CALCOLO

, Volume 43, Issue 4, pp 287–306 | Cite as

A discrete de Rham complex with enhanced smoothness

  • Xue–Cheng Tai
  • Ragnar Winther
Open Access
Article

Abstract

Discrete de Rham complexes are fundamental tools in the construction of stable elements for some finite element methods. The purpose of this paper is to discuss a new discrete de Rham complex in three space dimensions, where the finite element spaces have extra smoothness compared to the standard requirements. The motivation for this construction is to produce discretizations which have uniform stability properties for certain families of singular perturbation problem. In particular, we show how the spaces constructed here lead to discretizations of Stokes type systems which have uniform convergence properties as the Stokes flow approaches a Darcy flow.

Keywords: Discrete exact sequences, nonconforming finite elements, Darcy–Stokes flow, uniform error estimates.

Mathematics Subject Classification (1991): Primary 65N12, 65N15, 65N30

References

  1. 1. Arnold, D.N., Falk, R.S., Winther, R.: Differential complexes and stability of finite element methods. I. The de Rham complex. In: Arnold, D.N. et al. (eds.): Compatible spatial discretizations. (The IMA Volumes in Mathematics and its Applications 142) Berlin: Springer (2006), pp. 23–46Google Scholar
  2. 2. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1–155 (2006)MATHCrossRefMathSciNetGoogle Scholar
  3. 3. Brenner, S.C.: Korn's inequalities for piecewise H 1 vector fields. Math. Comp. 73, 1067–1087 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4. Brezzi, F.: On the existence, uniqueness and approximation of saddle–point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationelle Sér. Rouge 8, 129–151 (1974)MathSciNetGoogle Scholar
  5. 5. Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47, 217–235 (1985)MATHCrossRefMathSciNetGoogle Scholar
  6. 6. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Berlin: Springer 1991Google Scholar
  7. 7. Brezzi, F., Fortin, M., Marini, L.D.: Mixed finite element methods with continuous stress. Math. Models Methods Appl. Sci. 3, 275–287 (1993)MATHCrossRefMathSciNetGoogle Scholar
  8. 8. Dauge, M.: Stationary Stokes and Navier-Stokes systems on two- and three-dimensional domains with corners. I. Linearized equations. SIAM J. Math. Anal. 20, 74–97 (1989)MATHCrossRefMathSciNetGoogle Scholar
  9. 9. Girault, V., Raviart, P.-A.: Finite element methods for Navier–Stokes equations. Berlin: Springer 1986Google Scholar
  10. 10. Mardal, K.-A., Tai, X-C, Winther, R.: A robust finite element method for Darcy–Stokes flow. SIAM J. Num. Anal. 40, 1605–1631 (2002)MATHCrossRefMathSciNetGoogle Scholar
  11. 11. Mardal, K.-A., Winther, R.: An observation on Korn's inequality for nonconforming finite element methods. Math. Comp. 75, 1–6 (2005)CrossRefMathSciNetGoogle Scholar
  12. 12. Nédélec, J.C.: Mixed finite elements in R 3. Numer. Math. 35, 315–341 (1980)MATHCrossRefMathSciNetGoogle Scholar
  13. 13. Nédélec, J.C.: A new family of mixed finite elements in R 3. Numer. Math. 50, 57–81 (1986)MATHCrossRefMathSciNetGoogle Scholar
  14. 14. Nilssen, T.K., Tai, X.–C., Winther, R.: A robust nonconforming H 2-element. Math. Comp. 70, 489–505 (2001)MATHCrossRefMathSciNetGoogle Scholar
  15. 15. Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for 2nd order elliptic problems. In: Galligani, I.; Magenes, E. (eds.): Mathematical aspects of finite element methods. (Lecture Notes in Mathematics 606) Berlin: Springer (1977), pp. 292–315Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Xue–Cheng Tai
    • 1
  • Ragnar Winther
    • 2
  1. 1.Department of Mathematics, University of Bergen, 5007 BergenNorway
  2. 2.Centre of Mathematics for Applications and Department of Informatics, University of Oslo, 0316 OsloNorway

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