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

This paper studies finite element approximations of the stochastic Allen–Cahn equation with gradient-type multiplicative noise that is white in time and correlated in space. The sharp interface limit—as the diffuse interface thickness vanishes—of the stochastic Allen–Cahn equation is formally a stochastic mean curvature flow which is described by a stochastically perturbed geometric law of the deterministic mean curvature flow. Two fully discrete finite element methods which are based on different time-stepping strategies for the nonlinear term are proposed. Strong convergence with sharp rates for both fully discrete finite element methods is proved. This is done with the crucial help of the Hölder continuity in time with respect to the spatial L2-norm and H1-seminorm for the strong solution of the stochastic Allen–Cahn equation. It also relies on the fact that high moments of the strong solution are bounded in various spatial and temporal norms. Numerical experiments are provided to gauge the performance of the proposed fully discrete finite element methods and to study the interplay of the geometric evolution and gradient-type noise.