Two Classes Of Finite Element Methods For Evolution Equation

Evolution equations are a class of PDEs which are used to describe a time-dependent process. They combine mathematics with many fields of nat-ural sciences closely and represent a restrictive relationship between the change of spatial and temporal variables. The evolution equations are not only con-venient for people to know basic rules of natural phenomenon, but also play a very important role in designing engineering problems and predicting the changes of the natural phenomenon. In this paper, we focus on two kinds of evolution equations which arc Sobolev equation and convection diffusion equation.This dissertation is organized as follows. In chapter1, we introduce the backgrounds and current research situations of Sobolev equation and convec-tion diffusion equation. Preliminary knowledge is given in chapter2, including conservation laws and invariants of MRLW equation. Chapter3and Chapter4are the main body of this thesis.Chapter3considers the numerical simulation of the modified regular-ized long wave (MRLW) equation by split least-squares mixed finite element method (SLSMEM). MRLW equation is as follows, ut+ux+6u2ux-μuxxt=0, MRLW equation is the special case of the generalized long wave (GRLW) equation, which has the form ut+ux+δurux-μuxxt=0. The advantages of the present SLSMFEM in the solution of the MRLW (or GRLW) equation lie in its simplicity and efficiency. Using this approach, it is feasible to convert the original equation, which is high order PDE. to two first order systems, of which one only includes temporal derivatives and the other only includes spatial derivatives. For the system only including spatial deriva-tive, we present the least-squares mixed finite element procedure which can be split into two independent symmetric positive definite sub-schemes and solved separately. Obviously, the resulting scheme is easy to implement parallel com-puting. Numerical examples, based on (1.3), show that the proposed scheme is conservative. Specifically, this chapter consists of four sections. In§1, We present the analytical solution and invariants of the MRLW equation, In§2. Frist we give finite element space, and then we introduce the Semi-discrete and fully-discrete SLSMFEM scheme. In§3, we give three numerical examples to verify the feasibility of the proposed scheme. Finally, in§4, some remarks are given.In chapter4,we introduce the DDG method(The direct discontinuous Galerkin method) of the following convection diffusion equation the DDG method is proposed on the basis of direct weak formulation for so-lutions of equations in each computational cell, and each computational cell is communicate by the numerical flux only. This method doesn’t change the weak formulation of equations but changes the scheme of the numerical flux. The DDG method can keep the conservation relationship among physical quantities and has the virtues of easier formulation and implementation, less calculation and efficient computation of the solution. Specifically, this chapter consists of three sections. In§1, we establish the DDG scheme of the problem (1.5). define a general numerical flux formula for the solution derivative and intro-duce the semi-discrete and fully-discrete DDG scheme; In§2, after introducing concepts of admissibility and energy stability,we discuss the stability of the scheme. In§3, we give some numerical examples to verify the feasibility of the proposed scheme.