Stability estimates for solving Stokes problem with nonconforming finite elements

We propose to analyse the discretization of the Stokes problem with nonconforming finite elements in light of the T-coercivity. First we exhibit a family of operators to prove T-coercivity and we show that the stability constant is equal to the classical one up to a constant which depends on the Babuška–Aziz constant. Then we explicit the stability constants with respect to the shape regularity parameter for order 1 in 2 or 3 dimensions, and order 2 in 2 dimensions. In this last case, we improve the result of the original Fortin–Soulie paper. Second, we illustrate the importance of using a divergence-free velocity reconstruction on some numerical experiments.