Mathematical Sciences: Finite Element Method for Nonlinear Partial Differential Equations

本研究项目将研究非线性四阶演化Cahn-Hilliard方程。一个主要的问题将是发展一种适当的数值方法来精确地逼近这个方程在一个和两个空间维度上的解。我们将开发一种有效的数值格式来获得二维方程的长时间解。本文还将分析亚稳解的可能性。卡恩-希利亚德方程在二元合金中的相分离等现象中建立了模型。该研究项目的成果将在这个和其他相关的物理问题中找到应用。

This research project will study the nonlinear fourth order evolutionary Cahn-Hilliard equation. A main concern will be the development of an appropriate numerical method for accurate approximation of solutions of this equation for both one and two spatial dimensions. A efficient numerical scheme will be developed to obtain long time solutions of the two dimensional equation. The possibility of metastable solutions will also be analyzed. The Cahn Hilliard equation models, among other phenomena, phase separation in binary alloys. The results of the research project will find applications in this and other related physical problems.