Optimal and Pressure-Independent \(L^2\) Velocity Error Estimates for a Modified Crouzeix-Raviart Element with BDM Reconstructions
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 analytical results.
This research has been partially funded by 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.
- 2.Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7(R-3), 33–75 (1973)Google Scholar