A Fully Discrete Mixed Finite Element Method for the Stochastic Cahn–Hilliard Equation with Gradient-Type Multiplicative Noise

This paper develops and analyzes a fully discrete mixed finite element method for the stochastic Cahn–Hilliard equation with gradient-type multiplicative noise that is white in time and correlated in space. The stochastic Cahn–Hilliard equation is formally derived as a phase field formulation of the...

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Vydáno v:Journal of scientific computing Ročník 83; číslo 1; s. 23
Hlavní autoři: Feng, Xiaobing, Li, Yukun, Zhang, Yi
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.04.2020
Springer Nature B.V
Témata:
Algorithms
Approximation
Computational Mathematics and Numerical Analysis
Continuity (mathematics)
Error analysis
Estimates
Finite element analysis
Finite element method
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Noise
Numerical analysis
Theoretical
Stochastic Cahn–Hilliard equation
Mixed finite element methods
65N15
Strong convergence
Gradient-type multiplicative noise
65N30
Stochastic Hele–Shaw flow
Phase transition
ISSN:0885-7474, 1573-7691
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Popis
Shrnutí:This paper develops and analyzes a fully discrete mixed finite element method for the stochastic Cahn–Hilliard equation with gradient-type multiplicative noise that is white in time and correlated in space. The stochastic Cahn–Hilliard equation is formally derived as a phase field formulation of the stochastically perturbed Hele–Shaw flow. The main result of this paper is to prove strong convergence with optimal rates for the proposed mixed finite element method. To overcome the difficulty caused by the low regularity in time of the solution to the stochastic Cahn–Hilliard equation, the Hölder continuity in time with respect to various norms for the stochastic PDE solution is established, and it plays a crucial role in the error analysis. Numerical experiments are also provided to validate the theoretical results and to study the impact of noise on the Hele–Shaw flow as well as the interplay of the geometric evolution and gradient-type noise.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-020-01202-3