Iterative Substructuring Methods for Elliptic Problems & Related Algorithms

本项目继续研究偏微分方程组的迭代子结构方法和其他区域分解算法。这些算法是并行求解在油藏有限元/差分模拟、计算流体力学、弹性力学等领域中出现的非常大的代数方程组的迭代方法。在每次迭代中，给定问题对许多子区域的约束被近似地解决，并且在相邻的子域之间交换数据。当使用多个子域时，还需要一个全局问题，每个子域的自由度很少，以获得快速收敛。研究将集中在分析、改进和实现几类区域分解算法，特别是那些适用于新兴大规模并行系统的区域分解算法。其最终目标是解决大型、病态的连续介质力学问题，对于这些问题，传统的迭代方法不能令人满意，而直接方法太慢且需要太多的内存。在这些快速算法的最新发展中，已被证明是核心的基本理论将得到进一步发展，并在系统的数值实验中测试这些想法。

This project continues research on iterative substructuring methods and other domain decomposition algorithms for partial differential equations. These algorithms are iterative methods for the parallel solution of very large systems of algebraic equations that arise in finite element/difference modeling of petroleum reservoirs, computational fluid dynamics, elasticity, etc. In each iteration, the restriction of the given problem to each of many subregions is solved approximately and data is exchanged between neighboring subdomains. A global problem, with few degrees of freedom per subdomain, is also required to obtain fast convergence when many subdomains are used. The research will focus on analyzing, improving and implementing several classes of domain decomposition algorithms, in particular, those which are suitable for emerging massively parallel systems. The ultimate goal is to solve very large, ill-conditioned problems of continuum mechanics for which traditional iterative methods are unsatisfactory and for which direct methods are too slow and require too much memory. The underlying theory, which has proven central in the recent development of these fast algorithms, will be further developed and the ideas tested in systematic numerical experiments.