The Time Two-Mesh Finite Element Method For Allen-Cahn Equation

In order to solve the time-consuming problem caused by the nonlinear term of the Allen-Cahn equation,this paper studies time two-mesh(TT-M)finite element(FE)methods based on the linear interpolation and parabolic interpolation.The main idea of TT-M FE method is:in the coarse time-mesh,Newton iteration method is used to solve the fully discrete nonlinear equation;In the fine time-mesh,based on the interpolation solution obtained by the linear or parabolic interpolation method,the nonlinear problem is converted into a linear equation.The TT-M FE method is compared with the traditional Galerkin FE method and it is evident that the TT-M FE method can save the calculation time.The structure of this paper is as follows.Firstly,the background and significance of the Allen-Cahn equation are introduced.Secondly,the basic definition and lemma are given.Next,the time two-mesh(TT-M)finite element(FE)algorithm based on linear interpo-lation method is used to solve the nonlinear Allen-Cahn equation.The FE method is used for spatial discretization and the second-order backward difference(BDF)scheme is used for temporal discretization.The stability analysis and error estimation of numerical solutions are verified by theoretical analysis,and numerical examples are given to demonstrate the validity of the method.In addition,the time two-mesh(TT-M)finite element(FE)algorithm based on parabolic interpolation method is used to solve the nonlinear Allen-Cahn equation.The FE method is used for spatial discretization and the second-order ? scheme is used for temporal dis-cretization.The stability analysis and error estimation of numerical solutions are verified 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.