Volume 33, Issue 2
Optimal and Pressure-Independent $L^2$ Velocity Error Estimates for a Modified Crouzeix-Raviart Stokes Element with BDM Reconstructions

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

Published online: 2015-04

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• 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.

• Keywords

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

65N30, 65N15, 76D07.

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@Article{JCM-33-191, author = {C. and Brennecke and cbrenne@student.ethz.ch and 6997 and Eidgenössische Technische Hochschule Zürich, Departement Mathematik, Rämistr. 101, 8092 Zürich, Switzerland and C. Brennecke and A. and Linke and Alexander.Linke@wias-berlin.de and 6998 and Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany and A. Linke and C. and Merdon and Christian.Merdon@wias-berlin.de and 6999 and Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany and C. Merdon and J. and Schöberl and joachim.schoeberl@tuwien.ac.at and 7000 and TU Wien, Institut f¨ur Analysis und Scientific Computing, Wiedner Hauptstr. 8-10/101,1040 Wien, Austria and J. Schöberl}, title = {Optimal and Pressure-Independent $L^2$ Velocity Error Estimates for a Modified Crouzeix-Raviart Stokes Element with BDM Reconstructions}, journal = {Journal of Computational Mathematics}, year = {2015}, volume = {33}, number = {2}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1411-m4499}, url = {http://global-sci.org/intro/article_detail/jcm/9836.html} }
TY - JOUR T1 - Optimal and Pressure-Independent $L^2$ Velocity Error Estimates for a Modified Crouzeix-Raviart Stokes Element with BDM Reconstructions AU - Brennecke , C. AU - Linke , A. AU - Merdon , C. AU - Schöberl , J. JO - Journal of Computational Mathematics VL - 2 SP - 191 EP - 208 PY - 2015 DA - 2015/04 SN - 33 DO - http://doi.org/10.4208/jcm.1411-m4499 UR - https://global-sci.org/intro/article_detail/jcm/9836.html KW - Variational crime, Crouzeix-Raviart finite element, Divergence-free mixed method, Incompressible Navier-Stokes equations, A priori error estimates. AB -

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.

C. Brennecke, A. Linke, C. Merdon & J. Schöberl. (2020). Optimal and Pressure-Independent $L^2$ Velocity Error Estimates for a Modified Crouzeix-Raviart Stokes Element with BDM Reconstructions. Journal of Computational Mathematics. 33 (2). 191-208. doi:10.4208/jcm.1411-m4499
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