In this article, we develop and analyze a full discretization, based on the spatial spectral Galerkin method and the temporal drift implicit Euler scheme, for the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise. By introducing an appropriate decomposition of the numerical approximation, we first use the factorization method to deduce the a priori estimate and regularity estimate of the proposed full discretization. With the help of the variation approach, we then obtain the sharp spatial and temporal convergence rate in negative Sobolev space in mean square sense. Furthermore, the sharp mean square convergence rates in both time and space are derived via the Sobolev interpolation inequality and semigroup theory. To the best of our knowledge, this is the first result on the convergence rate of temporally and fully discrete numerical methods for the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise.
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