A divergence-free stabilised finite element method for the evolutionary Navier-Stokes equations

This work is devoted to the finite element discretisation of the incompressible Navier-Stokes equations. The starting point is a low-order stabilised finite element method using piecewise linear continuous discrete velocities and piecewise constant pressures. This pair of spaces needs to be stabilised, and, as such, the continuity equation is modified by adding a stabilising bilinear form based on the jumps of the pressure. This modified continuity equation can be rewritten in a standard way involving a modified different velocity field, which is as a consequence divergence-free. This modified velocity field is then fed back to the momentum equation making the convective term skew-symmetric. Thus, the discrete problem can be proven stable without the need to rewrite the convective field in its skew-symmetric way. Error estimates with constant independent of the viscosity are proven. Numerous numerical experiments confirm the theoretical results.