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

Regularization effect of noise on fully discrete approximation for stochastic reaction-diffusion equation near sharp interface limit