Robust error estimates for stabilized finite element approximations of the two dimensional Navier–Stokes’ equations at high Reynolds number

We consider error estimates for stabilized finite element approximations of the two-dimensional Navier–Stokes’ equations on the unit square with periodic boundary conditions. The estimates for the vorticity are obtained in a weak norm that can be related to the norms of filtered quantities. L2-norm estimates are obtained for the velocities. Under the assumption of the existence of a certain decomposition of the solution, into large eddies and small fine scale fluctuations, the constants of the estimates are proven to be independent of the Reynolds number. Instead they depend on the L∞-norm of the initial vorticity and an exponential with factor proportional to the L∞-norm of the gradient of the large eddies. The main error estimates are on a posteriori form, but for certain stabilized methods the residuals may be upper bounded uniformly, leading to robust a priori error estimates.