  The effect of non-magnetic impurities on 2D s-wave superconductors is studied
beyond the weak disorder regime. Within the Bogoliubov-de Gennes (BdG)
framework, the local pairing amplitude develops a broad distribution with
significant weight near zero with increasing disorder. Surprisingly, the
density of states continues to show a finite spectral gap. The persistence of
the spectral gap at large disorder is shown to arise from the break up of the
system into superconducting "islands". Superfluid density and off-diagonal
correlations show a substantial reduction at high disorder.A simple analysis of
phase fluctuations in the highly inhomogeneous BdG state is shown to lead to a
transition to a non-superconducting state.
