一类粘弹性流体模型与数值分析的分析

一类粘弹性流体模型与数值分析的分析

摘要

粘弹性流体问题一直是流体力学和理论数学研究的一个重要问题．本文主

要研究一类粘弹性流体的数学模型．耳POldroyd—B型流体的数学模型．这类数

学模型一直以来都是众多科学家感兴趣的研究内容，均归结为偏微分方程(组)的求解，因此，研究具有高效率高精度的算法是很有必要的．在本文

中我们提供了几种解决两类偏方程的数学方法．文章主要内容如下j

本文第一章介绍了非牛顿流体力学及相关数值分析综述．第二章着重讨论

了基于Oldroyd随体时间导数的01droyd-B型流体的数学模型的本构方程的

建立、求解，并最终给出了此类方程l级、2级变分一解析解，同时，我们还在

两个特殊情形(常压力梯度和周期性压力梯度)下，讨论了该变分一解析解具体表

达形式．

第三章主要工作是应用混合有限元、最小二乘混合有限元和V循环多重网格

法去解决Oldroyd B型流体流动问题．一方面，我们将混合有限元方法应用于求

解非定常型的服从Oldroyd B型本构律的黏弹性流体流动问题．另一方面，我们将

运用混合有限元方法、最小二乘混合有限元方法和Y循环多重网格法去逼近

Oldroyd B型流体流动问题，并讨论了逼近解与真解的误差估计和收敛性．其主要

内容如下：讨论用混合有限元方法去研究01droyd B型流体流动问题的解的存在

唯一性，并给出了逼近解的误差估计；介绍应用混合有限元的最小二乘法去逼近01droyd B型流体流动问题，并讨论了逼近解的收敛性；讨论01droyd B型流体

流动问题的V循环多重网格格式，并给出了迭代解的存在唯一性和误差估计．本文第四章的主要目的就是研究一类非对称椭圆问题的最小二乘混合有限

元方法的超收敛现象．特别是对一般的非自共扼二阶椭圆边值问题，我们讨论了其最小二乘混合元解的存在唯一性及超收敛性．在第五章中，我们分别对半线性反应扩散问题和非线性反应扩散问题的扩张混合有限元方法给出了几个两层网格方法，并对它们的收敛性进行了分析．关键词：Oldroyd—B型流体，反应扩散方程，有限元，混合有限元，超收敛，误差估计

Abstract

Lately,the viscoelastic fluid flow is one of the most important questions in Hydro-and theoretic mathematics．In this paper,we study one kind of viscoelastic fluid dynamics

flow model．It is Oldroyd-B type fluid，the models Can express using PDEs in math，and their solutions have received a great deal of attention．So，it is necessary to study

efficient and highly accurate algorithms for PDEs，we present some kinds of highly

method for solving PDEs using computational symbolic manipulation and finite element method．This paper includes five parts．

In chapter 1，we introduce the theoretic basis of non-Newtonian fluid mechanics and

Method．In rheology,and we introduce the mathematical foundation of Finite Element

chapter 2，we set up the constitutive equation of Oldroyd—B type fluid based on Oldroyd self-time differential，present the grade 1，grade 2 variational

analytical solutions respectively，specially,we present the concrete variational analytical solutions under constant pressure grade and periodic pressure grade．In chapter 3，our essential work is to solve viscoelastic fluid flow obeying an Oldroyd B type constitutive law by apply the mixed finite element method，least—square mixed finite element method and V-cycle multi-grid method．We study the mixed finite element method of Oldroyd B type viscoelastic fluid flow model，and we give the existence and uniqueness of approximation solution and error bounded；by applying the least-square

mixed finite element methodto study Oldroyd B type viscoelastic fluid flow

model，we discuss the existence and uniqueness of approximation solution and convergence；we analysis the V-cycle multi·grid formulation of Oldroyd B type viscoelastic fluid flow model，and we give the existence and uniqueness of iterative solution and error estimates．

In chapter 4，by using least-squares mixed finite element method over quadrilaterals，we investigate super convergence phenomena sep·-arately for boundary··value problems of un-symmetric elliptic equations，we obtain the super convergence result

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