  We consider the two-dimensional Cahn-Hilliard equation with logarithmic
potentials and periodic boundary conditions. We employ the standard
semi-implicit numerical scheme which treats
  the linear fourth-order dissipation term implicitly and the nonlinear term
explicitly. Under natural
  constraints on the time step we prove strict phase separation and energy
stability of the semi-implicit scheme. This appears to be the first rigorous
result for the semi-implicit discretization
  of the Cahn-Hilliard equation with singular potentials.
