Convergence of the Cahn-Hilliard equation to the Hele-Shaw model

Alikakos, Nicholas D.; Bates, Peter W.; Chen, Xinfu

doi:10.1007/bf00375025

Cited by 331 publications

(564 citation statements)

References 32 publications

Supporting

Mentioning

557

Contrasting

Unclassified

Order By: Relevance

“…with the basic mesh size k 0 , and some positive constants γ,t 0 = O (1). Notice that both meshes require asymptotically the same amount of computational cost (cf.…”

Section: F (S)mentioning

confidence: 99%

“…page 638 of [2] and page 1374 of [15]). It was shown in [1] that the solution to the CahnHilliard problem (1)-(3) indeed satisfies the required profile (cf. Theorems 4.12 and 2.1 of [1]) for sufficiently small ε; the statement of the subsequent proposition then follows from combining these two results.…”

Section: Where U Is the Solution To The Cahn-hilliard Problem (1)-(3)mentioning

confidence: 99%

“…At the end of the first stage, fine-scaled phase regions are formed, which are separated by a thin region, usually considered as a hypersurface called the interface. At the end of the second stage, the solution will generically tend to a stable state, which minimizes the energy functional associated with (1). For more physical background, derivation, and discussion of the Cahn-Hilliard equation and related equations, we refer to [12,7,11,34,35,3] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…In addition to the reason that the Cahn-Hilliard equation is a good model to describe the phase separation and coarsening phenomena in a melted alloy, it has been extensively studied in the past decade due to its connection to an interesting and complicated free boundary problem which is known as the Mullins-Sekerka problem arising from studying solidification/melting of materials of zero specific heat, which is also known as the (two-phase) HeleShaw problem arising from the study of the pressure of immiscible fluids in the air [36,1,16,13,11,33,32]. It was first formally shown by Pego [36] that, as ε 0, the function w := −ε u + ε −1 f (u), known as the chemical potential, tends to a limit, which, together with a free boundary := ∪ 0≤t≤T ( t ×{t}), satisfies the following Hele-Shaw (Mullins-Sekerka) problem:…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

2004

View full text Add to dashboard Cite

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].

show abstract

“…with the basic mesh size k 0 , and some positive constants γ,t 0 = O (1). Notice that both meshes require asymptotically the same amount of computational cost (cf.…”

Section: F (S)mentioning

confidence: 99%

Section: Where U Is the Solution To The Cahn-hilliard Problem (1)-(3)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

2004

View full text Add to dashboard Cite

show abstract

“…For a scalar u, set and W»=±(u 2 -1) 2 , /(u) = W'(u) = 2u(u 2 -1) , Then z f solves (observe that g(<p e ) = g{q(z e /e)) = q'(z ( /e) and 9" = 2(^>V)i…”

Section: Equationsmentioning

confidence: 99%

Convergence of the Phase-Field Equations to the Mullins-Sekerka Problem with Kinetic Undercooling

Soner

1999

Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids

View full text Add to dashboard Cite

I prove that the solutions of the phase field equations, on a subsequence, converge to a weak solution of the Mullins-Sekerka problem with kinetic undercooling. The method is based on energy estimates, a monotonicity formula, and the equipartition of the energy at each time. I also show that the limiting interface is (d -1)-rectifiable for almost all t with a square integrable mean curvature vector.

show abstract

On the convergence of stable phase transitions

Padilla

Tonegawa

1998

Comm. Pure Appl. Math.

View full text Add to dashboard Cite

Convergence of the Cahn-Hilliard equation to the Hele-Shaw model

Cited by 331 publications

References 32 publications

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

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

Convergence of the Phase-Field Equations to the Mullins-Sekerka Problem with Kinetic Undercooling

On the convergence of stable phase transitions

Contact Info

Product

Resources

About