This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization combines the Euler scheme for temporal approximation and the finite element method for spatial approximation. A pathwise uniform convergence rate is derived, encompassing general spatial \( L^q \)-norms, by using discrete versions of deterministic and stochastic maximal \( L^p \)-regularity estimates. Additionally, the theoretical convergence rate is validated through numerical experiments. The primary contribution of this work is the introduction of a technique to establish the pathwise uniform convergence of finite element-based full discretizations for nonlinear stochastic parabolic equations within the framework of general spatial \( L^q \)-norms.

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。该离散格式结合了时间近似中的欧拉格式与空间近似中的有限元方法。通过使用确定性和随机极大\( L^p \)-正则性估计的离散形式，推导出了包含一般空间\( L^q \)-范数在内的路径一致收敛速率。此外，通过数值实验验证了理论收敛速率。本研究的主要贡献在于提出了一种技术，用于在一般空间\( L^q \)-范数框架下，建立基于有限元的非线性随机抛物型方程全离散格式的路径一致收敛性。