The Study Of Time Two-Mesh Mixed Finite Element Method For The Cahn-Hilliard Equation

A time two-mesh mixed finite element method is proposed for the solving the time-consuming problem caused by nonlinear term in the Cahn-Hilliard equation.The main idea of this method is as follows: a nonlinear mixed finite element system on the time coarse mesh is solved by Newton iterative method,where the mixed finite element method is used for spatial discretization and the first-order Backward-Euler(B-E),the second-order Crank-Nicolson(C-N)and the second-order ? scheme are used for time discretization.On the fine time mesh,based on the initial iterative numerical solution and the Lagrange's interpolation formula,a linear mixed finite element system is solved.The details are as follows:Firstly,the background,the mixed finite element method and the time two-mesh method of the Cahn-Hilliard equation are introduced.Secondly,aiming at the Cahn-Hilliard equation,the time two-mesh mixed finite element method in the first-order B-E and second-order C-N scheme are studied.Aiming at the viscous Cahn-Hilliard equation,the time two-mesh mixed finite element method in the second-order ? scheme is studied.Finally,the stability and error estimates of the numerical method under three time discrete scheme are analyzed respectively.The theory is verified by numerical examples.The results show that,compared with the traditional mixed finite element method,the numerical method proposed in this paper can save calculation time while ensuring accuracy.