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Space–time discontinuous Galerkin methods for the ε-dependent stochastic Allen–Cahn equation with mild noise
Antonopoulou, Dimitra
Antonopoulou, Dimitra
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2019-04-08
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IMA_Antonopoulou.pdf
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Abstract
We consider the $\eps$-dependent stochastic Allen-Cahn equation with mild space-time noise posed on a bounded domain of R^2. The positive parameter $\eps$ is a measure for the inner layers width that are generated during evolution. This equation, when the noise depends only on time, has been proposed by Funaki in [15]. The noise although smooth becomes white on the sharp interface limit as $\eps$ tends to zero. We construct a nonlinear dG scheme with space-time finite elements of general type which are discontinuous in time. Existence of a unique discrete solution is proven by application of Brouwer's Theorem. We first derive abstract error estimates and then for the case of piece-wise polynomial finite elements we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter $\eps$. Finally, we present a linear approximation of the nonlinear scheme for which we prove existence of solution and optimal error in expectation in piece-wise linear finite element spaces. The novelty of this work is based on the use of a finite element formulation in space and
in time in 2+1-dimensional subdomains for a nonlinear parabolic problem. In addition, this problem involves noise. These type of schemes avoid any Runge-Kutta type discretization for the evolutionary variable and seem to be very effective when applied to equations of such a difficulty.
Citation
Antonopoulou, D. (2019). Space–time discontinuous Galerkin methods for the ε-dependent stochastic Allen–Cahn equation with mild noise. IMA Journal of Numerical Analysis, 40(3), 2076–2105. https://doi.org/10.1093/imanum/drz019
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Oxford University Press
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IMA Journal of Numerical Analysis
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Article
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en
Description
This is a pre-copyedited, author-produced PDF of an article accepted for publication in IMA Journal of Numerical Analysis following peer review. The version of record Antonopoulou, D. (2019-forthcoming). Space-Time Discontinuous Galerkin Methods for the '\eps'-dependent Stochastic Allen-Cahn Equation with mild noise. IMA Journal of Numerical Analysis is available online at: https://doi.org/10.1093/imanum/drz019
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ISSN
0272-4979
EISSN
1464-3642