An Expanded Mixed Finite Element Method And An Expanded Mixed Covolume Method For Sobolev Equation

This master dissertation consists of four parts. In chapter1, there is some preliminary knowledge. In section1.1, we briefly introduce the wide range of practical applications of the nonlinear Sobolev equation talked about in this paper. In section1.2and section1.3, we mainly introduce the development history of the expanded mixed finite element method (EMFEM) and the ex-panded mixed covolume method(EMCM), respectively. Also the advantages of using these two methods to solve the class of Sobolev equation are discussed.In chapters2and3of this article, we specifically discussed the EMFEM and EMCM for the Sobolev equation.In chapter2, we think of the Sobolev equation as following By defining three auxiliary variables w=ut,p=wx,q=ux, and transforming spatial derivative into temporal derivative, the main computation changed to solve an elliptic equation. With that, we shall present the semi-discrete and fully discrete schemes of EMFEM, draw the existence and uniqueness of solutions and obtain the optimal L2error estimates for these two schemes.The overall thinking of chapter3is similar to chapter2, we present the semi-discrete and fully discrete schemes of EMCM by using the lowest order Raviart-Thomas mixed element space, analyze the existence and uniqueness of solutions, and get the optimal L2error estimates for two schemes. In chapter4, some numerical experiments including single solitary wave, interaction of two and three waves are presented to verify our error estimates are optimal and the three invariants are well conserved. The relevant conclu-sions are made in the final of this chapter.