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)


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.



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.


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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

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