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

Authors

C. Brennecke Eidgenössische Technische Hochschule Zürich, Departement Mathematik, Rämistr. 101, 8092 Zürich, Switzerland
A. Linke Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
C. Merdon Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
J. Schöberl TU Wien, Institut f¨ur Analysis und Scientific Computing, Wiedner Hauptstr. 8-10/101,1040 Wien, Austria

DOI:

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 $H^1$ 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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Published

2018-08-22

Issue

Vol. 33 No. 2 (2015)

Section

Articles