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 $\\C^{-\\alpha}$ 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 $\\C^{-\\alpha}$ for $\\alpha\\in(0,2/9)$ and $\\delta>0$ arbitrarily small.