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A uniformly stable Fortin operator for the Taylor–Hood element

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Abstract

We construct a new Fortin operator for the lowest order Taylor–Hood element, which is uniformly stable both in \(L^2\) and \(H^1\). The construction, which is restricted to two space dimensions, is based on a tight connection between a subspace of the Taylor–Hood velocity space and the lowest order Nedelec edge element. General shape regular triangulations are allowed for the \(H^1\)-stability, while some mesh restrictions are imposed to obtain the \(L^2\)-stability. As a consequence of this construction, a uniform inf–sup condition associated the corresponding discretizations of a parameter dependent Stokes problem is obtained, and we are able to verify uniform bounds for a family of preconditioners for such problems, without relying on any extra regularity ensured by convexity of the domain.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bramble, J.H., Pasciak, J.E.: Iterative techniques for time dependent Stokes problems. Comput. Math. Appl. 33(1–2), 13–30 (1997) (approximation theory and applications)

    Google Scholar 

  3. Bramble, J.H., Pasciak, J.E., Steinbach, O.: On the stability of the \(L^2\) projection in \(H^1(\Omega )\). Math. Comp. 71, 147–156 (2001)

    Article  MathSciNet  Google Scholar 

  4. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from L0 multipliers. RAIRO Anal. Numér. 8, 129–151 (1974)

    MathSciNet  MATH  Google Scholar 

  5. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  6. Cahouet, J., Chabard, J.-P.: Some fast 3D finite element solvers for the generalized Stokes problem. Internat. J. Numer. Methods Fluids 8(8), 869–895 (1988)

    Article  MathSciNet  Google Scholar 

  7. Clement, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9, 77–84 (1975)

    MathSciNet  MATH  Google Scholar 

  8. Costabel, M., McIntosh, A.: On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains. http://arxiv.org/abs/0808.2614v1

  9. Falk, R.S.: A Fortin operator for two-dimensional Taylor–Hood elements. Math. Model. Numer. Anal. 42, 411–424 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations, vol 1. In: Linearized Steady problems. Springer Tracts in Natural Philosophy, vol. 38. Springer, New York (1994)

  11. Geissert, M., Heck, H., Hieber, M.: On the equation \(\text{ div} u = g\) and Bogovskiĭ’s operator in Sobolev spaces of negative order. Oper. Theory Adv. Appl. 168, 113–121 (2006)

    Article  MathSciNet  Google Scholar 

  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)

    Book  Google Scholar 

  13. Kolev, T., Vassilevski, P.: Regular Decomposition of H(div) Spaces. Comput. Methods Appl. Math. (to appear)

  14. Lay, M.-J., Schumaker, L.L.: Spline functions on triangulations. In: Encyclopedia of Mathematics and its Applications. Cambridge University Press, London (2007).

  15. Mardal, K.-A., Schöberl, J., Winther, R.: A uniform inf-sup condition with applications to preconditioning. arXiv:1201.1513v1,2012.

  16. Mardal, K.-A., Winther, R.: Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebra Appl. 18, 1–40 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mardal, K.-A., Winther, R.: Uniform preconditioners for the time dependent Stokes problem. Numer. Math. 98(2), 305–327 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mardal, K.-A., Winther, R.: Erratum: “Uniform preconditioners for the time dependent Stokes problem” [Numer. Math. 98(2), 2004, pp. 305–327]. Numer. Math. 103(1), 171–172 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Olshanskii, M.A., Peters, J., Reusken, A.: Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations. Numer. Math. 105, 159–191 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Turek, S.: Efficient Solvers for Incompressible Flow Problems. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Informatics, Centre for Biomedical Computing at Simula Research Laboratory, Norway, University of Oslo, Oslo, Norway

    Kent-Andre Mardal

  2. Institute for Analysis and Scientific Computing, Wiedner Hauptstrasse 8-10, 1040 , Wien, Austria

    Joachim Schöberl

  3. Department of Informatics, Centre of Mathematics for Applications, University of Oslo, 0316 , Oslo, Norway

    Ragnar Winther

Authors
  1. Kent-Andre Mardal
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  2. Joachim Schöberl
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  3. Ragnar Winther
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Corresponding author

Correspondence to Ragnar Winther.

Additional information

K.-A. Mardal was supported by the Research Council of Norway through Grant 209951 and a Centre of Excellence grant to the Centre for Biomedical Computing at Simula Research Laboratory. R. Winther was supported by the Research Council of Norway through a Centre of Excellence grant to the Centre of Mathematics for Applications.

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Mardal, KA., Schöberl, J. & Winther, R. A uniformly stable Fortin operator for the Taylor–Hood element. Numer. Math. 123, 537–551 (2013). https://doi.org/10.1007/s00211-012-0492-6

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  • DOI: https://doi.org/10.1007/s00211-012-0492-6

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