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<p>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 for general spatial <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( L^q \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>-norms, by using the discrete deterministic and stochastic maximal <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( L^p \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-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 fully discrete finite element approximations for nonlinear stochastic parabolic equations within the framework of general spatial <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( L^q \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>-norms.</p>

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

  • 期刊论文

  • 发表日期：

    2025年10月16日

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  • Binjie Li,
  • Qin Zhou

摘要

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 for general spatial \( L^q \) L q -norms, by using the discrete deterministic and stochastic maximal \( L^p \) 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 fully discrete finite element approximations for nonlinear stochastic parabolic equations within the framework of general spatial \( L^q \) L q -norms.