具有对数势和浓度迁移率的Cahn-Hilliard方程的有限元数值分析

The Numerical Analysis of Finite Element For the Cahn-Hilliard Equation with Logarithmic Flory-Huggins Potential and Concentration Mobility

研究具有对数势和浓度迁移率的Cahn-Hilliard方程.首先通过正则化方法,将对数势函数F(u)的域从(-1,1)拓展到了(-∞,∞);其次提出具有浓度迁移率的Cahn-Hilliard方程的半离散格式和全离散格式,其中时间上采用二阶Crank-Nicolson格式,空间上采用了混合有限元方法;然后证明了该格式是无条件能量稳定的,并对其进行了先验误差估计分析,最后通过数值算例验证了该方法的有效性.

In this paper,we present a numerical scheme for the Cahn-Hilliard equation with logarithmic potential and concentration mobility.First of all,the domain of the logarithmic Flory-Huggins potential F(u)is extended from(—1,1)to(-∞,∞) by regularization.Secondly,the semi-discrete and fully discrete versions of the Cahn-Hilliard equation with concentration mobility are proposed.In the space,the mixed finite element method is adopted for discretization,and the second order scheme is adopted for discretization in the time.In addition,we show the scheme is energy dissipative and perform the error estimates in detail.The verification and validation of the proposed scheme is conducted via numerical examples in the end.