Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy

Copetti, M. I. M.; Elliott, Charles M.

doi:10.1007/bf01385847

Cited by 221 publications

(189 citation statements)

References 18 publications

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“…The data used in each experiment on the coarse meshes were Ω = (0, 1), γ = 1.5 × 10 −3 , θ = 0.3, θ c = 1.0, T = 0.4, ∆t = 0.32h, h = 1/(J − 1), where J = 2 k + 1 (k = 6, 7, 8, 9), b max = 1, tol = 1 × 10 −7 and µ = 0.1. The last two quantities were parameters used to vary the degree and speed of convergence in the iterative method (method II of [10]) to solve for U n at each time level in (P h,∆t ). The data were the same for the fine mesh except J = 2 12 + 1.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility

1999

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Abstract. We consider the Cahn-Hilliard equation with a logarithmic free energy and non-degenerate concentration dependent mobility. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally some numerical experiments are presented.

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Section: Numerical Experimentsmentioning

confidence: 99%

“…However, this approximation is invalid if the quench is deep, i.e., θ θ c . For a fuller discussion of the model, see [10] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility

1999

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“…We like to remark that nonsmooth potentials have also been considered in the literature for the Cahn-Hilliard equation, for that we refer to [23,19,5,6] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Error analysis of a mixed finite element method for the Cahn-Hilliard equation

2004

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Summary.We propose and analyze a semi-discrete and a fully discrete mixed finite element method for the Cahn-Hilliard equation u t + (ε u − ε −1 f (u)) = 0, where ε > 0 is a small parameter. Error estimates which are quasi-optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain. In particular, it is shown that all error bounds depend on 1 ε only in some lower polynomial order for small ε. The cruxes of our analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [2], and Chen [15] to prove a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on a stretched time grid. The ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem as ε → 0 in [29].

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“…On the other hand, the truncated quartic potential 195) and the truncated logarithmic potential, which is defined e.g. in (Copetti and Elliott, 1992), satisfy both (A1) (with a = −∞ and b = ∞) and (A2).…”

Section: Energy-dissipative Time-integration Schemesmentioning

confidence: 99%

Computational Phase‐Field Modeling

Gómez

Zee

2017

Encyclopedia of Computational Mechanics Second Edition

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Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community.

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Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy

Cited by 221 publications

References 18 publications

Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility

Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility

Error analysis of a mixed finite element method for the Cahn-Hilliard equation

Computational Phase‐Field Modeling

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