Skip to main content
Log in

Local Mass Conservation of Stokes Finite Elements

Journal of Scientific Computing Aims and scope Submit manuscript

Cite this article


In this paper we discuss the stability of some Stokes finite elements. In particular, we consider a modification of Hood–Taylor and Bercovier–Pironneau schemes which consists in adding piecewise constant functions to the pressure space. This enhancement, which had been already used in the literature, is driven by the goal of achieving an improved mass conservation at element level. The main result consists in proving the inf-sup condition for the enhanced spaces in a general setting and to present some numerical tests which confirm the stability properties. The improvement in the local mass conservation is shown in a forthcoming paper (Boffi et al. In: Papadrakakis, M., Onate, E., Schrefler, B. (eds.) Coupled Problems 2011. Computational Methods for Coupled Problems in Science and Engineering IV, Cimne, 2011) where the presented schemes are used for the solution of a fluid-structure interaction problem.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others


  1. Bercovier, M., Pironneau, O.: Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math. 33(2), 211–224 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boffi, D.: Stability of higher order triangular Hood-Taylor methods for the stationary Stokes equations. Math. Models Methods Appl. Sci. 4(2), 223–235 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boffi, D.: Three-dimensional finite element methods for the Stokes problem. SIAM J. Numer. Anal. 34(2), 664–670 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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, vol. 1939, pp. 45–100. Springer, Berlin (2008)

    Chapter  Google Scholar 

  5. Boffi, D., Gastaldi, L., Heltai, L., Peskin, C.S.: On the hyper-elastic formulation of the immersed boundary method. Comput. Methods Appl. Mech. Eng. 197(25–28), 2210–2231 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boffi, D., Cavallini, N., Gardini, F., Gastaldi, L.: Immersed boundary method: performance analysis of popular finite element spaces. In: Papadrakakis, M., Onate, E., Schrefler, B. (eds.) Coupled Problems 2011. Computational Methods for Coupled Problems in Science and Engineering IV, Cimne (2011)

    Google Scholar 

  7. Brezzi, F., Falk, R.S.: Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal. 28(3), 581–590 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)

    Book  MATH  Google Scholar 

  9. Chapelle, D., Bathe, K.-J.: The inf-sup test. Comput. Struct. 47(4–5), 537–545 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  11. Glowinski, R.: Finite element methods for incompressible viscous flow. In: Handbook of Numerical Analysis. Handb. Numer. Anal., vol. IX, pp. 3–1176. North-Holland, Amsterdam (2003)

    Google Scholar 

  12. Gresho, P.M., Lee, R.L., Chan, S.T., Leone, J.M.: A new finite element for Boussinesq fluids. In: Pro. Third Int. Conf. on Finite Elements in Flow Problems, pp. 204–215. Wiley, New York (1980)

    Google Scholar 

  13. Griffiths, D.F.: The effect of pressure approximation on finite element calculations of compressible flows. In: Morton, K.W., Baines, M.J. (eds.) Numerical Methods for Fluid Dynamics, pp. 359–374. Academic Press, San Diego (1982)

    Google Scholar 

  14. Pierre, R.: Local mass conservation and C 0-discretizations of the Stokes problem. Houst. J. Math. 20(1), 115–127 (1994)

    MathSciNet  MATH  Google Scholar 

  15. Qin, J., Zhang, S.: Stability of the finite elements 9/(4c+1) and 9/5c for stationary Stokes equations. Comput. Struct. 84(1–2), 70–77 (2005)

    Article  MathSciNet  Google Scholar 

  16. Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. Modél. Math. Anal. Numér. 19(1), 111–143 (1985)

    MathSciNet  MATH  Google Scholar 

  17. Stenberg, R.: Analysis of mixed finite elements methods for the Stokes problem: a unified approach. Math. Comput. 42(165), 9–23 (1984)

    MathSciNet  MATH  Google Scholar 

  18. Taylor, C., Hood, P.: A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1(1), 73–100 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Thatcher, R.W.: Locally mass-conserving Taylor-Hood elements for two- and three-dimensional flow. Int. J. Numer. Methods Fluids 11(3), 341–353 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tidd, D.M., Thatcher, R.W., Kaye, A.: The free surface flow of Newtonian and non-Newtonian fluids trapped by surface tension. Int. J. Numer. Methods Fluids 8(9), 1011–1027 (1988)

    Article  MathSciNet  Google Scholar 

  21. Verfürth, R.: Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO. Anal. Numér. 18(2), 175–182 (1984)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to F. Gardini.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Boffi, D., Cavallini, N., Gardini, F. et al. Local Mass Conservation of Stokes Finite Elements. J Sci Comput 52, 383–400 (2012).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: