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

Santiago Badia, Juan Vicente Gutiérrez-Santacreu

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2 Citations (Scopus)

Abstract

Residual-based stabilized finite element (FE) 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 (VMS) framework, based on the decomposition of the fluid velocity into a resolvable FE component plus a modelled subgrid-scale component. The subgrid closure acts as a large eddy simulation turbulence model, leading to accurate under-resolved simulations. However, even though VMS formulations are increasingly used in the applied FE 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.

Original languageEnglish
Pages (from-to)1193-1221
Number of pages29
JournalIMA Journal of Numerical Analysis
Volume34
Issue number3
DOIs
Publication statusPublished - 1 Jan 2014
Externally publishedYes

Keywords

  • convergence
  • Navier-Stokes equations
  • stability
  • stabilized finite element methods
  • subgrid scales
  • variational multiscale methods

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