We consider the stochastic Cahn-Hilliard equation with additive noise term
$\varepsilon^\gamma g\, \dot{W}$ ($\gamma >0$) that scales with the interfacial
width parameter $\varepsilon$. We verify strong error estimates for a gradient
flow structure-inheriting time-implicit discretization, where
$\varepsilon^{-1}$ only enters polynomially; the proof is based on
higher-moment estimates for iterates, and a (discrete) spectral estimate for
its deterministic counterpart. For $\gamma$ sufficiently large, convergence in
probability of iterates towards the deterministic Hele-Shaw/Mullins-Sekerka
problem in the sharp-interface limit $\varepsilon \rightarrow 0$ is shown.
These convergence results are partly generalized to a fully discrete finite
element based discretization.
We complement the theoretical results by computational studies to provide
practical evidence concerning the effect of noise (depending on its 'strength'
$\gamma$) on the geometric evolution in the sharp-interface limit. For this
purpose we compare the simulations with those from a fully discrete finite
element numerical scheme for the (stochastic) Mullins-Sekerka problem. The
computational results indicate that the limit for $\gamma\geq 1$ is the
deterministic problem, and for $\gamma=0$ we obtain agreement with a (new)
stochastic version of the Mullins-Sekerka problem.