Spectral Method Analysis For Cahn-Hilliard Equation

Cahn-Hilliard equation is an important class of fourth-order nonlinear partial differential equation, Since the Cahn-Hilliard equation in chemistry, chemical engineering and materials science and other aspects of important background, much attention in recent years, the results a lot. There are three basic methods of partial differential equations solved: Spectral method, Finite difference method and Finite element method. This article will use Spectral methods to analyze a given solution of the problem Cahn-Hilliard equation. Spec--tral method regards trigonometric polynomial as basic functions and is a form of Rize-Galerkin method.. Due to the nature of trigonometric functions with a very good appr-oximation, it makes Spectral method has become a high-precision numerical methods. Its biggest advantage is that the so-called "infinite order convergence", and can be imple-mented fast Fourier calculations. Thus, Spectral method can obtain high accuracy with less computational cost, and have incomparable advantages which the finite difference and finite element method can not match.This paper conducted a spectral method of nonlinear evolution type Cahn-Hilliard equa tion analysis and discussion, First, It established an equivalent variational form of Cahn-Hilliard equation with cycle and boundary value problem.Then, it dispersed the space vari able x to get a semi-discrete approximation of the Cahn-Hilliard equation. Then we prove the existence and uniqueness, and the stability, boundedness and convergence of the solu-tions of semi-discrete problem. Finally, on the basis of the solution of semi-discrete format we discussed fully discrete format of Cahn-Hilliard equation, construct implicit Euler scheme, and gives the stability and convergence of fully discrete format.