Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier-Stokes flows

TL;DR

This paper presents a divergence-free finite element method for the time-dependent incompressible Navier-Stokes equations, providing error estimates independent of pressure and Reynolds number, with numerical validation.

Contribution

It introduces a pressure-robust analysis for conforming FEM with divergence-free approximations, including optional edge stabilization for convection-dominated flows.

Findings

Error estimates independent of pressure and Reynolds number

Numerical simulations confirm theoretical predictions

Edge stabilization improves convection-dominated flow accuracy

Abstract

This article focusses on the analysis of a conforming finite element method for the time-dependent incompressible Navier-Stokes equations. For divergence-free approximations, in a semi-discrete formulation, we prove error estimates for the velocity that hold independently of both pressure and Reynolds number. Here, a key aspect is the use of the discrete Stokes projection for the error splitting. Optionally, edge-stabilisation can be included in the case of dominant convection. Emphasising the importance of conservation properties, the theoretical results are complemented with numerical simulations of vortex dynamics and laminar boundary layer flows.