Advertisement

Optimal and Pressure-Independent \(L^2\) Velocity Error Estimates for a Modified Crouzeix-Raviart Element with BDM Reconstructions

  • Christian Brennecke
  • Alexander Linke
  • Christian Merdon
  • Joachim Schöberl
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 77)

Abstract

Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a-priori estimates for the velocity error become pressure-dependent, while divergence-free mixed finite elements deliver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modified Crouzeix-Raviart element, obeying an optimal pressure-independent discrete \(H^1\) velocity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressure-independent \(L^2\) velocity error. Numerical examples confirm the analytical results.

Notes

Acknowledgments

This research has been partially funded by the project “Macroscopic Modeling of Transport and Reaction Processes in Magnesium-Air-Batteries” (Grant 03EK3027D) under the research initiative “Energy storage” of the German Federal government.

References

  1. 1.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)CrossRefMATHGoogle Scholar
  2. 2.
    Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7(R-3), 33–75 (1973)Google Scholar
  3. 3.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)MATHGoogle Scholar
  4. 4.
    Linke, A.: On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Engrg. 268, 782–800 (2014)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Christian Brennecke
    • 1
  • Alexander Linke
    • 2
  • Christian Merdon
    • 2
  • Joachim Schöberl
    • 3
  1. 1.Eidgenössische Technische Hochschule ZürichDepartement MathematikZürichSwitzerland
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  3. 3.TU Wien, Institut für Analysis und Scientific ComputingWienAustria

Personalised recommendations