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

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.