In this talk I will review some recent results (detailed in [1, 2]) on the stabilisation of linearised incom-

pressible 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:

|| u − u_h k ||_{L 2} ≤ C h^k+ 2 |u|_H^{k+1},

(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, this is the

first time this type of estimate is obtained in a pressure-robust way.

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 H_1 -conforming divergence-free elements,

with the Scott-Vogelius element as the prime example. In this case, due to the H_1 -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. Guzmán (Brown, USA), and A. Linke and C. Merdon, from WIAS, Berlin.

References

[1] G.R. Barrenechea, E. Burman, and J. Guzmán, Well-posedness and H(div)-conforming fi

nite element approximation of a linearised model for inviscid incompressible flow, Math. Mod.

Meth. Appl. S., 30(5), pp. 847–865, 2020.

[2] N. Ahmed, G.R. Barrenechea, E. Burman, J. Guzmán, A. Linke, and C. Merdon,A pressure-robust

discretization of Oseen’s equation using stabilization in the vorticity equation, SIAM Journal on

Numerical Analysis, 59 (5), pp. 2746–2774, 2021.

The work of GRBwas supported by the Leverhulme Trust through the Research Fellowship No. RF-

2019-510.