10th International Workshop on Variational Multiscale and Stabilized Finite Elements (VMS2015)

for 10th International Workshop on Variational Multiscale and Stabilized Finite Elements (VMS2015) Some open problems of inf-sup stable FEM for incompressible flow problems G. Lube∗ Georg-August University Göttingen, Institute for Numerical and Applied Mathematics [email protected] In this talk, I will address some open problems occuring in the numerical approximation of incompressible flow problems using inf-sup stable FEM, in particular in case of large Reynolds numbers. In particular, I will discuss the following topics: • The numerical analysis of incompressible flow problems with no-slip boundary conditions is not convincing for applications. An improvement is possible with mixed boundary conditions including the case of directional do-nothing conditions recently suggested by Braack and Mucha in [1] (as opposed to ”classical” do-nothing conditions). • The theoretical foundation of grad-div stabilization (eventually as pressure subgrid model) for the improvement of local mass conservation is not convincing so far. As a remedy, the approach of A. Linke et al. in [2] to exactly divergence-free and inf-sup stable FEM for the Stokes problem deserves an extension to the time-dependent Navier-Stokes problem. • The theoretical foundation of appropriate velocity subgrid models for high Reynolds number flows is not convincing. We briefly consider the case of local projection stabilization of the velocity gradient. Are such subgrid models appropriate as implicit LES turbulence model ? • Realistic Gronwall constants in semidiscrete error estimates of the time-dependent Navier-Stokes problem are required. Again we consider the case of local projection stabilization for the velocity. • What can be recommended for practice: inf-sup stable or equal-order interpolation of velocitypressure ?