SeMA Journal

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

  • Philipp W. Schroeder
  • Christoph Lehrenfeld
  • Alexander Linke
  • Gert Lube


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.


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 


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Copyright information

© Sociedad Española de Matemática Aplicada 2018

Authors and Affiliations

  1. 1.Institute for Numerical and Applied MathematicsGeorg-August-University GöttingenGöttingenGermany
  2. 2.Weierstrass InstituteBerlinGermany

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