and with an appropriate choice of the stabilization parameter, estimates for the \(H^{1}(\Omega )\) error of the velocity are obtained that do not directly depend on the viscosity and the pressure. The minimization of the \(L^{2}(\Omega )\) error of the pressure requires in many cases smaller stabilization parameters than the minimization of the \(H^{1}(\Omega )\) velocity error. Altogether, depending on the situation, the optimal stabilization parameter could range from being very small to very large. The analytic results are supported by numerical examples. Applying the analysis to the MINI element leads to proposals for the stabilization parameter which seem to be new.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Arnold, D., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo 21(4), 337–344 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Arnold, D., Qin, J.: Quadratic velocity/linear pressure Stokes elements. In: Vichnevetsky, R., Knight, D., Richter, G. (eds.) Advances in Computer Methods for Partial Differential Equations VII, pp. 28–34. IMACS (1992)

  3. 3.

    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  MATH  MathSciNet  Google Scholar 

  4. 4.

    Brenner, S., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994)

    Book  MATH  Google Scholar 

  5. 5.

    Bychenkov, Y., Chizonkov, E.V.: Optimization of one three-parameter method of solving an algebraic system of Stokes type. Russ. J. Numer. Anal. Math. Model. 14, 429–440 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Case, M., Ervin, V., Linke, A., Rebholz, L.: A connection between Scott-Vogelius elements and grad-div stabilization. SIAM J. Numer. Anal. 49(4), 1461–1481 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Dorok, O., Grambow, W., Tobiska, L.: Aspects of finite element discretizations for solving the Boussinesq approximation of the Navier-Stokes equations. Notes on numerical fluid mechanics; numerical methods for the Navier-Stokes equations. In: Hebeker, F.-K., Rannacher, R., Wittum, G. (eds.) Proceedings of the International Workshop held at Heidelberg, October 1993, vol. 47, pp. 50–61. (1994)

  8. 8.

    Franca, L., Hughes, T.: Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Eng. 69(1), 89–129 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Franca, L.P., Oliveira, S.P.: Pressure bubbles stabilization features in the Stokes problem. Comput. Methods Appl. Mech. Eng. 192(16-18), 1929–1937 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    freeFEM.org. freeFEM. http://www.freefem.org/

  11. 11.

    Galvin, K., Linke, A., Rebholz, L., Wilson, N.: Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection. Comput. Methods Appl. Mech. Eng. 237, 166–176 (2012)

    Article  MathSciNet  Google Scholar 

  12. 12.

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

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Glowinski, R., Le Tallec, P.: Augmented Lagrangian and operator splitting methods in nonlinear mechanics. SIAM, Studies in Applied Mathematics, Philadelphia (1989)

  14. 14.

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

    Article  MathSciNet  Google Scholar 

  15. 15.

    John, V., Kindl, A.: Numerical studies of finite element variational multiscale methods for turbulent flow simulations. Comput. Methods Appl. Mech. Eng. 199(13-16), 841–852 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    John, V., Matthies, G.: MooNMD - a program package based on mapped finite element methods. Comput. Vis. Sci. 6(2–3), 163–170 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. 17.

    Layton, W.: An Introduction to the Numerical Analysis of Viscous Incompressible Flows, SIAM (2008)

  18. 18.

    Layton, W., Manica, C., Neda, M., Olshanskii, M.A., Rebholz, L.: On the accuracy of the rotation form in simulations of the Navier-Stokes equations. J. Comput. Phys. 228(9), 3433–3447 (2009)

    Article  MathSciNet  Google Scholar 

  19. 19.

    Linke, A., Rebholz, L., Wilson, N.: On the convergence rate of grad-div stabilized Taylor-Hood to Scott-Vogelius solutions for incompressible flow problems. J. Math. Anal. Appl. 381, 612–626 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Lube, G., Olshanskii, M.: Stable finite element calculations of incompressible flows using the rotation form of convection. IMA J. Num. Anal. 22, 437–461 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. 21.

    Lube, G., Rapin, G.: Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math. Models Methods Appl. Sci. 16(7), 949–966 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Manica, C., Neda, M., Olshanskii, M.A., Rebholz, L.: Enabling accuracy of Navier-Stokes-alpha through deconvolution and enhanced stability. M2AN: Math. Model. Numer. Anal. 45, 277–308 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    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. Meth. Appl. Mech. Eng. 191, 5515–5536 (2002)

    Article  MathSciNet  Google Scholar 

  24. 24.

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

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

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

    Article  MATH  MathSciNet  Google Scholar 

  26. 26.

    Pierre, R.: Simple C 0 approximations for the computation of incompressible flows. Comput. Methods Appl, Mech. Eng. 68(2), 205–227 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  27. 27.

    Qin, J.: On the convergence of some low order mixed finite elements for incompressible fluids. Pennsylvania State University, PhD thesis (1994)

    Google Scholar 

  28. 28.

    Roos, H.-G., Stynes, M., Tobiska, L.: Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems. In: Springer Series in Computational Mathematics, vol. 24. Springer-Verlag, Berlin (1996)

  29. 29.

    Roos, H.-G., Stynes, M., Tobiska, L. Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems. In: Springer Series in Computational Mathematics, vol. 24, 2nd edn. Springer-Verlag, Berlin (2008)

  30. 30.

    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  MATH  MathSciNet  Google Scholar 

  31. 31.

    Zhang, S.: A new family of stable mixed finite elements for the 3d Stokes equations. Math. Comput. 74(250), 543–554 (2005)

    Article  MATH  Google Scholar 

  32. 32.

    Zhang, S.: On the P1 Powell-Sabin divergence-free finite element for the Stokes equations. J. Comput. Math. 26(3), 456–470 (2008)

    MATH  MathSciNet  Google Scholar 

  33. 33.

    Zhang, S.: Bases for C0-P1 divergence-free elements and for C1-P2 finite elements on union jack grids. Submitted (2012)

Download references

Author information

Affiliations

Authors

Corresponding authors

Correspondence to Eleanor W. Jenkins or Alexander Linke.

Additional information

A. Linke supported by the DFG Research Center MATHEON, project D27. L. G. Rebholz partially supported by NSF grant DMS1112593.

Communicated by: M. Stynes

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Jenkins, E.W., John, V., Linke, A. et al. On the parameter choice in grad-div stabilization for the Stokes equations. Adv Comput Math 40, 491–516 (2014). https://doi.org/10.1007/s10444-013-9316-1

Download citation

Keywords

  • Incompressible Stokes equations
  • Mixed finite elements
  • Grad-div stabilization
  • Error estimates
  • Pointwise divergence-free subspace

Mathematics Subject Classification (2010)

  • 65N30
  • 76M10