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


Optimal and Pressure-Independent L² Velocity Error Estimates for a Modified Crouzeix-Raviart Stokes Element with BDM Reconstructions

C. Brennecke 1, A. Linke 2, C. Merdon 2, J. Schöberl 3

1 Eidgenössische Technische Hochschule Zürich, Departement Mathematik, Rämistr. 101, 8092 Zürich, Switzerland
2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
3 TU Wien, Institut für Analysis und Scientific Computing, Wiedner Hauptstr. 8-10/101,1040 Wien, Austria

Received 2014-3-3 Accepted 2014-11-17
Available online 2015-3-13
doi:10.4208/jcm.1411-m4499

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.

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

AMS subject classifications: 65N30, 65N15, 76D07.


Email: cbrenne@student.ethz.ch, Alexander.Linke@wias-berlin.de, Christian.Merdon@wias-berlin.de, joachim.schoeberl@tuwien.ac.at
 

The Global Science Journal