Optimal and pressure-independent $L^2$ velocity error estimates for a modified Crouzeix--Raviart Stokes element with BDM reconstructions
- Brennecke, Christian
- Linke, Alexander
- Merdon, Christian
- Schöberl, Joachim
- variational crime, Crouzeix-Raviart finite element, divergence-free mixed method, incompressible Navier-Stokes equations, a priori error estimates
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.
- J. Comput. Math., 33 (2015) pp. 191--208.