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In two recent publications [Kov{\'a}cs, Larsson, and Mesforush, SIAM J.Numer. Anal. 49(6), 2407-2429, 2011] and [Furihata, et al., SIAM J. Numer.Anal. 56(2), 708-731, 2018], strong convergence of the semi-discrete and fullydiscrete finite element methods is, respectively, proved for theCahn-Hilliard-Cook (CHC) equation, but without convergence rates revealed. Thepresent work aims to fill the left gap, by recovering strong convergence ratesof (fully discrete) finite element methods for the CHC equation. Moreaccurately, strong convergence rates of a full discretization are obtained,based on Galerkin finite element methods for the spatial discretization and thebackward Euler method for the temporal discretization. It turns out that theconvergence rates heavily depend on the spatial regularity of the noiseprocess. Different from the stochastic Allen-Cahn equation, the presence of theunbounded elliptic operator in front of the cubic nonlinearity in theunderlying model makes the error analysis much more challenging and demanding.To address such difficulties, several new techniques and error estimates aredeveloped. Numerical examples are finally provided to confirm the previousfindings.

Error estimates of semi-discrete and fully discrete finite element methods for the Cahn-Hilliard-Cook equation