  In this paper, we study the asymptotic behavior of a semi-linear slow-fast
stochastic partial differential equation with singular coefficients. Using the
Poisson equation in Hilbert space, we first establish the strong convergence in
the averaging principe, which can be viewed as a functional law of large
numbers. Then we study the stochastic fluctuations between the original system
and its averaged equation. We show that the normalized difference converges
weakly to an Ornstein-Uhlenbeck type process, which can be viewed as a
functional central limit theorem. Furthermore, rates of convergence both for
the strong convergence and the normal deviation are obtained, and these
convergence are shown not to depend on the regularity of the coefficients in
the equation for the fast variable, which coincides with the intuition, since
in the limit systems the fast component has been totally averaged or
homogenized out.
