WIAS Preprint No. 1929, (2014)

Optimal and pressure-independent $L^2$ velocity error estimates for a modified Crouzeix--Raviart Stokes element with BDM reconstructions



Authors

  • Brennecke, Christian
  • Linke, Alexander
    ORCID: 0000-0002-0165-2698
  • Merdon, Christian
  • Schöberl, Joachim

Keywords

  • variational crime, Crouzeix-Raviart finite element, divergence-free mixed method, incompressible Navier-Stokes equations, a priori error estimates

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 H1 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 L2 velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the Navier-Stokes case.

Appeared in

  • J. Comput. Math., 33 (2015) pp. 191--208.

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