【作者】 曲慧宁；

【导师】 顾海明；

【作者基本信息】 青岛科技大学 ， 应用数学， 2011， 硕士

【摘要】 本文主要讨论了在实际问题中遇到的两类偏微分方程的数值解法.主要研究了Burgers方程的隐-显多步有限元方法和分裂型最小二乘混合有限元格式,并且对一类反应扩散方程的隐-显多步有限元方法进行了研究.隐-显多步有限元方法结合了隐显式多步法和有限元方法,首先在空间层上利用有限元方法得到方程的弱形式,然后对时间变量运用隐显式多步法进行离散,从而得到稳定的、相容的、有效的演绎格式.这种格式能够得到时间方向的高精度离散,可以更广泛的应用于对时间精度要求较高的实际问题中.分裂型最小二乘混合有限元方法首先在时间层上将方程进行离散,引入未知变量后,便可得到一个耦合的方程组,然后进行区域分解,把方程组分成两个独立的子系统,这样就会使原问题的求解规模和难度得到很大程度的降低.通过理论上的分析,本文得到了该方法对原有未知变量具有L2 (Ω)最优阶误差估计.本文重点进行了隐-显多步有限元方法和分裂型最小二乘混合有限元方法的理论研究,研究结果表明本文所运用的两种数值模拟方法是可行的.本文共分四章:第一章绪论部分简要介绍了有限元方法的发展历程以及本文所用到的基本理论知识.第二章研究了Burgers方程的隐-显多步有限元格式.第三章研究了Burgers方程的分裂型最小二乘混合有限元格式.第四章研究了一类非线性反应扩散方程的隐-显多步有限元格式.更多还原

【Abstract】 This paper focuses on the numerical solutions to two partial differential equations. This paper studies implicit-explicit multistep finite element method and two splitting least squares mixed finite element method for Burgers equation, and use implicit-explicit multistep finite element method to study reaction-diffusion equations. Implicit-explicit multistep finite element method combines the implicit and explicit multistep method and finite element method, One part of the equation is discretized implicitly and the other explicitly. The resulting schemes are stable, consistent and very efficient. Two splitting least squares mixed finite element method discretizes the equations in time layer firstly, then introduce an unknown variable to get a coupled equations. And then decompose the domain, the coupled system can be splited into two independent sub-systems, so reduce the difficulty and scale of primal problems. Theoretical analysis shows that the method can get approximate solutions for the primal problems with optimal accuracy in ( )L2Ωnorm. This paper theoretical analysis on implicit-explicit multistep finite element method and two splitting least squares mixed finite element method, and verify the reasonableness of the two structures.The main results of this paper are outlined as follows: In chapter one, the paper briefly introduces the development process of finite element method and some basic knowledge used in the latter part of this paper.In chapter two, the paper studies implicit-explicit multistep finite element method for Burgers equation.In chapter three, the paper studies two splitting least squares mixed finite element method for Burgers equation. In chapter four, the paper studies implicit-explicit multistep finite element method for some nonlinear reaction-diffusion equations.更多还原