This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization uses the Euler scheme for temporal discretization and the finite element method for spatial discretization. A key contribution of this work is the introduction of a novel stability estimate for a discrete stochastic convolution, which plays a crucial role in establishing pathwise uniform convergence estimates for fully discrete approximations of nonlinear stochastic parabolic equations. By using this stability estimate in conjunction with the discrete stochastic maximal $L^p$-regularity estimate, the study derives a pathwise uniform convergence rate that encompasses general general spatial $L^q$-norms. Moreover, the theoretical convergence rate is verified by numerical experiments.

翻译：本文分析了一个带有乘性噪声的三维随机Allen-Cahn方程的全离散格式。该离散格式采用欧拉格式进行时间离散，并利用有限元方法进行空间离散。本工作的一个关键贡献是提出了一种离散随机卷积的新型稳定性估计，该估计在建立非线性随机抛物型方程全离散逼近的路径一致收敛估计中起着至关重要的作用。通过将该稳定性估计与离散随机极大值$L^p$-正则性估计结合使用，本研究推导出了一个覆盖一般空间$L^q$-范数的路径一致收敛速率。此外，理论收敛速率通过数值实验得到了验证。