The Finite Element Solution Of Sine-Gordon Equation

The one-dimensional sine-gordon equation was first introduced in differential geometry, when it was used to describe surfaces with a constant negative Gaussian curvature. By later, it was investigated to be used in many fields of science, such as Josephson junctions between two superconductors, the motion of rigid pendular attached to a stretched wire, solid state physics, nonlinear optics and so on. It is regarded as one of the most important nonlinear equations in applied science. For a long time, researchers have tried a variety of methods to solve it, some people gave the analytical solution, some used the finite difference method, both methods have been applied successfully to solve it, but there is no much good results for solving SG equation by means of finite element method. In this paper, we attempt to apply finite element method to solve SG equation.The finite element method is a very important numerical method to solve partial differential equations. Based on the variational principle, it is not only convenient to process boundary condi-tions, but also show great advantage in adaptiveness, so, it has been used in many fields of physical mechanics and engineering.This paper consists of four parts. In the preface part introduces the history viewing of soli-ton and the development of sine-Gordon equation in early and recently and solving steps of finite element method. The second part and the third part we use finite element method to solve sine-Gordon equation.Firstly we proposed the semidiscrete schemes and the fully discrete schemes secondly given the error analysis. In the last part we get the discontinuous Galerkin method of equation.