Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noise

TL;DR

This paper develops a novel error analysis approach for finite element methods solving the stochastic Allen-Cahn equation with additive noise, achieving optimal convergence rates without high regularity assumptions.

Contribution

It introduces an original error decomposition method that handles non-globally Lipschitz nonlinearities, establishing convergence rates for semi-discrete and fully discrete schemes in multiple dimensions.

Findings

Convergence rate of order O(h^γ) in space and O(τ^{γ/2}) in time under noise regularity conditions.

Classical convergence rate of O(h^2 + τ) achieved when noise regularity condition with γ=2 holds.

Numerical experiments validate the theoretical error estimates.

Abstract

Strong approximation errors of both finite element semi-discretization and spatio-temporal full discretization are analyzed for the stochastic Allen-Cahn equation driven by additive noise in space dimension $d \leq 3$. The full discretization is realized by combining the standard finite element method with the backward Euler time-stepping scheme. Distinct from the globally Lipschitz setting, the error analysis becomes rather challenging and demanding, due to the presence of the cubic nonlinearity in the underlying model. By introducing two auxiliary approximation processes, we propose an appropriate decomposition of the considered error terms and introduce a novel approach of error analysis, to successfully recover the convergence rates of the numerical schemes. The approach is original and does not rely on high-order spatial regularity properties of the approximation processes. It is shown that the fully discrete scheme possesses convergence rates of order $ O(h^{\gamma} ) $ in space and order $ O( \tau^{ \frac{\gamma}{2} } )$ in time, subject to the spatial correlation of the noise process, characterized by $ \|A^{\frac{\gamma-1}2}Q^{\frac12}\|_{\mathcal{L}_2}<\infty, \, \gamma \in[\frac d3,2] $, $ d\in\{1,2,3\}$. In particular, a classical convergence rate of order $O(h^2 +\tau)$ is reachable, even in multiple space dimensions, when the aforementioned condition is fulfilled with $ \gamma = 2 $. Numerical examples confirm the previous findings.