Optimal Strong Rates of Convergence for a Space-Time Discretization of
the Stochastic Allen–Cahn Equation with Multiplicative Noise

Majee, Ananta K.; Prohl, Andreas

doi:10.1515/cmam-2017-0023

Cited by 41 publications

(36 citation statements)

References 10 publications

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“…As a typical example of parabolic SPDEs with non-globally Lipschitz nonlinearity, stochastic Allen-Cahn equations, perturbed by additive or multiplicative noises, have received increasing attention in the last few years. Recently, several research works were reported on numerical approximations of such equations [3,5,10,17,18,23,24,26,27,30]. The present work makes further contributions in this direction, by successfully recovering optimal strong convergence rates for finite element semi-discretization and spatio-temporal full discretization of stochastic Allen-Cahn equations with additive trace-class noise.…”

Section: Introductionmentioning

confidence: 79%

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

Wang

2019

J Sci Comput

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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

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Section: Introductionmentioning

confidence: 79%

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

Wang

2019

J Sci Comput

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Chen,

Dang,

Hong

2021

Preprint

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It is known in [1] that a regular explicit Euler-type scheme with a uniform timestep, though computationally efficient, may diverge for the stochastic Allen-Cahn equation. To overcome the divergence, this paper proposes an adaptive time-stepping full discretization, whose spatial discretization is based on the spectral Galerkin method, and temporal direction is the adaptive exponential integrator scheme. It is proved that the expected number of timesteps is finite if the adaptive timestep function is bounded suitably. Based on the stability analysis in C(O, R)-norm of the numerical solution, it is shown that the strong convergence order of this adaptive time-stepping full discretization is the same as usual, i.e., order β in space and β 2 in time under the correlation assumptionon the noise. Numerical experiments are presented to support the theoretical results.

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“…Our main motivation for the Itô-Alekseev-Gröbner formula are strong convergence rates for time-discrete numerical approximations of SEEs. In the literature, positive strong convergence rates have been established for SEEs with monotone nonlinearities; see, e.g., [12,26,23,4,3,6,5,33,40] for the case of additive noise and [34,32] for the case of multiplicative noise. To the best of our knowledge, strong convergence rates for time-discrete approximations of SEEs with non-monotone superlinearly growing nonlinearities remain an open problem.…”

Section: Introductionmentioning

confidence: 99%

On the Itô-Alekseev-Gröbner formula for stochastic differential equations

Hudde,

Hutzenthaler,

Jentzen

et al. 2018

Preprint

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In this article we establish a new formula for the difference of a test function of the solution of a stochastic differential equation and of the test function of an Itô process. The introduced formula essentially generalizes both the classical Alekseev-Gröbner formula from the literature on deterministic differential equations as well as the classical Itô formula from stochastic analysis. The proposed Itô-Alekseev-Gröbner formula is a powerful tool for deriving strong approximation rates for perturbations and approximations of stochastic ordinary and partial differential equations.

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Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen–Cahn Equation with Multiplicative Noise

Cited by 41 publications

References 10 publications

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

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

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

On the Itô-Alekseev-Gröbner formula for stochastic differential equations

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