Variational multiscale modeling with discretely divergence-free subscales: Non-divergence-conforming discretizations

TL;DR

This paper extends a variational multiscale finite element method for incompressible Navier-Stokes equations to arbitrary inf-sup-stable pairs, ensuring convergence and robustness without divergence-conforming restrictions.

Contribution

It generalizes the convergence analysis of a stabilized finite element method to non-divergence-conforming discretizations, maintaining stability in advection-dominated flows.

Findings

Numerical convergence demonstrated with Taylor-Hood elements.

Method remains robust in advection-dominated regimes.

Extended analysis applies to arbitrary inf-sup-stable pairs.

Abstract

A recent paper [J. A. Evans, D. Kamensky, Y. Bazilevs, "Variational multiscale modeling with discretely divergence-free subscales", Computers & Mathematics with Applications, 80 (2020) 2517-2537] introduced a novel stabilized finite element method for the incompressible Navier-Stokes equations, which combined residual-based stabilization of advection, energetic stability, and satisfaction of a discrete incompressibility condition. However, the convergence analysis and numerical tests of the cited work were subject to the restrictive assumption of a divergence-conforming choice of velocity and pressure spaces, where the pressure space must contain the divergence of every velocity function. The present work extends the convergence analysis to arbitrary inf-sup-stable velocity-pressure pairs (while maintaining robustness in the advection-dominated regime) and demonstrates the convergence of the method numerically, using both the traditional and isogeometric Taylor-Hood elements.