Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits
Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits
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DOI：
10.1090/s0025-5718-03-01588-6
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发表时间：
2003-07
期刊：
影响因子：
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通讯作者：
Xiaobing H. Feng;A. Prohl
中科院分区：
文献类型：
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作者：
Xiaobing H. Feng;A. Prohl
针对材料科学中描述凝固过程的相场模型，提出并分析了一种全离散有限元格式。本文的主要目标是建立一些有用的先验误差估计所提出的数值方法，特别是，通过专注于依赖的误差界的参数e，被称为测量的界面厚度。在合理的网格尺寸h和时间步长k的约束下，给出了全离散格式的最优阶误差界。特别地，它表明，所有的误差界取决于1 a只有在一些低多项式阶小e。分析的关键是建立离散解的稳定性估计，使用Chen的谱估计结果，并建立线性化相场算子的离散对应物以处理非线性效应。最后，作为一个非平凡的副产品，误差估计是用来建立收敛的解决方案的完全离散计划的解决方案的尖锐的界面限制的相场模型在不同的缩放系数。尖锐的界面限制包括经典Stefan问题，广义Stefan问题与表面张力和表面动力学，运动的平均曲率流，和Hele-Shaw模型。
We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter e, known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size h and the time step size k. In particular, it is shown that all error bounds depend on 1 a only in some lower polynomial order for small e. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.