(abgesagt) Dr. Gabriel Barrenechea: Divergence-free finite element methods for an inviscid flow model

Dr. Gabriel Barrenechea, Univ. Strathclyde, Glasgow, Scotland, UK
Ansprechpartner: PD Dr. Sebastian Franz

*** Der Vortrag muss leider abgesagt werden ***

Abstract: In this talk I will review some recent results on the stabilisation of  linearised incompressible invisid flows (or, with a very small viscosity). The partial differential equation  is a linearised incompressible equation similar to Euler's equation, or Oseen's equation in the vanishing viscosity limit. In the first part of the talk I will present results on the well-posedeness of the partial differential equation itself. From a numerical methods'  perspecitve, the common point of the two works is the aim of proving the following type of estimate:

(1)    ∥u - u_h∥_L^2 ≤ C h^{(k + 1/2)} ∥u∥_H^(k+1)

where u is the exact velocity and u_h is its finite element approximation. In the estimate above, the constant C is independent of the viscosity (if the problem has a viscosity), and, more importantly, independent of the pressure. This estimate mimicks what has been achieved for stabilised methods for the convection-diffusion equation in the past. Nevertheless, up to the best of our knowledge, had only been achieved for Oseen's equation using equal-order elements, and assuming a regular pressure.

I will first present results of discretisations  using H(div)-conforming spaces, such as Raviart-Thomas, or Brezzi-Douglas-Marini where an estimate of the type (1) is proven (besides an optimal estimate for the pressure). In the second part of the talk I will move on to H¹-conforming divergence-free elements, with the Scott-Vogelius element as the prime example. In this case, due to the H¹-conformity, the need of an extra control of the vorticity equation, and some appropriate jumps, appears. So, a new stabilised finite element method adding control on the vorticity equation is proposed. The method is independent of the pressure gradients, which makes it pressure-robust and leads to pressure-independent error estimates such as (1). Finally, some numerical results will be presented and the present approach will be compared to the  classical residual-based SUPG stabilisation.

This work is a collaboration with N. Ahmed (Gulf University for Science and Technology, Kuwait), E. Burman (UCL, UK), J. Guzman (Brown, USA), and A. Linke and C. Merdon, from WIAS, Berlin.