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

TL;DR

This paper develops robust error estimates for stabilized finite element methods applied to the 2D Navier-Stokes equations, with implications for implicit large eddy simulation, ensuring independence from Reynolds number and solution norms.

Contribution

It introduces error estimates in weak norms related to filtered quantities, demonstrating robustness and independence from key parameters under certain solution decompositions.

Findings

Constants are independent of Reynolds number and solution norms.

Error estimates are robust and can be upper bounded uniformly.

Residuals can be controlled using stabilization properties.

Abstract

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.