  A lattice boson model is used to study ordering phenomena in regular 2D array
of superconductive mesoscopic granules, Josephson junctions or pores filled
with a superfluid helium. Phase diagram of the system, when quantum
fluctuations of both the phase and local superfluid density are essential, is
analyzed both analytically and by quantum Monte Carlo technique. For the system
of strongly interacting bosons it is found that as the boson density $n_0$ is
increased the boundary of ordered superconducting state shifts to {\it lower
temperatures} and at $n_0 > 8$ approaches its limiting position corresponding
to negligible relative fluctuations of moduli of the order parameter (as in an
array of "macroscopic" granules). In the region of weak quantum fluctuations of
phases mesoscopic phenomena manifest themselves up to $n_0 \sim 10$. The mean
field theory and functional integral $1/n_0$ - expansion results are shown to
agree with that of quantum Monte Carlo calculations of the boson Hubbard model
and its quasiclassical limit, the quantum XY model.
