In this paper, we consider the numerical approximation of a time-fractional stochastic Cahn--Hilliard equation driven by an additive fractionally integrated Gaussian noise. The model involves a Caputo fractional derivative in time of order $\alpha\in(0,1)$ and a fractional time-integral noise of order $\gamma\in[0,1]$. The numerical scheme approximates the model by a piecewise linear finite element method in space and a convolution quadrature in time (for both time-fractional operators), along with the $L^2$-projection for the noise. We carefully investigate the spatially semidiscrete and fully discrete schemes, and obtain strong convergence rates by using clever energy arguments. The temporal H\"older continuity property of the solution played a key role in the error analysis. Unlike the stochastic Allen--Cahn equation, the presence of the unbounded elliptic operator in front of the cubic nonlinearity in the underlying model adds complexity and challenges to the error analysis. To overcome these difficulties, several new techniques and error estimates are developed. The study concludes with numerical examples that validate the theoretical findings.

翻译：本文研究由加性分数阶积分高斯噪声驱动的时间分数阶随机Cahn-Hilliard方程的数值逼近问题。该模型涉及时间阶数为$\alpha\in(0,1)$的Caputo分数阶导数和阶数为$\gamma\in[0,1]$的分数阶时间积分噪声。数值格式采用空间分片线性有限元法和时间卷积求积法（对两个时间分数阶算子）对模型进行逼近，并结合噪声的$L^2$投影。我们仔细研究了空间半离散和全离散格式，并通过巧妙的能量论证获得了强收敛速度。解的时间Hölder连续性在误差分析中起到了关键作用。与随机Allen-Cahn方程不同，底模型中立方非线性项前的无界椭圆算子给误差分析增加了复杂性和挑战。为克服这些困难，本文发展了若干新技巧和误差估计。最后通过数值算例验证了理论结果的正确性。