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This paper studies finite element approximations of the stochastic Allen-Cahnequation with gradient-type multiplicative noises that are white in time andcorrelated in space. The sharp interface limit as the parameter $\epsilon\rightarrow 0$ of the stochastic equation formally approximates a stochasticmean curvature flow which is described by a stochastically perturbed geometriclaw of the deterministic mean curvature flow. Both the stochastic Allen-Cahnequation and the stochastic mean curvature flow arise from materials science,fluid mechanics and cell biology applications. Two fully discrete finiteelement methods which are based on different time-stepping strategies for thenonlinear term are proposed. Strong convergence with sharp rates for both fullydiscrete finite element methods is proved. This is done with a crucial help ofthe H\"{o}lder continuity in time with respect to the spatial $L^2$-norm and$H^1$-seminorm for the strong solution of the stochastic Allen-Cahn equation,which are key technical lemmas proved in paper. It also relies on the fact thathigh moments of the strong solution are bounded in various spatial and temporalnorms. Numerical experiments are provided to gauge the performance of theproposed fully discrete finite element methods and to study the interplay ofthe geometric evolution and gradient-type noises.

Finite Element Methods for the Stochastic Allen-Cahn Equation with Gradient-type Multiplicative Noises