Iterative Substructuring Methods for Elliptic Problems and Related Algorithms

本项目延续了以往区域分解的研究成果 椭圆型偏微分方程的方法 这些是 迭代法求解大型线性或非线性系统 椭圆问题离散化时出现的代数方程 有限元或有限差分，它们提供了非常 高效并行解决方案的巨大潜力。 的方法 在这项调查中考虑的可以被视为分而治之 算法 在迭代的每一步中， 椭圆问题是解决的子区域，其中原来的 区域被细分。 不同的人之间的相互作用 区域的一部分通过传送适当类型的数据来处理 通过细分创建的界面。 在 除了局部问题外，还必须解决一些全局的、粗糙的问题。 加入预处理器，以获得快速的 就许多次区域而言， 对并行计算感兴趣。 将开展进一步工作，为 区域分解方法的研究 分区域和那些没有。 这样做有两个主要目的 研究：（1）开发新的和改进的方法，并将其扩展到 非均匀情形，混合有限元模型，有限体积 （2）开发工具和经验，使其 为解决这些问题提出具体建议 弹性力学、流体动力学的重要问题， 改变系数的值等。

This project continues the past research on domain decomposition methods for elliptic partial differential equations. These are iterative methods for solving large linear or nonlinear systems of algebraic equations which arise when elliptic problems are discretized by finite elements or finite differences, and they offer very considerable promise for efficient parallel solution. The methods considered in this investigation can be regarded as divide and conquer algorithms. In each step of an iteration, the original discrete elliptic problem is solved on the subregions into which the original region has been subdivided. The interaction between the different parts of the region is handled by transferring data of a suitable type across the interfaces which are created by the subdivision. In addition to the local problems, some global, coarse problem must be incorporated into the preconditioner in order to obtain a fast rate of convergence in the case of many subregions, the case of primary interest in parallel computing. Further work will be carried out to develop a general framework for the study of domain decomposition methods both those which use overlapping subregions and those which do not. There are two main aims of this research: (1) To develop new and improved methods and to extend them to the nonconforming case, mixed finite element models, the finite volume case, etc. (2) To develop tools and experience which will make it possible to provide specific recommendations for the solution of important problems of elasticity, fluid dynamics, problems with widely varying values of the coefficients, etc.