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A posteriori error analysis of space-time discontinuous Galerkin methods for the ε-stochastic Allen-Cahn equation
Antonopoulou, Dimitra Egwu, Bernard Yan, Yubin
Antonopoulou, Dimitra
Egwu, Bernard
Yan, Yubin
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2023-08-02
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Abstract
In this work, we apply an \textit{a posteriori} error analysis for the space-time, discontinuous in time, Galerkin scheme which has been proposed in \cite{AIMA} for the $\eps$-dependent stochastic Allen-Cahn equation with mild noise $\dot{W}^\eps$ tending to rough as $\eps\rightarrow 0$. Our results are derived under low regularity since the noise even smooth in space, is assumed only one-time continuously differentiable in time, according to the minimum regularity properties of \cite{Fun99}. We prove \textit{a posteriori} error estimates for the $m$-dimensional problem, $m\leq 4$ for a general class of space-time finite element spaces. The \textit{a posteriori} bound is growing only polynomially in $\eps^{-1}$ if the step length $h$ is bounded by
a positive power of $\eps$. This agrees with the restriction posed so far in the \textit{a priori} error analysis of
continuous finite element schemes for the $\eps$-dependent deterministic Allen-Cahn or deterministic and stochastic
Cahn-Hilliard equation. As an application we examine tensorial elements where the discrete solution is approximated by
polynomial functions of separated space and time variables; the \textit{a posteriori} estimates there involve dimensions, and the space, time discretization parameters. We then consider the special case of the mild noise $\dot{W}^\eps$ as defined in
\cite{weber1} through the convolution of a Gaussian process with a proper mollifying kernel, which is then numerically
constructed. Finally, we provide some useful insights for the numerical algorithm, and present for the first time some numerical experiments of the scheme for both one and two-dimensional problems in various cases of interest, and compare with the deterministic ones.
Citation
Antonopoulou, D., Egwu, B., & Yan, Y. (2023). A posteriori error analysis of space-time discontinuous Galerkin methods for the ε- stochastic Allen-Cahn equation. IMA Journal of Numerical Analysis, 44(3), 1862-1902. https://doi.org/10.1093/imanum/drad052
Publisher
Oxford University Press
Journal
IMA Journal of Numerical Analysis
Research Unit
DOI
10.1093/imanum/drad052
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Article
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This is a pre-copyedited, author-produced version of an article accepted for publication in [IMA Journal of Numerical Analysis] following peer review. The version of record [Antonopoulou, D., Egwu, B., & Yan, Y. (2023). A posteriori error analysis of space-time discontinuous Galerkin methods for the ε- stochastic Allen-Cahn equation. IMA Journal of Numerical Analysis, 44(3), 1862-1902] is available online at: https://academic.oup.com/imajna/article/44/3/1862/7234090
Series/Report no.
ISSN
0272-4979
EISSN
1464-3642