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 spatiotemporal full discretization are analyzed for the stochastic Allen-Cahn equation driven by additive trace-class noise in space dimension d ≤ 3. The full discretization is realized by combining the standard finite element method with the backward Euler timestepping 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 do appropriate decomposition of the considered error terms and propose a novel approach of error analysis, to successfully recover the expected convergence rates of the numerical schemes. The approach is ingenious and does not rely on high-order spatial regularity properties of the approximation processes. It is shown that the full discrete scheme possesses convergence rates of order h γ in space and order τ γ 2 in time, subject to the spatial correlation of the noise process, characterized by AIn particular, a classical convergence rate of order O(h 2 + τ ) is reachable, even in multiple spatial dimensions, when the aforementioned condition is fulfilled with γ = 2. Numerical examples confirm the previous findings.Here · L 2 stands for the Hilbert-Schmidt norm,Ḣ α := D(A α 2 ), α ∈ R and the parameter γ ∈ [1, 2] coming from (1.2) quantifies the spatial regularity of the covariance operator Q of the

Strong and Weak Convergence Rates of a Spatial Approximation for Stochastic Partial Differential Equation with One-sided Lipschitz Coefficient

“…The stochastic process {W (t)} t≥0 is a generalized Q-Wiener process on a filtered probability space (Ω, F , {F t } t≥0 , P). Under further assumptions on X 0 , Q, f and A β−1 2 L 0 2 < ∞, β ∈ (0, 1], similar arguments in [6,26] yield that there is a unique mild solution X of Eq. (1), which possesses the optimal spatial regularity E X(t) p H β ≤ C(T, Q, X 0 , p), p ≥ 1.…”

Weak convergence and invariant measure of a full discretization for non-globally Lipschitz parabolic SPDE

“…Furthermore, the following strong error estimate holds. Its proof is similar to the proofs of [26,Theorem 4.1] and [13,Theorem 3.1].…”

An adaptive time-stepping full discretization for stochastic Allen--Cahn equation

“…Numerical approximations for stochastic partial differential equations (SPDEs) with globally Lipschitz coefficients have been studied in recent decades (see e.g., [8], [9], [10], [17], [19], [29], [31] and references therein). In contrast, numerical analysis of SPDEs with non-globally Lipschitz coefficients, for example the stochastic Allen-Cahn equation, has been considered (see e.g., [2], [4], [5], [11], [12], [15], [18], [21], [24], [25], [27], [30] and references therein) and is still not fully understood. It is pointed out in [1] that the explicit, the exponential and the linear-implicit Euler-type methods given by the uniform timestep fail to converge for SPDEs with superlinearly growing coefficients.…”