OPTIMAL AND PRESSURE-INDEPENDENT L2 VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS

doi:10.4208/jcm.1411-m4499

Journal of Computational Mathematics ›› 2015, Vol. 33 ›› Issue (2): 191-208.DOI: 10.4208/jcm.1411-m4499

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OPTIMAL AND PRESSURE-INDEPENDENT L² VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS

C. Brennecke¹, A. Linke², C. Merdon², J. Schöberl³

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:2014-03-03 Revised:2014-11-17 Online:2015-03-15 Published:2015-03-13
Supported by:
The research documented in this paper has been partially funded in the framework of the project “Macroscopic Modeling of Transport and Reaction Processes in Magnesium-Air-Batteries” (Grant 03EK3027D) under the research initiative “Energy storage” of the German Federal government.

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C. Brennecke, A. Linke, C. Merdon, J. Schöberl. OPTIMAL AND PRESSURE-INDEPENDENT L² VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS[J]. Journal of Computational Mathematics, 2015, 33(2): 191-208.

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 pressureindependent 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

CLC Number:

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OPTIMAL AND PRESSURE-INDEPENDENT L2 VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS

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OPTIMAL AND PRESSURE-INDEPENDENT L² VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS