In the study of geometric surface evolutions, stochastic reaction-diffusion equation provides a powerful tool for capturing and simulating complex dynamics. A critical challenge in this area is developing numerical approximations that exhibit error bounds with polynomial dependence on $\vv^{-1}$, where the small parameter $\vv>0$ represents the diffuse interface thickness. The existence of such bounds for fully discrete approximations of stochastic reaction-diffusion equations remains unclear in the literature. In this work, we address this challenge by leveraging the asymptotic log-Harnack inequality to overcome the exponential growth of $\vv^{-1}$. Furthermore, we establish the numerical weak error bounds under the truncated Wasserstein distance for the spectral Galerkin method and a fully discrete tamed Euler scheme, with explicit polynomial dependence on $\vv^{-1}$.

翻译：在几何曲面演化的研究中，随机反应-扩散方程为捕捉和模拟复杂动力学提供了有力工具。该领域的一个关键挑战在于构建数值近似方法，使其误差界具有关于 $\vv^{-1}$ 的多项式依赖性，其中小参数 $\vv>0$ 表示扩散界面厚度。现有文献中，关于随机反应-扩散方程全离散近似的此类误差界的存在性尚不明确。本工作通过利用渐近对数哈纳克不等式来克服 $\vv^{-1}$ 的指数增长，从而应对这一挑战。此外，我们针对谱伽辽金方法和全离散驯服欧拉格式，在截断瓦瑟斯坦距离下建立了数值弱误差界，并明确给出了其对 $\vv^{-1}$ 的多项式依赖关系。