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arXiv · 2405.03016

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

Binjie Li·Qin Zhou
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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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Binjie Li, Qin Zhou. 2024-05-05. Pathwise uniform convergence of a full discretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. https://arxiv.org/abs/2405.03016

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