  We numerically model fragmentation of a gravitationally unstable gaseous disc
under conditions that may be appropriate for the formation of the young massive
stars observed in the central parsec of our Galaxy. In this study, we adopt a
simple prescription with a locally constant cooling time. We find that, for
cooling times just short enough to induce disc fragmentation, stars form with a
top-heavy Initial Mass Function (IMF), as observed in the Galactic Centre (GC).
For shorter cooling times, the disc fragments much more vigorously, leading to
lower average stellar masses. Thermal feedback associated with gas accretion
onto protostars slows down disc fragmentation, as predicted by some analytical
models. We also simulate the fragmentation of a gas stream on an eccentric
orbit in a combined Sgr A* plus stellar cusp gravitational potential. The
stream precesses, self-collides and forms stars with a top-heavy IMF. None of
our models produces large enough co-moving groups of stars that could account
for the observed ``mini star cluster'' IRS13E in the GC. In all of the
gravitationally unstable disc models that we explored, star formation takes
place too fast to allow any gas accretion onto the central super-massive black
hole. While this can help to explain the quiescence of `failed AGN' such as Sgr
A*, it poses a challenge for understanding the high gas accretion rates infered
for many quasars.
