Iterative Substructuring Methods for Elliptic Problems and Related Algorithms

椭圆问题的迭代子结构方法及相关算法

• 批准号: 8703768
• 负责人: Olof Widlund
• 金额: $ 14.35万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-15 至 1989-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8703768&HistoricalAwards=false
• 关键词: Iterative; Substructuring; Methods; Elliptic; Problems

This project continues to work on iterative substructuring methods. These are iterative methods for large linear systems of algebraic equations which arise when elliptic problems are discretized by finite elements or finite differences. The region of the problem is divided into subregions (substructures) and the problems on the subregions are solved repeatedly while using a preconditioned conjugate gradient algorithm to satisfy the continuity requirements across the curves or surfaces that divide the region. Main current research issues are the extension of these algorithms to nonconforming finite elements, mixed methods, etc., and the development of new algorithms, particularly in three dimensions. This program will combine theoretical work with systematic numerical experiments using both model problems and industrial finite element codes. The best of these algorithms often represent an improvement over previously known methods on the uni processors and they hold considerable promise for multi processor systems. In addition, new algorithms for certain queuing network problems will be developed. Kolmogorov's balance equations are generated and the goal is to compute the steady state probability distribution in terms of the null element of the corresponding very large, sparse, nonsymmetric matrix. The basic idea, which has proven quite successful in a number of cases, is to use suitably chosen nearby decomposable problems as preconditioners. These algorithms will be further developed, the theory will be refined to try to explain the interesting distribution of the singular values of the iteration matrices and an attempt will be made to extend the applicability of methods to new classes of problems. This research addresses new and faster methods for solving large systems of linear equations, a computational task that is fundamental to many scientific and engineering problems.

该项目继续致力于迭代子结构方法。 这些都是大型线性代数系统的迭代方法， 当椭圆问题被有限差分离散化时， 元素或有限差分。 问题的区域被划分为 划分为子区域（子结构），子区域上的问题是 反复求解，同时使用预处理共轭梯度 满足曲线上连续性要求的算法，或 分割区域的曲面。 当前的主要研究问题是 将这些算法推广到有限元，混合 方法等，以及新算法的发展，特别是在 三维空间 该计划将联合收割机理论工作与 系统的数值实验，使用模型问题和工业 有限元程序 这些算法中最好的通常代表 对UNI处理器上的先前已知方法的改进， 对多处理器系统具有相当大的希望。 此外，某些排队网络问题的新算法将在 开发 Kolmogorov的平衡方程被生成，目标是 为了计算零的稳态概率分布， 对应的非常大的，稀疏的，非对称矩阵的元素。 基本的想法，这已被证明是相当成功的，在一些情况下， 是使用适当选择的附近可分解的问题， 预处理器 这些算法将得到进一步的发展，理论 将被细化，试图解释有趣的分布， 迭代矩阵的奇异值，并尝试 将方法的适用性扩展到新的问题类别。 这项研究提出了解决大型系统的新的更快的方法 线性方程组，这是一个计算任务，是基本的许多 科学和工程问题。