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

TL;DR

This paper establishes the convergence rates of Galerkin approximations for stochastic Allen-Cahn equations on a 2D torus driven by space-time white noise, advancing understanding of numerical methods for such stochastic PDEs.

Contribution

It provides the first known convergence rate results for Galerkin approximations of stochastic Allen-Cahn equations in 2D with space-time white noise.

Findings

Convergence rate for stochastic 2D heat equation is of order α−δ in Besov space C^{−α}.

Convergence rate for stochastic Allen-Cahn equations is of order α−δ in C^{−α} for α∈(0,2/9).

Results hold for arbitrarily small δ>0.

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.