Convergence Rate for Galerkin Approximation of the Stochastic Allen—Cahn Equations on 2D Torus

In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations driven by space-time white noise on T². First we prove that the convergence rate for stochastic 2D heat equation is of order α — δ in Besov space C^−α for α ∈ (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 ∈ (0,2/9) and δ > 0 arbitrarily small.