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The inf-sup condition for low order elements on anisotropic meshes

CALCOLO volume 41, pages 89–113 (2004)Cite this article

Abstract

On a large class of two-dimensional anisotropic meshes, the infsup condition (stability) is proved for the triangular and quadrilateral finite element pairs suggested by Bernardi/Raugel and Fortin. As a consequence the pairs\(\mathcal{P}_2 - \mathcal{P}_0 ,\mathcal{Q}_2 - \mathcal{P}_0 \), and\(\mathcal{Q}'_2 - \mathcal{P}_0 \) turn out to be stable independent of the aspect ratio of the elements.

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References

  1. Acosta, G., Durán R.G.: The maximum angle condition for mixed and non-conforming elements: application to the Stokes equations. SIAM J. Numer. Anal.37, 18–36 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ainsworth, M., Coggins, P.: The stability of mixedhp-finite element methods for Stokes flow on high aspect ratio elements. SIAM J. Numer. Anal.38, 1721–1761 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Apel, Th.: Anisotropic finite elements: local estimates and applications. (Advances in Numerical Mathematics) Stuttgart: Teubner 1999

    MATH  Google Scholar 

  4. Apel, Th., Nicaise, S., Schöberl, J.: A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges. IMA J. Numer. Anal.21, 843–856 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Apel, Th., Randrianarivony, H.M.: Stability of discretizations of the Stokes problem on anisotropic meshes. Math. Comput. Simulation61, 437–447 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Becker, R.: An adaptive finite element method for the incompressible Navier-Stokes equations on time-dependent domains. Dissertation. Heidelberg: Ruprecht-Karls-Universität 1995

    Google Scholar 

  7. Becker, R., Rannacher, R.: Finite element solution of the incompressible Navier-Stokes equations on anisotropically refined meshes. In: Hackbusch, W., Wittum, G. (eds.): Fast solvers for flow problems. (Notes on numerical fluid mechanics,49) Wiesbaden: Vieweg 1995, pp. 52–62.

    Google Scholar 

  8. Bernardi, C., Raugel, G.: Analysis of some finite elements for the Stokes problem. Math. Comp.44, 71–79 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. New York: Springer 1991

    MATH  Google Scholar 

  10. Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978. Reprinted: Philadelphia SIAM 2002

    MATH  Google Scholar 

  11. Durán, R.G., Hernández, E., Hervella-Nieto, L., Liberman, E., Rodríguez, R.: Error estimates for low-order isoparametric quadrilateral finite elements for plates. SIAMJ, Numer. Anal.41, 1751–1772 (2003)

    Article  MATH  Google Scholar 

  12. Durán, R.G., Liberman, E.: On mixed finite element methods for the Reissner-Mindlin plate model. Math. Comp.58, 561–573 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Durán, R.G., Hervella-Nieto, L., Liberman, E., Rodríguez, R., Solomin, J.: Approximation of the vibration modes of a plate by Reissner-Mindlin equations. Math. Comp.68, 1447–1463 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fortin, M.: Old and new finite elements for incompressible flows. Internat. J. Numer. Methods Fluids1, 347–364 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  15. Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. Theory and algorithms. Springer Series in Computational Mathematics5 Berlin: Springer 1986

    MATH  Google Scholar 

  16. Randrianarivony, M.: Stability of mixed finite element methods with anisotropic meshes. Master's thesis. Chemnitz: TU Chemnitz 2001

    Google Scholar 

  17. Schötzau, D., Schwab, Ch.: Mixedhp-FEM on anisotropic meshes. Math. Models Methods Appl. Sci.8, 787–820 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Schötzau, D., Schwab, Ch., Stenberg, R.: Mixedhp-FEM on anisotropic meshes. II. Hanging nodes and tensor products of boundary layer meshes. Numer. Math.83, 667–697 (1999)

    MATH  MathSciNet  Google Scholar 

  19. Synge, J.L.: The hypercircle in mathematical physics. Cambridge: Cambridge University Press 1957

    MATH  Google Scholar 

  20. Toselli, A., Schwab, C.: Mixedhp-finite element approximations on geometric edge and boundary layer meshes in three dimensions. Numer. Math.94, 771–801 (2003)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München, Neubiberg, Germany

    T. Apel

  2. Institut des Sciences et Techniques de Valenciennes, Université de Valenciennes et du Hainaut Cambrésis, Valenciennes, France

    S. Nicaise

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  1. T. Apel
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  2. S. Nicaise
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Both the visit of the first author in Valenciennes and the visit of the second author in Chemnitz were financed by the DFG (German Research Foundation), Sonderforschungsbereich 393.

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Apel, T., Nicaise, S. The inf-sup condition for low order elements on anisotropic meshes. Calcolo 41, 89–113 (2004). https://doi.org/10.1007/BF02637257

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  • DOI: https://doi.org/10.1007/BF02637257

Keywords

  • Isotropic Element
  • Mixed Finite Element Method
  • Anisotropic Mesh
  • Hanging Node
  • Regular Node