Research Of Local Discontinuous Galerkin Method On Cahn-Hilliard Equations

The Cahn-Hillard equation is a very important fourth-order nonlinear diffusion equation,which has been widely used in practical problems such as thermodynamics and fluid mechanics since its introduction.On the other hand,the discontinuous finite element method and the local discontinuous finite element method have the same precision and effect in solving partial differential equations.In this paper,we study the numerical solution of the droplet infiltration equation of the solid surface with the potential term of the Cahn-Hillard equation.The LDG method is proposed for the two equations with zero or not,The fourth order nonlinear equation with non-constant mobility is rewritten as the first-order system by introducing the auxiliary variables,and then the discrete method is used to solve the ordinary differential equation.In this paper,we discuss the theoretical and numerical aspects.In the second chapter,we study the special case of the droplet infiltration equation of the solid surface with the potential term of zero,and give the three weak schemes for the discrete direction of space.By introducing the Soblev space And the concept of the tool and the stability of the study,while the introduction of DG discrete operator using the projection method gives the error estimate.The third chapter consists of the previous chapter,which mainly studies the Cahn-Hillard equation with zero potential.At this time,the existence of the solution changes due to the increase of the potential term.Therefore,On the basis of the existence of the generalized solution,the stability of the solution is discussed,and the error estimation is given similarly.In the fourth chapter,the main contents are numerical discretization and experimental calculation.In this paper,the traditional real-time third-order TVD Runge-Kutta method and semi-implicit predictive correction method are used to compare time.