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

## 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 \(\nabla u \in L^1(0,T;L^\infty (\varOmega ))\) 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.

## Keywords

Time-dependent incompressible flow*Re*-semi-robust error estimates Pressure–robustness Inf-sup stable methods Exactly divergence-free FEM

## Mathematics Subject Classification

35Q30 65M15 65M60 76D17 76M10## References

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