  We perform an extensive suite of N-body simulations of the matter power
spectrum, incorporating massive neutrinos in the range M = 0.15-0.6 eV, probing
the non-linear regime at scales k < 10 hMpc-1 at z < 3. We extend the widely
used HALOFIT approximation to account for the effect of massive neutrinos on
the power spectrum. In the strongly non-linear regime HALOFIT systematically
over-predicts the suppression due to the free-streaming of the neutrinos. The
maximal discrepancy occurs at k ~ 1 hMpc-1, and is at the level of 10% of the
total suppression. Most published constraints on neutrino masses based on
HALOFIT are not affected, as they rely on data probing the matter power
spectrum in the linear or mildly non-linear regime. However, predictions for
future galaxy, Lyman-alpha forest and weak lensing surveys extending to more
non-linear scales will benefit from the improved approximation to the
non-linear matter power spectrum we provide. Our approximation reproduces the
induced neutrino suppression over the targeted scales and redshifts
significantly better. We test its robustness with regard to changing
cosmological parameters and a variety of modelling effects.
