【作者】 秦树杰；

【导师】 孙澈；

【作者基本信息】 南开大学 ， 计算数学， 2001， 博士

【摘要】 间断有限元方法（the Discontinuous Galerkin Method，简称DG方法）是由Leasaint和Raviart([3])在1974年求解中子迁移方程时，针对中子迁移运动的特点提出的一种新型数值方法。随后C.Johnson，G.R.Richter等人对这种方法作了进一步的研究（[4],[8],[9],[10],[28]），并且得到了改进的误差分析结果。由于这种方法具有沿流线从“上游”到“下游”逐层逐元进行计算的显式求解的特点，并且可以进行并行计算，因此被相当广泛地应用于双曲问题中，如双曲守恒律方程及可压的N-S方程（[11]-[15]）等。 G.R.Richter等人针对对流占优扩散问题在很大程度上呈现的是双曲性质这一特点，于1992年提出利用间断有限元方法求解定常对流扩散问题（[18]），并且证明当网格剖分满足一定的条件及扩散系数之取值很小以致与网格参数同阶时，数值解具有良好的稳定性和高阶精度。 能否以及如何利用间断有限元方法求解发展型对流占优扩散问题是本文研究的核心内容。基于时空元的传统有限元方法可以使时间和空间方向的精度很好地协调起来，但却增加了实际计算的复杂程度。在二十世纪九十年代，孙澈教授（[26]）对非定常一阶双曲问题提出了差分间断有限元方法（finit difference discontinuous Galerkin method，简称FDDG方法）：对时间方向采用差分离散，对空间方向采用DG离散。本文首先将这种方法推广到发展型对流占优扩散问题，比较系统地阐明了FDDG方法的构造及有关理论结果，在此基础上，应用算子分裂的观点，提出了间断有限元算子分裂方法及间断有限元分数步方法，给出了它们的理论分析并应用于非线性对流占优扩散问题。 本文内容主要分为两部分。第一部分（前两章）简要介绍了一阶双曲问题的DG方法和FDDG方法的格式构造，并将其推广应用到线性对流占优扩散问题，阐明了FDDG方法的理论分析及具体实现。第二部分（后三章）则针对对流占优扩散问题的具体特点，构造了基于算子分裂观点而提出的两个耦合型有限元格式；间断有限元算子分裂格式与间断有限元分数步格式，并对格式的理论分析和具体实现进行了适当的探讨。 一阶双曲问题的间断有限元方法是本文工作的基础和背景，所以在第一章我们介绍了一阶双曲问题的DG方法和FDDG方法，首先以线性问题为模型，介绍了DG方中文摘要 ＊1法的基本思想和算法实现：从人流边界仅在求解区域的入流边界的单元开始，沿流线方向逐个单元进行计算，得到在整个求解区域上的数值解．对发展型的问题，则采用时空有限元空间．理论结果表明，间断有限元方法具有良好的稳定性和收敛阶，第一章的重点是以非定常一阶线性双曲问题为模型，介绍了作者参与的一项前期工作UqX该文提出了求解非定常一阶双曲问题的差分间断有限元（＊＊＊q方法：对时间方向采用有限差分离散，对空间方向采用间断有限元离散．理论分析表明，＊＊＊G格式保持了＊G方法的主要优点，具有良好的数值稳定性和高阶精度，并且由于其显式性，使得具体实现更加方便快捷． 第二章探讨了将求解非定常一阶双曲问题的＊＊DG格式及mchter对定常对流扩散问题提出的 DG格式O18］）推广应用到非定常线性对流占优扩散问题的可行性．本章提出了适用于非定常问题的 Euler型 FDDG格式和 C－N型 FDDG格式，应用与卜8］完全不同的理论分析方法，在关于空间网格剖分与文［18］所给出的相同条件下，建立了该格式的稳定性与精度阶估计．理论分析与数值算法表明：”在一定的条件下，作为一种迎风型显式有限元方法的始自数值求解一阶双曲问题的＊＊＊G方法，确可应用于作为二阶抛物问题的发展型对流占优扩散问题． 第三章提出了求解非定常对流扩散问题的间断有限元算子分裂方法．在第二章的讨论中，我们发现，只有当扩散系数很小且空间网格剖分满足某种条件时，才能保证＊DDG格式的稳定性和高阶精度．为克服用＊＊＊G格式数值求解非定常对流占优扩散问题的这些不足，在本章中，我们提出了间断有限元算子分裂方法，即在每一个时间段内，先利用 FDDG方法求解一个双曲问题，然后以其数值解作为初值，用标准有限元方法求解一个热传导问题，这种算法是半显式的．其基本思想是：在＊＊＊G格式的基础上，求解一个易于计算的纯扩散问题，对＊D＊G之解进行修正．这样，使用算子分裂技术就十分自然了．理论分析表明：这种耦合型的间断有限元算子分裂格式确可解除第二章中单纯采用 FDDG格式时施加于扩散系数以及网格剖分 Th上的较苛刻的限制，且仍然具有与＊＊＊G格式相同的稳定性与较高的精度阶估计．本项工作也受到应隆安教授工作U叫）的启发．关于同类有限元分裂格式的构造及分析，在国内外文献中，作者尚未见到，本章对线性问题的间断有限元算子分裂格式进行了较详尽的讨论． 第四章利用线性化和归纳假设论证技巧，将第三章的方法推广到非线性问题．利用更多还原

【Abstract】 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更多还原