Application Of The Finite　Element Method On Nonlinear Evolution Equation

The evolution equation plays an important role in the studies of physics, mechanics and other natural sciences. In the development of science and technology, kinds of evolution equations have been raised. However, in the overwhelming majority situations, solutions of these problems cannot be expressed in analysis-theformula form or the form is much too complex, thus we need adopt numerical methods to get approximate solutions of these partial-difficiential equations. In the latter part of the twentieth century, finite element method became one of the extremely effective methods to get numerical solutions of partial-difficiential equations which also include evolution equations. Finite element method has remarkable advantages when solving problems with irregular regions or general bound conditions. This paper is based on finite element method and devote to studying the characteristic finite element method and fourier spectral method.Firstly, we not only introduce the research context and development of finite element method but also exhibit how to solve physics-mathmetics problem by general finite element method. Secondly, we introduce the characteristic finite element method and Fourier spectral method which will be involved in the following part .After that we expound some related concepts and theories on characteristic finite element method and fourier spectral method .Next, we set up the characteristic scheme for the convection-dominated diffusion problems. And derive optimal L2 and H 1 error estimates. Through the analysis we have proved the validity of this method.Finally, we develop fourier seimdiscrete and fully discrete schemes that inherit the conservation properties of differential equation to numerically solve the nolinear schr?dinger equation with periodic boundary conditions. We not only discuss the conditions under which the fully discrete form has an unique solution, but also analysis the error estimates. The implement of the fully discrete scheme requires the solution of a nonlinear system of equations at each time step. We use the predictor-corrector algorithm, and confirm our theoretical results by numerical experiments.