  We introduce a family of bi-grid schemes in finite elements for solving 2D
incompressible Navier-Stokes equations in velocity and pressure $(u,p)$. The
new schemes are based on projection methods and use two pairs of FEM spaces, a
sparse and a fine one. The main computational effort is done on the coarsest
velocity space with an implicit and unconditionally time scheme while its
correction on the finer velocity space is realized with a simple stabilized
semi-implicit scheme whose the lack of stability is compensated by a high mode
stabilization procedure; the pressure is updated using the free divergence
property. The new schemes are tested on the lid driven cavity up to $Re=7500$.
An enhanced stability is observed as respect to classical semi-implicit methods
and an important gain of CPU time is obtained as compared to implicit
projection schemes.
