文档介绍：摘要
论文题目:一类求解广义特征值问题的瀑布型多重网格法
学科专业:计算数学
研究生:白建军签名:
指导教师:赵凤群教授签名:
摘要
随着科学技术的发展,自然界的许多现象都可以用微分方程模型来描述,所以求解微
分方程就成为许多科技工作者的研究重点。有限差分法、有限元法等方法是微分方程经典
数值解法,得到空前的发展。而微分方程离散以后,求解代数方程也发展了许多方法,比
较经典的有多重网格法,由此发展形成了套迭代法,二重网格法,瀑布型多重网格法等等。
为了提高精度,本文对瀑布型多重网格法进行了改进,提出了新的处理方法,并将该方法
应用到了广义特征值的求解。具体有以下几方面的工作:
(1)提出了一种新的基于并行保域逆迭代法的外推瀑布型多重网格法,首先在求解区
域上设置若干层粗细不等的网格,然后在最粗的网格和次粗的网格上选取初值,再通过外
推技巧可以得较高精度的解, 通过插值延拓到细层,得到细层上的解,再利用并行保域逆
迭代法作为光滑迭代的磨光过程,从而得到精度更高的解。该方法优点在于既能有效避免
漏根、迭代过程中的不稳定等现象,又能达到有效提高解的精度和提高运算效率的目的。
(2)用新的瀑布型多重网格法求解在电磁学中的特征值问题。先将数学物理模型进行
有限元离散,得到广义特征方程,用本文提出的新的瀑布型多重网格法求解,并与并行保
域逆迭代法及瀑布型多重网格法进行了比较,验证了本文提出的方法具有精度较高,相对
误差较小,运算效率较高的特点。
关键词: 瀑布型多重网格法;广义特征值;外推方法;保域迭代法
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西安理工大学硕士学位论文
Title: A NEW WATERFALL MULTIGRID METHOD FOR SOLVING THE
GENERALIZED EIGENVALUE PROBLEM
Major: Computational Mathematics
Name: Jianjun Bai Signature:
Supervisor: Prof. Fengqun Zhao Signature:
Abstract
With the development of science and technology, a number ofnatural phenomena can be
described by using the model of differential equation, which has e the focus of many
scientific and technological workers. The finite difference method and finite element method,
which are classical theories for differential equations, e to unprecedented development.
After being discretized, the algebraic equation has developed many methods. The classical
method including iterations, the double grid method, waterfalls multigrid method and so on. In
order to improve the accuracy, in this paper, the waterfall multigrid method is improved during
solving the generalized eigenvalue, and the new processing method is proposed to deal with the
generalized eigenvalue. Specifically the following aspects:
(1) A new waterfall multigrid method, based on collateral conservative reverse iterative
method, is suggested. First, layers of the grid of different thickness are arranged in the domain.
Second, initial value are selected in the thickest grid and the one second to it, which can lead to
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