Pathwise uniform convergence of a full discretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise

2024/05/05 by Li, Binjie, Zhou, Qin
#35R60 #60H15 #60H35 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2405.03016

Abstract

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.

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