Research Of A Least-squares Mixed Finite Element Method With Characteristics For Two Kinds Of Parabolic Equations

There has been a great development for the method of finite element since 1950s in practical application, for example: least-squares mixed finite element method, characteristics finite element and moving mesh finite element. These methods have been generally applied in many industry fields and many problems in real life have been solved. The Least-squares mixed finite element methods have been formulated to solve convection-diffusion parabolic differential equations. The advantage of this methods is that it is not subject to the LBB condition. The convergence analysis showed that the least-squares mixed element schemes yield the approximate solution with optimal accuracy in H ( div;Ω)×H~1 (Ω).The main task in this paper is to solve convection-diffusion parabolic equations by making use of the methods of least-squares mixed finite element, characteristics finite element and moving mesh finite element. According to the actual needs of problem for convection-diffusion parabolic equations, together least-squares mixed finite element schemes with the method of characteristics, we discuss two least-squares characteristic mixed finite element procedures for one kind of convection-dominated parabolic equations. On the other hand, the least-squares characteristic mixed finite element scheme with moving mesh is formulated to solve first-order time-dependent convection-diffusion systems. Error estimate show that the method have optimal convergent rate in some sense. Its main contents are as follows:In the first chapter, preface gives the background and the status of recent research of the finite element method.In the second chapter, some notes and three kinds of finite element method are presented which we will use in the paper.In the third chapter, we study one kind of convection-dominated parabolic differential equation by the method of least-squares characteristic mixed finite element and get optimal estimation for two different least-squares characteristic mixed finite element schemes in H ( div;Ω)×H~1 (Ω).In the fourth chapter, the least-squares characteristic mixed finite element scheme with moving mesh is formulated to solve the other kind convection-diffusion system and error estimate is given also.