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

This work is devoted to the finite element discretization of the incompressible Navier--Stokes equations. The starting point is a low order stabilized finite element method using piecewise linear continuous discrete velocities and piecewise constant pressures. This pair of spaces needs to be stabilized, and, as such, the continuity equation is modified by adding a stabilizing 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 constants independent of the viscosity are proven. Numerous numerical experiments confirm the theoretical results.