Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem

Abstract

This paper concerns numerical approximations for the Cahn-Hilliard equation u t + ∆(ε∆u − ε −1 f (u)) = 0 and its sharp interface limit as ε 0, known as the Hele-Shaw problem. The primary goal of this paper is to establish the convergence of the solution of the fully discrete mixed finite element scheme proposed in [29] to the solution of the Hele-Shaw (Mullins-Sekerka) problem, provided that the Hele-Shaw (Mullins-Sekerka) problem has a global (in time) classical solution. This is accomplished by establishing some improved a priori solution and error estimates, in particular, an L ∞ (L ∞) error estimate, and making full use of the convergence result of [2]. The cruxes of the analysis are to establish stability estimates for the discrete solutions, use a spectrum estimate result of Alikakos and Fusco [3] and Chen [15], and establish a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term.

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

“…A direct consequence of (27) are the following stability estimates for the scheme (25)-(26) which hold on both meshes J 1 k and J 2 k . Moreover, additional estimates in strong norms are shown for the stretched mesh, which indicates its stabilizing effect.…”

“…On the other hand, we show in the following that the estimates will improve drastically if the initial data u 0 ∈ H 3 ( ) and the boundary ∂ ∈ C 2,1 are considered. Alternatively, the subsequent results also hold for convex polygonal domains in the case N = 2; see [27] for a short proof.…”

“…For verifications of the estimates (i)-(v), (viii), (ix), see [27]. Here, we only give proofs of (vi), (vii) on the time-mesh J 2 k .…”

“…This paper is a condensed version of [27], where one can found more details and helpful discussions which could not be included here due to page limitation.…”

Finite Element Approximation of a Three Dimensional Phase Field Model for Void Electromigration

“…Of course, given that (P γ ) is a phase field model for the original sharp interface problem (1.1a,b) and (1.2), the ultimate goal would be to show convergence of the discrete solutions to the sharp interface solutions as γ, h → 0. To our knowledge, the only result in this direction in the literature can be found in [18], where the authors show such a convergence for the finite element solutions of the nondegenerate Cahn-Hilliard equation, i.e. (Q γ ) with b(u) = 1 and a smooth double well potential , to the corresponding sharp interface limit, the so-called Hele-Shaw problem.…”