The Finite Element Algorithms Of Cahn-Hilliard Equation And Allen-Cahn Equation

In this paper,the finite element numerical algorithms of the Cahn-Hilliard equation and the Allen-Cahn equation are studied.On the one hand,a mixed finite element method for the viscous Cahn-Hilliard equation with the concentration mobility and the logarithmic potential energy is studied,and a two-grid mixed finite element method for solving the viscous Cahn-Hilliard equation with logarithmic potential energy is also studied in order to solve the time-consuming problem caused by nonlinear terms.On the other hand,a two-grid finite element method for the Allen-Cahn equation with logarithmic potential energy is studied.The main idea of the two-grid finite element method is as follows: the nonlinear finite element system on the coarse grid is solved by Newton iterative method,where the finite element method is used for spatial discretization and the one-step semi-implicit scheme or fully-implicit scheme are used for time discretization.On the fine grid,based on the initial iterative numerical solution,a linear finite element system is solved.The details are as follows:In the first chapter,the background of the Cahn-Hilliard equation and the AllenCahn equation,the finite element method and the two-grid method are introduced.In the second chapter,the basic definition and lemma are given.In the third chapter,the mixed finite element method is used to solve the viscous Cahn-Hilliard equation with the concentration mobility and the logarithmic potential energy.The mixed finite element method is used for spatial discretization and the onestep semi-implicit scheme are used for temporal discretization.The stability analysis and the convergence analysis of numerical solutions are given by theoretical analysis,and numerical examples are given to demonstrate the validity of the method.In the fourth chapter,the two-grid mixed finite element method is used to solve the nonlinear Cahn-Hilliard equation with the logarithmic potential.The mixed finite element method is used for spatial discretization and the one-step semi-implicit scheme are used for temporal discretization.The stability analysis and the convergence analysis of numerical solutions are given by theoretical analysis,and numerical examples are given to demonstrate the validity of the method.In the fifth chapter,the two-grid finite element method is used to solve the nonlinear Allen-Cahn equation with the logarithmic potential.The finite element method is used for spatial discretization and the one-step fully-implicit scheme are used for temporal discretization.The stability analysis and the convergence analysis of numerical solutions are given by theoretical analysis,and numerical examples are given to demonstrate the validity of the method.Finally,the summary and some expections of this paper are given.