Optimal Error Estimates of a Discontinuous Galerkin Method for Stochastic Allen-Cahn Equation Driven by Multiplicative Noise

Abstract

In this paper, we develop and analyze an efficient discontinuous Galerkin method for stochastic Allen-Cahn equation driven by multiplicative noise. The proposed method is realized by symmetric interior penalty discontinuous Galerkin finite element method for space domain and implicit Euler method for time domain. Several new estimates and techniques are developed. Under some suitable regularity assumptions, we rigorously establish strong convergence results for the proposed fully discrete numerical scheme and obtain optimal convergence rates in both space and time. Numerical experiments are also carried out to validate our theoretical results and demonstrate the effectiveness of the proposed method.