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

摘要

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