Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows

Abstract

We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation u t − u + ε −2 f (u) = 0 arising from phase transition in materials science, where ε is a small parameter known as an "interaction length". The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on ε. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u 0 . In particular, all our error bounds depend on 1 ε only in some lower polynomial order for small ε. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18,19] and Chen [12] and to establish a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow.

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

“…We remark that, using a similar approach parallel studies were also carried out by the authors in [28] for the Allen-Cahn equation and the related curvature driven flows, and in [30] for the classical phase field model and the related Stefan problems. On the other hand, unlike the Allen-Cahn equation which is a gradient flow for (4) in L 2 , the Cahn-Hilliard equation is a gradient flow for (4) in H −1 , which makes the analysis for the Cahn-Hilliard equation in this paper more delicate and complicated than that for the Allen-Cahn equation.…”

“…Another strategy is to use a certain combination of f evaluated at different time steps (cf. [28]). Despite the advantage of having a discrete energy law without parameter restrictions, we prefer the scheme (25)- (26) for its simpler structure and simpler subsequent error analysis.…”

“…which was originally introduced by Allen and Cahn [3] to describe the motion of antiphase boundaries in crystalline solids (see [12,35,28] and references therein). It is easy to check that the Cahn-Hilliard problem (1)- (3) conserves the total mass because its solution satisfies d dt u(x, t)dx = 0, however, the corresponding Allen-Cahn problem does not have this mass conservation property.…”

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

“…Explicit Euler discretizations for Allen-Cahn obstacle problems have been used for example in [12,20,23,24]. Numerical analysis for (semi-) implicit discretizations of the Allen-Cahn model has been performed in the papers [15,22,31,32,33,34] and in works cited in these papers. Fully implicit discretizations are the most accurate, see e.g.…”

Multi-Branched Blood Vessels Segmentation Based on Phase-Field and Statistical Model

“…Recently, the Allen-Cahn equation [17], also known phase field model, has attracted the attention of some scholars, and it has also proved efficiency in image segmentation problems [18]- [23]. For example, Jung et al [21] proposed a phase-field method to solve mutliphase piecewise constant segmentation problem.…”