粘性Cahn-Hilliard方程的时间双层网格混合有限元方法

Two Time-Mesh Finite Element Method for Viscous Cahn-Hilliard Equation

主要针对在求解粘性Cahn-Hilliard方程时非线性项引起的时间耗时问题,提出了时间双层网格混合有限元方法.在空间上采用混合有限元方法进行离散,时间上采用Crank-Nicolson格式.首先在时间粗网格上,通过非线性牛顿迭代方法求解非线性混合有限元系统.其次基于初始迭代数值解和拉格朗日插值公式在时间细网格上求解线性混合有限元系统,然后证明了该方法的稳定性和误差估计,并通过数值算例对理论部分进行验证.结果表明,理论与数值算例相一致.

In this paper,the time-consuming problem caused by non-linear term in the numerical solution of the viscous Cahn-Hilliard equation is solved,and the two time-mesh finite element method method is proposed.In the space,the mixed finite element method is adopted for discrete,and the Crank-Nicolson format is adopted for the time.firstly,on a time coarse grid,a non-linear mixed finite element system is solved by a non-linear Newton iteration method,secondly,a linear mixed finite element system is solved on a time fine grid based on the initial iterative numerical solution and the Lagrange interpolation formula,The stability and error estimation of the method are then proved,and the theoretical part is verified by a numerical example.Finally,numerical results are presented to support our theoretical analysis.