  We present a numerical study of the cosmic density vs. velocity divergence
relation (DVDR) in the mildly non-linear regime. We approximate the dark matter
as a non-relativistic pressureless fluid, and solve its equations of motion on
a grid fixed in comoving coordinates. Unlike N-body schemes, this method yields
directly the volume-averaged velocity field. The results of our simulations are
compared with the predictions of the third-order perturbation theory (3PT) for
the DVDR. We investigate both the mean `forward' relation (density in terms of
velocity divergence) and the mean `inverse' relation (velocity divergence in
terms of density), with emphasis on the latter. On scales larger than about 20
megaparsecs, our code recovers the predictions of 3PT remarkably well,
significantly better than recent N-body simulations. On scales of a few
megaparsecs, the DVDR predicted by 3PT differs slightly from the simulated one.
In particular, approximating the inverse DVDR by a third-order polynomial turns
out to be a poor fit. We propose a simple analytical description of the inverse
relation, which works well for mildly non-linear scales.
