This paper proposes and analyzes a novel fully discrete finite element scheme with the interpolation operator for stochastic Cahn-Hilliard equations with functional-type noise. The nonlinear term satisfies a one-side Lipschitz condition and the diffusion term is globally Lipschitz continuous. The novelties of this paper are threefold. First, the $L^2$-stability ($L^\infty$ in time) and the discrete $H^2$-stability ($L^2$ in time) are proved for the proposed scheme. The idea is to utilize the special structure of the matrix assembled by the nonlinear term. None of these stability results has been proved for the fully implicit scheme in existing literature due to the difficulty arising from the interaction of the nonlinearity and the multiplicative noise. Second, the higher moment stability in $L^2$-norm of the discrete solution is established based on the previous stability results. Third, the H\"older continuity in time for the strong solution is established under the minimum assumption of the strong solution. Based on these, the discrete $H^{-1}$-norm of the strong convergence is discussed. Several numerical experiments including stability and convergence are also presented to validate our theoretical results.

翻译：本文提出并分析了一种基于插值算子的新型全离散有限元格式，用于求解含函数型噪声的随机Cahn-Hilliard方程。非线性项满足单侧Lipschitz条件，扩散项全局Lipschitz连续。本文的创新性体现在三个方面：首先，证明了所提格式的$L^2$-稳定性（时间$L^\infty$）和离散$H^2$-稳定性（时间$L^2$），其核心思想是利用非线性项组装矩阵的特殊结构。由于非线性项与乘性噪声相互作用的复杂性，现有文献中的全隐式格式尚未证明这些稳定性结果。其次，基于前述稳定性结果，建立了离散解在$L^2$-范数下的高阶矩稳定性。第三，在强解的最小假设条件下，证明了强解在时间方向上的Hölder连续性。基于上述结果，讨论了离散$H^{-1}$-范数下的强收敛性。文中还通过包含稳定性和收敛性在内的多项数值实验验证了理论结果。