Convergence towards weak solutions of the Navier–Stokes equations for a finite element approximation with numerical subgrid-scale modelling

Residual-based stabilized nite element techniques for the Navier-Stokes equations lead to numerical discretizations that provide convection stabilization as well as pressure stability without the need to satisfy an inf-sup condition. They can be motivated by using a variational multiscale framework, based on the decomposition of the uid velocity into a resolvable nite element component plus a modeled subgrid scale component. The subgrid closure acts as a large eddy simulation turbulence model, leading to accurate under-resolved simulations. However, even though variational multiscale formulations are increasingly used in the applied nite element community, their numerical analysis has been restricted to a priori estimates and convergence to smooth solutions only, via a priori error estimates. In this work we prove that some versions of these methods (based on dynamic and orthogonal closures) also converge to weak (turbulent) solutions of the Navier-Stokes equations. These results are obtained by using compactness results in Bochner-Lebesgue spaces. Navier-Stokes equations; stability; convergence; stabilized nite element methods; subgrid scales; variational multiscale methods.

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