A Finite Element Method via Noise Regularization for the Stochastic Allen-cahn Problem

We study finite element approximations of stochastic partial differential equations of Ginzburg-Landau type and the main paradigm considered in this paper is the stochastic Allen-Cahn model. We first demonstrate that the constructed stochastic finite element approximations are within an arbitrary level of tolerance from the corresponding one-dimensional stochastic partial differential equation; secondly we show that the finite element approximation is close to the most probable deterministic trajectory of the stochastic Allen-Cahn equation, even in large time intervals where interfaces form and evolve according to macroscopic mean curvature-dependent evolutions.