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

• DOI: 10.4208/jcm.1411-m4499
• Publication Date: 2015-03-13T08:38:44Z
• Authors: C. Brennecke C. B...; A. Linke A. Linke; C. Merdon C. Merdon; J. Schöberl J. Sc...
• Keywords: ddc:510; a priori error estimates; article; Crouzeix-Raviart finite element; 16. Peace & justice; variational crime -- Crouzeix-Raviart finite element -- divergence-free mixed method -- incompressible Navier-Stokes equations -- a priori error estimates; 01 natural sciences; 510; divergence-free mixed method; variational crime; incompressible Navier-Stokes equations; 0101 mathematics

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.