  This work proposes a new stabilized $P_1\times P_0$ finite element method for
solving the incompressible Navier--Stokes equations. The numerical scheme is
based on a reduced Bernardi--Raugel element with statically condensed face
bubbles and is pressure-robust in the small viscosity regime. For the Stokes
problem, an error estimate uniform with respect to the kinematic viscosity is
shown. For the Navier--Stokes equation, the nonlinear convection term is
discretized using an edge-averaged finite element method. In comparison with
classical schemes, the proposed method does not require tuning of parameters
and is validated for competitiveness on several benchmark problems in 2 and 3
dimensional space.
