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

• Published in: arXiv.org
• Main Authors: Schroeder, Philipp W, Lehrenfeld, Christoph, Linke, Alexander, Lube, Gert
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 16.04.2018
• Subjects: Best practice; Computational fluid dynamics; Convection; Divergence; Energy dissipation; Estimates; Fluid flow; Galerkin method; Incompressible flow; Mathematics - Mathematical Physics; Mathematics - Numerical Analysis; Navier-Stokes equations; Physics - Computational Physics; Physics - Fluid Dynamics; Physics - Mathematical Physics; Reynolds number; Robustness; Time dependence; Velocity errors
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1709.03063

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.