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

Feng, Xiaobing; Li, Yukun; Zhang, Yi

doi:10.1007/s10915-020-01202-3

Cited by 13 publications

(11 citation statements)

References 57 publications

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“…Recent PDE studies in this direction can be found in [364,142,396,442,441], those stochastic PDEs involve gradient-type multiplicative noise and thus have stronger nonlinearity. Finite element approximations have been carried out in [198] for a stochastic Allen-Cahn equation with gradient-type multiplicative noise and in [200] for a related stochastic Cahn-Hilliard equation with gradient-type multiplicative noise. Not surprisingly, their sharp interface models are expected to be a stochastic mean curvature flow [364] and a stochastic Hele-Shaw model [200,19], respectively.…”

Section: Fluid and Solid Mechanicsmentioning

confidence: 99%

“…Finite element approximations have been carried out in [198] for a stochastic Allen-Cahn equation with gradient-type multiplicative noise and in [200] for a related stochastic Cahn-Hilliard equation with gradient-type multiplicative noise. Not surprisingly, their sharp interface models are expected to be a stochastic mean curvature flow [364] and a stochastic Hele-Shaw model [200,19], respectively. However, their rigorous convergence proofs are still missing although some partial results were reported in [364,19] and positive numerical results were given in [196,198,200].…”

Section: Fluid and Solid Mechanicsmentioning

confidence: 99%

“…Not surprisingly, their sharp interface models are expected to be a stochastic mean curvature flow [364] and a stochastic Hele-Shaw model [200,19], respectively. However, their rigorous convergence proofs are still missing although some partial results were reported in [364,19] and positive numerical results were given in [196,198,200].…”

Section: Applications and Extensionsmentioning

confidence: 99%

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The phase field method for geometric moving interfaces and their numerical approximations

Feng

2020

Handbook of Numerical Analysis

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This chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modeling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state-of-the-art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modeling and their numerical methods in materials science, fluid mechanics, biology and image science. Key words and phrases. Phase field method, geometric law, curvature-driven flow, geometric nonlinear PDEs, finite difference methods, finite element methods, spectral methods, discontinuous Galerkin methods, adaptivity, coarse and fine error estimates, convergence of numerical interfaces, nonlocal and stochastic phase field models, microstructure evolution, biology and image science applications.

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Section: Fluid and Solid Mechanicsmentioning

confidence: 99%

Section: Fluid and Solid Mechanicsmentioning

confidence: 99%

Section: Applications and Extensionsmentioning

confidence: 99%

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The phase field method for geometric moving interfaces and their numerical approximations

Feng

2020

Handbook of Numerical Analysis

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“…The paper [15] presented the analysis of strong convergence rate of an implicit full discretization applied to stochastic Cahn-Hilliard equation with unbounded noise diffusiton in dimension one. The authors of [23] derived strong convergence rates of a fully discrete mixed FEM for the stochastic Cahn-Hilliard equation with gradient-type multiplicative noise, where the noise process is a real-valued Wiener process.…”

Section: Introductionmentioning

confidence: 99%

Strong convergence rates of a fully discrete scheme for the Cahn-Hilliard-Cook equation

Qi¹,

Chen²,

Wang³

2022

Preprint

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The first aim of this paper is to examine existence, uniqueness and regularity for the Cahn-Hilliard-Cook (CHC) equation in space dimension d ≤ 3. By applying a spectral Galerkin method to the infinite dimensional equation, we elaborate the well-posedness and regularity of the finite dimensional approximate problem. The key idea lies in transforming the stochastic problem with additive noise into an equivalent random equation. The regularity of the solution to the equivalent random equation is obtained, in one dimension, with the aid of the Gagliardo-Nirenberg inequality and done in two and three dimensions, by the energy argument. Further, the approximate solution is shown to be strongly convergent to the unique mild solution of the original CHC equation, whose spatio-temporal regularity can be attained by similar arguments. In addition, a fully discrete approximation of such problem is investigated, performed by the spectral Galerkin method in space and the backward Euler method in time. The previously obtained regularity results of the problem help us to identify strong convergence rates of the fully discrete scheme.

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“…The stochastic wave equations with Lipschitz continuous nonlinear drift term and diffusion term were studied using the semi-group approach in [2,17,19,43]. As a comparison, the multiplicative noise is considered in this paper based on the variational approach (see [27,28,29,30]), and the nonlinear drift term is not Lipschitz continuous.…”

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confidence: 99%

Finite Element Approximations of a Class of Nonlinear Stochastic Wave Equation with Multiplicative Noise

Xing

2021

Preprint

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Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite element method for a class of nonlinear stochastic wave equations, where the diffusion term is globally Lipschitz continuous while the drift term is only assumed to satisfy weaker conditions as in [11]. The novelties of this paper are threefold. First, the error estimates cannot not be directly obtained if the numerical scheme in primal form is used. The numerical scheme in mixed form is introduced and several Hölder continuity results of the strong solution are proved, which are used to establish the error estimates in both L 2 norm and energy norms. Second, two types of discretization of the nonlinear term are proposed to establish the L 2 stability and energy stability results of the discrete solutions. These two types of discretization and proper test functions are designed to overcome the challenges arising from the stochastic scaling in time issues and the nonlinear interaction. These stability results play key roles in proving the probability of the set on which the error estimates hold approaches one. Third, higher order moment stability results of the discrete solutions are proved based on an energy argument and the underlying energy decaying property of the method. Numerical experiments are also presented to show the stability results of the discrete solutions and the convergence rates in various norms.

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

Cited by 13 publications

References 57 publications

The phase field method for geometric moving interfaces and their numerical approximations

The phase field method for geometric moving interfaces and their numerical approximations

Strong convergence rates of a fully discrete scheme for the Cahn-Hilliard-Cook equation

Finite Element Approximations of a Class of Nonlinear Stochastic Wave Equation with Multiplicative Noise

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