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
This article offers sharp spatial and temporal mean-square regularity results for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by infinite dimensional fractional Brownian motion with the Hurst parameter greater than one-half. In addition, mean-square numerical approximation of such problem are investigated, performed by the spectral Galerkin method in space and the linear implicit Euler method in time. The obtained sharp regularity properties of the problems enable us to identify optimal mean-square convergence rates of the full discrete scheme. These theoretical findings are accompanied by several numerical examples.
In this paper, we consider a semi-linear stochastic strongly damped wave equation driven by additive Gaussian noise. Following a semigroup framework, we establish existence, uniqueness and space-time regularity of a mild solution to such equation. Unlike the usual stochastic wave equation without damping, the underlying problem with space-time white noise (Q = I) allows for a mild solution with a positive order of regularity in multiple spatial dimensions. Further, we analyze a spatio-temporal discretization of the problem, performed by a standard finite element method in space and a well-known linear implicit Euler scheme in time. The analysis of the approximation error forces us to significantly enrich existing error estimates of semidiscrete and fully discrete finite element methods for the corresponding linear deterministic equation. The main results show optimal convergence rates in the sense that the orders of convergence in space and in time coincide with the orders of the spatial and temporal regularity of the mild solution, respectively. Numerical examples are finally included to confirm our theoretical findings.< ∞ under the same assumptions [2,23]. This benefits from smoothing effect of the analytic semigroup S(t) generated by the dominant linear operator A. In particular, different from both the stochastic heat equation and the stochastic wave equation, the strongly damped problem driven by space-time white noise (Q = I) allows for a mild solution with a positive order of regularity in multiple spatial dimensions (d > 1). For example, the space-time white noise case when d = 2 admits a mild solution u ∈ L ∞ [0, T ]; L 2 (Ω,Ḣ α ) for any α < 1 (consult Remark 2.5 for more details).As the second contribution of this article, we analyze the mean-square approximation errors caused by finite element spatial semi-discretization and space-time full-discretization of (1.1). More precisely,
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