  We revisit the periodic homogenization of Dirichlet problems for the Laplace
operator in perforated domains, and establish a unified proof that works for
different regimes of hole-cell ratios, that is the ratio between the scaling
factor of the holes and that of the periodic cells. The approach is then made
quantitative and it yields correctors and error estimates for vanishing
hole-cell ratios. For positive volume fraction of holes, the approach is just
the standard oscillating test function method; for vanishing volume fraction of
holes, we study asymptotic behaviors of a properly rescaled cell problems and
use them to build oscillating test functions. Our method reveals how the
different regimes are intrinsically connected through the cell problems and the
connection with periodic layer potentials.
