Iterative Substructuring Methods for Elliptic Problems and Related Algorithms

本课题继续了以往椭圆型偏微分方程域分解方法的研究。这些是求解大型线性或非线性代数方程组的迭代方法，当椭圆问题被有限元或有限差分离散时，它们为高效并行求解提供了非常大的希望。研究中考虑的方法可以看作是分而治之的算法。在迭代的每一步中，在原始区域被细分成的子区域上求解原始离散椭圆问题。该区域的不同部分之间的交互是通过在由细分创建的接口上传输适当类型的数据来处理的。除了局部问题外，为了在许多子区域的情况下获得快速的收敛速度，必须在预条件中加入一些全局的、粗略的问题，这是并行计算的主要兴趣。将进一步开展工作，制定研究区域分解方法的一般框架，包括使用重叠分区域的方法和不使用重叠分区域的方法。这项研究的主要目的有两个：(1)发展新的和改进的方法，并将它们推广到非协调情况、混合有限元模型、有限体积情况等。(2)开发工具和经验，使之有可能为解决重要的弹性、流体动力学、系数值变化很大的问题等问题提供具体的建议。

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.