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 $\alpha -\delta$ in Besov space $\C^{-α}$ for $\alpha \in (0,1)$ and $\delta >0$ arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations of order $\alpha -\delta$ in $\C^{-α}$ for $\alpha \in (0,2/9)$ and $\delta >0$ arbitrarily small.