Study On Two-grid Finite Volume Element Method For A Class Of Phase-field Equations

This paper mainly studies the two-grid finite volume element method for a class of phase field equations.As an important physical model in computational mathematics,phase field equation is essentially a kind of nonlinear partial differential equation.The phase field equation satisfies a nonlinear stability relationship called energy stability,and has nonlinear terms and high-order differential terms,which is difficult to solve numerically.The finite volume element method satisfies the energy stability numerical scheme for the fluid.The two-grid method can deal with the nonlinear term well and the mixed finite volume element method can effectively deal with the high-order differential term.Therefore,this paper mainly uses the two-grid mixed finite volume element method to study the numerical solution of the phase field equation.The main contents are as follows :1)A coupled two-grid finite volume element method for Cahn-Hilliard equation is presented.In this paper,the intermediate variable is introduced to transform the high-order differential term in the Cahn-Hilliard equation into a low-order mixed form,and the semidiscrete and full-discrete schemes of the coupled two-grid finite volume element based on θ scheme are given.The two-grid algorithm is designed.The algorithm first solves the nonlinear mixed finite volume element system obtained on the coarse grid,and then solves the coupled linearized mixed finite volume element system by projecting on the fine grid.The stability of the solution of the two-grid discrete system is proved,and the H~1 priori error estimation is given.The feasibility of the theoretical method is verified by numerical experiments.2)An uncoupled two-grid finite volume element method for the Cahn-Hilliard equation is presented.Based on the above mixed equation system,a two-grid θ format uncoupled mixed finite volume element algorithm is given.The algorithm is also a nonlinear mixed finite volume element system obtained on the coarse grid,but the elliptic linearized finite volume element system is solved by projection on the fine grid.The stability of the solution of the two-grid discrete system is proved,a priori error estimation is given,and the efficiency of the uncoupled algorithm is verified by numerical experiments.3)A two-grid finite volume element method for the Allen-Cahn equation is presented.The method uses θ format for time discretization and finite volume element for space discretization,and obtains a fully discrete format.A two-grid algorithm is designed.The algorithm solves the nonlinear finite volume element system on the coarse grid and solves the elliptic linearized finite volume element system at the final moment by projection on the fine grid.This paper proves the numerical stability of the finite volume element discrete solution,gives a priori error estimation,and verifies the accuracy and efficiency of the algorithm through numerical experiments.Compared with the traditional algorithm,the two-grid finite volume element method proposed in this paper can be used to solve the phase field problems,which can effectively improve the accuracy and speed of the problem and reduce the difficulty of solving nonlinear problems.