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Pathwise uniform convergence of a full discretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noise (2405.03016v5)

Published 5 May 2024 in math.NA and cs.NA

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 ( Lq )-norms, by using discrete versions of deterministic and stochastic maximal ( Lp )-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 ( Lq )-norms.

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