In this talk I will review some recent results [1,2] on the stabilisation of linearised incompressible inviscid 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-posedness of the partial differential equation itself. From a numerical methods’ perspective, the common point of the two worksis the aim of proving the following type of estimate:

\[\label{main} \|\boldsymbol{u}-\boldsymbol{u}_h^{}\|_{L^2}^{}\le C\,h^{k+\frac{1}{2}}|\boldsymbol{u}|_{H^{k+1}}^{},\]

where \(\boldsymbol{u}\) is the exact velocity and \(\boldsymbol{u}_h^{}\) 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 mimics 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 (very) regular pressure.

I will first present results of a discretisation using \(H(div)\)-conforming spaces, such as Raviart-Thomas, or Brezzi-Douglas-Marini, where an estimate of the type [main] 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, dueto 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 [main]. Finally, some numerical results will be presented and the present approach will be compared to the classical residual-based SUPG stabilisation.

References :

Barrenechea, G.R., Burman, E., and Guzmán, J.: Well-posedness and H(div)-conforming finite element approximation of a linearised model for inviscid incompressible flow. Mathematical Models and Methods in Applied Sciences (M3AS), 30(5), 847-865, (2020).

Ahmed, N., Barrenechea, G.R., Burman, E., Guzmán, J., Linke, A., and Merdon, C. A pressure-robust discretization of Oseen’s equation using stabilization in the vorticity equation. SIAM Journal on Numerical Analysis, to appear.