Advances in Numerical Analysis

Volume 2016, Article ID 1492812, 16 pages

http://dx.doi.org/10.1155/2016/1492812

## Remarks on Numerical Experiments of the Allen-Cahn Equations with Constraint via Yosida Approximation

^{1}Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan^{2}Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

Received 16 February 2016; Accepted 5 April 2016

Academic Editor: Yinnian He

Copyright © 2016 Tomoyuki Suzuki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider a one-dimensional Allen-Cahn equation with constraint from the viewpoint of numerical analysis. The constraint is provided by the subdifferential of the indicator function on the closed interval, which is the multivalued function. Therefore, it is very difficult to perform a numerical experiment of our equation. In this paper we approximate the constraint by the Yosida approximation. Then, we study the approximating system of the original model numerically. In particular, we give the criteria for the standard forward Euler method to give the stable numerical experiments of the approximating equation. Moreover, we provide the numerical experiments of the approximating equation.

#### 1. Introduction

In this paper, for each we consider the following Allen-Cahn equation with constraint from the viewpoint of numerical analysis: where and is given initial data. Also, the constraint is the subdifferential of the indicator function on the closed interval defined by More precisely, is a set-valued mapping defined by

The Allen-Cahn equation was proposed to describe the macroscopic motion of phase boundaries. In the physical context, the function in is the nonconserved order parameter that characterizes the physical structure. For instance, let be the local ratio of the volume of pure liquid relative to that of pure solid at time and position , defined by where is the ball in with center and radius and denotes its volume. Put for any . Then, we observe that is the nonconserved order parameter that characterizes the physical structure:

There is a vast amount of literature on the Allen-Cahn equation with or without constraint . For these studies, we refer to [1–16]. In particular, Chen and Elliott [5] considered the singular limit of as in the general bounded domain .

Also, there is a vast amount of literature on the numerical analysis of the Allen-Cahn equation without constraint . For these studies, we refer to [17–21].

Note that the constraint is the multivalued function. Therefore, it is very difficult to apply the numerical methods developed by [17–21] to . Hence, it is difficult to perform a numerical experiment of . Recently, Blank et al. [2] proposed as a numerical method a primal-dual active set algorithm for the local and nonlocal Allen-Cahn variational inequalities with constraint. Also, Farshbaf-Shaker et al. [8] gave the results of the limit of a solution and an element of , called the Lagrange multiplier, to as . Moreover, they [9] gave the numerical experiment to via the Lagrange multiplier in one dimension of space for sufficient small . Furthermore, they [9] considered the approximating method, called the Yosida approximation. More precisely, for , the Yosida approximation of is defined by where is the positive part of . Then, for each , they [9] considered the following approximation problem of : In [9, Remark ], Figure 1 shows the unstable numerical result to was given by the standard explicit finite difference scheme to .