Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus

Abstract

In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations driven by space-time white noise on $\T$. First we prove that the convergence rate for stochastic 2D heat equation is of order $α-δ$ in Besov space $\C^{-α}$ for $α\in(0,1)$ and $δ>0$ arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations of order $α-δ$ in $\C^{-α}$ for $α\in(0,2/9)$ and $δ>0$ arbitrarily small.