To capture and simulate geometric surface evolutions, one effective approach is based on the phase field methods. Among them, it is important to design and analyze numerical approximations whose error bound depends on the inverse of the diffuse interface thickness (denoted by $\frac 1\epsilon$) polynomially. However, it has been a long-standing problem whether such numerical error bound exists for stochastic phase field equations. In this paper, we utilize the regularization effect of noise to show that near sharp interface limit, there always exists the weak error bound of numerical approximations, which depends on $\frac 1\epsilon$ at most polynomially. To illustrate our strategy, we propose a polynomial taming fully discrete scheme and present novel numerical error bounds under various metrics. Our method of proof could be also extended to a number of other fully numerical approximations for semilinear stochastic partial differential equations (SPDEs).

翻译：为捕捉和模拟几何曲面演化，一种有效方法是基于相场方法。其中，设计和分析数值逼近方法至关重要，其误差界对扩散界面厚度的倒数（记为$\frac 1\epsilon$）呈多项式依赖。然而，对于随机相场方程，此类数值误差界是否存在一直是一个长期悬而未决的问题。本文利用噪声的正则化效应证明：在近尖锐界面极限下，数值逼近的弱误差界始终存在，且对$\frac 1\epsilon$至多为多项式依赖。为阐明这一策略，我们提出了一种多项式驯化全离散格式，并在多种度量下给出了新的数值误差界。本文的证明方法还可推广至若干其他半线性随机偏微分方程的全数值逼近。