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

Optimal and Pressure-Independent $L^2$ Velocity Error Estimates for a Modified Crouzeix-Raviart Stokes Element with BDM Reconstructions
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Year:    2015

Author:    C. Brennecke, A. Linke, C. Merdon, J. Schöberl

Journal of Computational Mathematics, Vol. 33 (2015), Iss. 2 : pp. 191–208

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 analysis and demonstrate the improved robustness in the Navier-Stokes case.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1411-m4499

Journal of Computational Mathematics, Vol. 33 (2015), Iss. 2 : pp. 191–208

Published online:    2015-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    18

Keywords:    Variational crime Crouzeix-Raviart finite element Divergence-free mixed method Incompressible Navier-Stokes equations A priori error estimates.

Author Details

C. Brennecke

A. Linke

C. Merdon

J. Schöberl

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