Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Allen–Cahn Equation with Additive Noise

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 $$d \\le 3$$d≤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 $$ O(h^{\\gamma }) $$O(hγ) in space and order $$ O( \\tau ^{ \\frac{\\gamma }{2} } )$$O(τγ2) in time, subject to the spatial correlation of the noise process, characterized by $$ \\Vert A^{\\frac{\\gamma -1}{2}}Q^{\\frac{1}{2}}\\Vert _{\\mathscr {L}_2}<\\infty , \\, \\gamma \\in [\\frac{d}{3},2] $$‖Aγ-12Q12‖L2<∞,γ∈[d3,2], $$d\\in \\{1,2,3\\}$$d∈{1,2,3}. In particular, a classical convergence rate of order $$O(h^2 +\\tau )$$O(h2+τ) is reachable, even in multiple space dimensions, when the aforementioned condition is fulfilled with $$ \\gamma = 2 $$γ=2. Numerical examples confirm the previous findings.