Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows
Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows
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DOI：
10.1007/s00211-002-0413-1
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发表时间：
2003-03
影响因子：
2.1
通讯作者：
Xiaobing H. Feng;A. Prohl
中科院分区：
文献类型：
--
作者：
Xiaobing H. Feng;A. Prohl
我们提出并分析了材料科学中由相变引起的Allen-Cahn方程u−Δu+ɛ−2f(U)=0的半离散(时间)格式和全离散格式，其中ɛ是一个称为“相互作用长度”的小参数。本文的主要目的是为所提出的数值方法建立一些有用的先验误差估计，特别是通过关注误差界对ɛ的依赖性。给出了半离散格式和全离散格式在不同的网格大小、时间步长和初始基准函数的不同正则性假设下的最优阶和准最优阶误差界。特别地，对于较小的ɛ，我们的所有误差界仅依赖于某个较低的多项式阶。分析的关键是建立离散解的稳定性估计，利用De Mottoni和Schatzman[18，19]和Chen[12]的谱估计结果，并为线性化的Allen-Cahn算子处理非线性项建立它的离散对应项。最后，作为一个非平凡的副产品，利用误差估计建立了运动的平均曲率流和广义运动的平均曲率流全离散解的零水平集的收敛和收敛速度。
We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equationut−Δu+ɛ−2f(u)=0 arising from phase transition in materials science, where ɛ 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 ɛ. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh sizehand the time step sizekand different regularity assumptions on the initial datum functionu0. In particular, all our error bounds depend on only in some lower polynomial order for small ɛ. 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.