Two Mixed Finite Element Methods On Changing Meshes For The Sobolev Equations

Sobolev equations have important applications in many mathematical and physical problems, such as the percolation theory when the fluid flows through the cracks, the transfer problem of the moisture in the soil, and the heat, conduction problem in different materials, and so on. So there is important and actual sig-nificance to research Sobolev equations. Mixed finite element method on changing meshes is used to reseach the Sobolev equations in this dissertation. This method deals with the problem by mixed method on space domain, using different meshes and different basic functions at different time level. This dissertation is divided into two chapters:Chapter 1 introduces the upwind-mixed finite element method on changing meshes for the Sobolev equations.Mixed finite element method changes the model into a first order system about unknown variable u and its fluxσ=▽u, using▽u as a unknown variable, then dis-cretes it by an upwind-mixed method on changing meshes. The approximating finite element spaces to variable u and fluxσrespectively are Wh×Vh.The convection term c·▽u in the Sobolev equations is approximated by the upwind method and the diffusion term by an expanded mixed finite element method. The upwind-mixed method is presented on changing meshes.According to the above methods, this chapter builds the upwind-mixed scheme for the Sobolev equations on changing meshes. Optimal error estimates in L2-norm can be obtained. The Sobolev equations have the mixed differential term▽·│a▽ut│, which increases the difficulty for the study, and it is also the significance of this paper.This chapter is divided into four sections. Section I introduces the upwind-mixed method on changing meshes briefly. In Sections II and III, we build the upwind-mixed finite element scheme on changing meshes for the linear and nonlinear mathematical models respectively. In Section IV, two numerical experiments are given to illustrate the efficiency of the method.Chapter 2 considers the Godunov-mixed finite element method on changing meshes for the Sobolev equations.Godunov-mixed finite element method combines the mixed finite element method and the Godunov method, and changes the model into a first order system, then discretes it by introducing a Godunov-type procedure. In this method, the approx-imating finite element spaces Wh×Vh, to variable u and fluxσ, are "lowest-order" Raviart-Thomas spaces. We give a gradient approximation as well as an approx-imation to the diffusion term. The diffusion term in the Sobolev equation is also approximated by an expanded mixed finite element method, but the convection term by a Godunov-type procedure. The Godunov-mixed finite element method is presented on changing meshes.This chapter researches Sobolev equations by the Godunov-mixed finite element method on changing meshes. To get higher precision, the scheme built in this chapter combines the advantages of two methods, which can simultaneously approximate the scalar unknown and the vector flux effectively.This chapter is also divided into four sections. Section I introduces the Godunov-mixed finite element method on changing meshes briefly. In Sections II and III, we build the Godunov-mixed finite element scheme on changing meshes for the lin-ear and nonlinear mathematical models respectively. In Section IV, two numerical experiments are given to illustrate the efficiency of the method.