Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier-Stokes equations

TL;DR

This paper develops robust error estimates for inf-sup stable finite element methods applied to time-dependent incompressible Navier-Stokes equations, ensuring pressure-robustness and Reynolds number semi-robustness for various divergence-free FEM schemes.

Contribution

It introduces pressure- and Reynolds number semi-robust error estimates for divergence-free FEM, including upwind stabilized methods for convection-dominated flows.

Findings

Pressure-robust error estimates are established for divergence-free FEM.

Reynolds number semi-robust estimates are derived under certain regularity assumptions.

Upwind stabilization is incorporated for convection-dominated problems.

Abstract

Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, $Re$-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption $\nabla u \in L^1(0,T;L^\infty(\Omega))$ which is discussed in detail. In the sense of best practice, we review and establish pressure- and $Re$-semi-robust estimates for pointwise divergence-free $H^1$-conforming FEM (like Scott-Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free $H$(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.