On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation

Kovács, Mihály; Larsson, Stig; Lindgren, Fredrik

doi:10.1239/jap/1437658601

Cited by 34 publications

(12 citation statements)

References 24 publications

(29 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. The proof is completely analogous to the proofs of [20, Proposition 5.1] and [23, Theorem 2.1], based on a discrete factorization method using the analyticity of the semigroup E and the deterministic error estimate [20] considers the stochastic Allen-Cahn equation where the semigroup in the stochastic convolution is generated by the Laplacian ∆ which has a weaker smoothing effect than in the present case, where the semigroup is generated by −∆ 2 . The proofs in [20] and [23] require that p is large, but the result is then valid for smaller p ≥ 1 as well.…”

Section: Convergencementioning

confidence: 77%

Strong Convergence of a Fully Discrete Finite Element Approximation of the Stochastic Cahn--Hilliard Equation

Furihata¹,

Kovács²,

Larsson³

et al. 2018

SIAM J. Numer. Anal.

Self Cite

View full text Add to dashboard Cite

We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension d ≤ 3. We discretize the equation using a standard finite element method in space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.2000 Mathematics Subject Classification. 60H15, 60H35, 65C30.

show abstract

Section: Convergencementioning

confidence: 77%

Strong Convergence of a Fully Discrete Finite Element Approximation of the Stochastic Cahn--Hilliard Equation

Furihata¹,

Kovács²,

Larsson³

et al. 2018

SIAM J. Numer. Anal.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The case of the Allen-Cahn equation has been treated in the recent work [3], with a scheme based on an exponential integrator and a tamed discretization of the nonlinear coefficient, see also the recent preprint [61]. References [48,49] present analysis of implicit schemes. Finally, a Wong-Zakai approximation has been considered in [52].…”

Section: Charles-edouard Bréhier and Ludovic Goudenègementioning

confidence: 99%

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

Bréhier¹,

Goudenège²

2019

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution. We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, L 2 (Ω)-convergence of order almost 1/4, localized on an event of arbitrarily large probability, then convergence in probability of order almost 1/4. The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.

show abstract

“…The existence of solutions to and their regularity has been studied in, for example, (variational solution). We summarise some known results in the following theorem and we also refer to the discussion preceding and after [, Proposition 3.1] for more details. To briefly describe the notion of a variational solution we consider the mapping

\tilde{A} : V \to V^{'}

defined by

〈\overset{A}{true} u, v〉 = - false⟨ \nabla u, \nabla v false⟩ - false⟨ f (u), v false⟩, u, v \in V .

A continuous, H ‐valued,

F_{t}

‐adapted process

X = {X false(t false)}_{t \in [0, T]}

is called a variational solution of , if for its

d t \otimes P

equivalence class

\overset{X}{true}

we have

\hat{X} \in L^{α} false([0, T] \times normalΩ, normald t \otimes double-struckP; V false)

, for some

α \geq 2

, and

0true X (t) = X_{0} + \int_{0}^{t} \overset{A}{true} \overset{X}{true} (s) normald s + W (t), t \in [0, T], double-struckP - as.

Theorem If

∥ A^{\frac{1}{2}}

…”

Section: Preliminariesmentioning

confidence: 99%

“…The smoothing of the operators allows us to use Itô's formula, paving the way for a situation where the one‐sided Lipschitz condition helps us to prove moment bounds and establish the correct convergence rate. In contrast to, for example, , we thus avoid the use of the mild form of , which only allows for a proof of the mere fact of strong convergence without rate. How the symmetry comes into play will be clear from the proof of Item (3) of Lemma .…”

Section: Introductionmentioning

confidence: 99%

“…In contrast, there are only few results on strong convergence of time discretisation schemes for SPDEs with superlinearly growing coefficients. Furthermore, almost all of these results are establishing the fact of strong convergence with no rate given . Only in a very recent paper do the authors obtain strong rates, by employing an exponential integrator scheme for a class of SPDEs without a linear growth condition on the coefficients.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the discretisation in time of the stochastic Allen–Cahn equation

Kovács

Larsson

Lindgren

2018

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

Abstract. We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a bounded spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretisation in time of the equation by an Euler type split-step method with step size k > 0. We show that the method converges strongly with a rate O(k 1 2 ). By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.

show abstract

On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation

Cited by 34 publications

References 24 publications

Strong Convergence of a Fully Discrete Finite Element Approximation of the Stochastic Cahn--Hilliard Equation

Strong Convergence of a Fully Discrete Finite Element Approximation of the Stochastic Cahn--Hilliard Equation

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

On the discretisation in time of the stochastic Allen–Cahn equation

Contact Info

Product

Resources

About