Robust error estimates for stabilized finite element approximations of the two dimensional Navier-Stokes equations with application to implicit large eddy simulation

We consider error estimates in weak parametrised norms for stabilized finite element approximations of the two-dimensional Navier-Stokes' equations. These weak norms can be related to the norms of certain filtered quantities, where the parameter of the norm, relates to the filter width. Under the assumption of the existence of a certain decomposition of the solution, into large eddies and fine scale fluctuations, the constants of the estimates are proven to be independent of both the Reynolds number and the Sobolev norm of the exact solution. Instead they exhibit exponential growth with a coefficient proportional to the maximum gradient of the large eddies. The error estimates are on a posteriori form, but using Sobolev injections valid on finite element spaces and the properties of the stabilization operators the residuals may be upper bounded uniformly, leading to robust a priori error estimates.