Erratum: Finite Element Approximation of the Cahn--Hilliard--Cook Equation

Kovács, Mihály; Larsson, Stig; Mesforush, Ali

doi:10.1137/140968161

Cited by 13 publications

(24 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Next we prove that

{\tilde{X}}_{k}

converges uniformly strongly to X but with no specific rate given. The proof is in the same spirit as the proof of [, Theorem 5.4] (see also ). Lemma Let

{\tilde{X}}_{k}

be the solution of and X of .…”

Section: Error Boundsmentioning

confidence: 90%

“…We omit the details of the proof. We also note that the rate of convergence is suboptimal in terms of the regularity of the noise but a sharper (almost order k instead of

k^{1 / 2}

) estimate is not needed for our purposes and would require and extended range for s and r in the deterministic error estimates in Lemma (compare with [, (2.2)]). Lemma Let

0true W_{A} (t) = \int_{0}^{t} E (t - s) normald W (s)

and

0true W_{{scriptA}_{k}} (t) = \int_{0}^{t} E_{k} (t - s) R_{\frac{k}{2}} normald W (s)

and suppose that

∥ A^{1 / 2} Q^{1 / 2} {true∥}_{HS}^{2} < \infty

and

2 p \geq 1

.…”

Section: Error Boundsmentioning

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

“…Next we prove that

{\tilde{X}}_{k}

converges uniformly strongly to X but with no specific rate given. The proof is in the same spirit as the proof of [, Theorem 5.4] (see also ). Lemma Let

{\tilde{X}}_{k}

be the solution of and X of .…”

Section: Error Boundsmentioning

confidence: 90%

“…We omit the details of the proof. We also note that the rate of convergence is suboptimal in terms of the regularity of the noise but a sharper (almost order k instead of

k^{1 / 2}

) estimate is not needed for our purposes and would require and extended range for s and r in the deterministic error estimates in Lemma (compare with [, (2.2)]). Lemma Let

0true W_{A} (t) = \int_{0}^{t} E (t - s) normald W (s)

and

0true W_{{scriptA}_{k}} (t) = \int_{0}^{t} E_{k} (t - s) R_{\frac{k}{2}} normald W (s)

and suppose that

∥ A^{1 / 2} Q^{1 / 2} {true∥}_{HS}^{2} < \infty

and

2 p \geq 1

.…”

Section: Error Boundsmentioning

confidence: 99%

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

show abstract

“…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: 99%

“…There it is required that the operator composition A γ 2 Q 1 2 for γ > 1 is Hilbert-Schmidt, while here we only require this with γ = 1. Therefore, the present work can be viewed as the (non-trivial) extension of [22,23] to a strongly convergent fully discrete scheme, still without a strong rate, but with improvements on the regularity requirement on the noise. Both here and in [22,23] the strategy is based on proving a priori moment bounds with large exponents and in higher order norms using energy arguments and bootstrapping followed by a pathwise Gronwall argument in the mild solution setting.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the present work can be viewed as the (non-trivial) extension of [22,23] to a strongly convergent fully discrete scheme, still without a strong rate, but with improvements on the regularity requirement on the noise. Both here and in [22,23] the strategy is based on proving a priori moment bounds with large exponents and in higher order norms using energy arguments and bootstrapping followed by a pathwise Gronwall argument in the mild solution setting. Finally, in connection to the Cahn-Hilliard-Cook equation we note that in [19,25] the linearized Cahn-Hilliard-Cook equation is treated numerically which requires a significantly simpler analysis.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Strong convergence for an explicit fully‐discrete finite element approximation of the Cahn‐Hillard‐Cook equation with additive noise

Lin

2023

Numerical Methods Partial

View full text Add to dashboard Cite

In this paper, we consider an explicit fully‐discrete approximation of the Cahn–Hilliard–Cook (CHC) equation with additive noise, performed by a standard finite element method in space and a kind of nonlinearity‐tamed Euler scheme in time. The main result in this paper establishes strong convergence rates of the proposed scheme. The key ingredient in the proof of our main result is to employ uniform moment bounds for the numerical approximations. To the best of our knowledge, the main contribution of this work is the first result in the literature which establishes strong convergence for an explicit fully‐discrete finite element approximation of the CHC equation. Finally, numerical results are finally reported to confirm the previous theoretical findings.

show abstract

Erratum: Finite Element Approximation of the Cahn--Hilliard--Cook Equation

Cited by 13 publications

References 3 publications

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

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

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

Strong convergence for an explicit fully‐discrete finite element approximation of the Cahn‐Hillard‐Cook equation with additive noise

Contact Info

Product

Resources

About