Pressure and convection robust bounds for continuous interior penalty divergence-free finite element methods for the incompressible Navier-Stokes equations

In this paper we analyze a pressure-robust method based on divergence-free mixed finite element methods with continuous interior penalty stabilization. The main goal is to prove an $O(h^{k+1\mathrm{/}2})$ error estimate for the $L^2$ norm of the velocity in the convection dominated regime. This bound is pressure robust (the error bound of the velocity does not depend on the pressure) and also convection robust (the constants in the error bounds are independent of the Reynolds number).