I am trying to solve the incompressible Navier-Stokes equations by multigrid (FAS) using fractional-step method. At the beginning of simulation everything goes fine, but after few hundreds or few thousands time steps, the simulation diverges. I recognized that the divergence is not free on the coarse grids and it makes a kind of instability in the code. Has anybody any idea, how is it possible to have divergence free on the coarse girds? Is it possible to apply constrains for the velocity and/or pressure correction to get free divergence on the corse grid? I thank you in advance.

From my knowledge, we should use multigrid method only for solving pressure Poisson equation which is a linear equation in Fractional-step method. Therefore you have to use linear multigrid (correction scheme). If you approaching this method I may help you.