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

Brennecke, C.; Linke, A.; Merdon, C.; Schöberl, J.

J. Comput. Math. 33, No. 2, 191-208 (2015).

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

Keywords:

Crouzeix-Raviart finite element; incompressible Navier-Stokes equations; a priori error estimates