## Optimal and pressure-independent $$L^2$$ velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions.(English)Zbl 1340.76024

Summary: 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.

### MSC:

 76D07 Stokes and related (Oseen, etc.) flows 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs
Full Text: