Finite Element Analysis For Cahn-hilliard Equation

Finite element analysis is a kind of very valid number method which solute differential equation. First we change differential equation into equal variational problems, then we use the limited dimension to approach unlimited dimension space. It is based on spline function, which provides a skill that it select approaching space's "local dimension function" or "divide slices'polynomial". Finite element method defines the differential equation on sample shapes of unit area. And take no account of the complicated boundary condition of whole area. It is one of the reasons that the finite element method is better than other methods.Cahn-Hilliard equation is a class of important nonlinear diffusion equation with fourth order, which was derived by Cahn and Hilliard when they studied the phenomenon of inter-diffusion between two materials (such as alloy) in energetic. later on, such mathematical model is also used by people in describing the phenomenon of the competition and blackball of biology grow groups, the process of riverbed transplant and diffusion of a tiny drop on the surface of a solid. Since 80'in this century, people have studied the Cahn-Hilliard equation systematically. Recently for the important background of Cahn-Hilliard equation in chemistry, materials, etc, there are great achievement in the research of Cahn-Hilliard equation.In this thesis we mainly discuss the Finite Element Analysis of Cahn-Hilliard equation. we consider the semi-discrete scheme and discrete scheme of Cahn-Hilliard equation by means of a finite element biharmonic projection approximation. We studied optimal order error estimates in L2 norms. Especially, for the third order Hermite finite element space, we establish the optimal order error estimates in derivative nodes. Then we discuss the discrete scheme of Cahn-Hilliard equation.We set up Euler scheme and Grank-Nicolson scheme and study the stability and error estimates for discrete scheme.