Some New Discontinuous Galerkin Method For Convection-Dominated Diffusion Problems

The Discontinuous Galerkin Method (DO method for short below)is a new finite. element method presented by Leasaint and Raviart in solving the netron transport equations according the feature of the netron transport, in 1974([3]). then C.Johnson and G.R.Richter et.al. made further study of this method and proved the improved error estimate ([4],[8],[9],[10],[28]). The DG method has been applied in some hyperbolic problems such as scalar Conservation Laws and N-S equations ([11]15]) because of its explicitiness that implies the numerical solution can be developed in an explicit fashion frompstreamoownstream?along the streamline. Since the convection-dominated diffusion problems present hyperbolic characteristics to a great extent, G.R.Richter proposed, in 1992, that DG method can be constructed to solve some hyperbolic problems with?diffusion (ie. convection-dominated diffusion problems) and proved the good stabilitis and high-order accuracy of this scheme when the triangulation of Q is conformed to some conditions and the value of the diffusion coefficient has the same order of the mesh size.The purpose of this thesis is to discuss whether and how to solve the timedependent convectionominated diffusion problems by the DO method. Although the traditional finite element method based on timepace element can coordinate the accuracy of both space and time very well,it always leads to a more complex computing programs corresponding the fulliscrete scheme. In 1990, professor Sun Che([26]) presented a simplified DO method Finite Difference Discontinuous Galerkin Method(FDDG for short below), that is using DO finite element discretization only in space variables and finite difference discretization in time variable. In this thesis, we will, firstly, discuss the construction and the theory of FDDG scheme in solving the convection?dominated diffusion problems. Secondlly, we will costruct two coupling schemes he DO operater splitting method and the DO fractional step method for the linear and nonlinear convectionominated diffusion problems and make some study of the theory of these method.This thesis is composed of two parts. In the first part, we review briefly thebasic idea and the develpoment of the DG method and the FDDG method of one or-der hyperbolic problems, then construct the FDDG scheme fOr the time-dependentconvection--diffusion problems and do detail theory analysis;In the second part, ac-coding the specific feature of the convection Afominated diffosion problems, wepresent two coupled schemes based on the operator splitting view--the DG opera-tor splitting method and the DG fractional step method, and do some researches inthe theory analysis and the implementation of these two schemes.The DG method of one-order hyperbolic problems is the base of this thesis. Inchapter one, we first introduce the main idea and the realization of the DG methodof one--order hyperbolic problems: starting from the units whose inflow boundaIyis on the inflow boundary of the domain,we secure the numerical solution unit byunit along the streamline until the solution in the whole'dOmain is obtained. Thetheoretical results indicate that DG method has both good stability and high-orderaccuracy. The key point in chapter one is the fountain and construction of theFDDG scheme of time-dependent one-order hyperbolic problems: using differencetackles in dealing the time variable and the space variables--finite difference dis-cretization in time and DG finite element discretization in space. we will provethat the FDDG method really maintains the essence charater of DG method,thatis, good stability and high-order accuracy. Because of the explicitness of FDDGmethod, the realization is more simple than some implicit schemes.In chapter two, we will discuss the feasibility of appling both of the FDDGmethod of time--dependent one order hyperbolic problems and the DG method ofthe steady convection~dominated diffusion problems, raised in [18] by Richter, tothe time--depende...