Absolute continuity and numerical approximation of stochastic Cahn--Hilliard equation with unbounded noise diffusion

In this article, we develop and analyze a full discretization, based on the spatial spectral Galerkin method and the temporal drift implicit Euler scheme, for the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise. By introducing an appropriate decomposition of the nu...

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Bibliographic Details
Published inarXiv.org
Main Authors Cui, Jianbo, Hong, Jialin
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 19.06.2020
Subjects
Convergence
Discretization
Galerkin method
Interpolation
Mathematical analysis
Noise
Numerical methods
Relativity
Sobolev space
Spacetime
White noise
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Summary:In this article, we develop and analyze a full discretization, based on the spatial spectral Galerkin method and the temporal drift implicit Euler scheme, for the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise. By introducing an appropriate decomposition of the numerical approximation, we first use the factorization method to deduce the a priori estimate and regularity estimate of the proposed full discretization. With the help of the variation approach, we then obtain the sharp spatial and temporal convergence rate in negative Sobolev space in mean square sense. Furthermore, the sharp mean square convergence rates in both time and space are derived via the Sobolev interpolation inequality and semigroup theory. To the best of our knowledge, this is the first result on the convergence rate of temporally and fully discrete numerical methods for the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise.
ISSN:2331-8422