Iterative Substructuring Methods for Elliptic Problems and Related Algorithms

大尺度问题数值方法的发展 系统是开发高效代码的核心， 流体动力学、弹性和其他核心问题 连续介质力学 这类代码中的许多其他任务相对并行化 轻易 因此，代数系统求解器的重要性随着 并行和分布式计算系统的出现， 大量的快速处理器，每个都有相对较大的内存。 的 改进方法的发展，加上更强大的计算机系统， 使其能够进行三维模拟，具有相当高的 解决方案，相对容易。 将继续努力， 迭代子结构和其它域分解方法 困难的椭圆问题 区域分解算法是迭代的 方法，往往共轭梯度型，为并行解决方案的 大型线性或非线性代数方程组，当 偏微分方程离散有限元，有限 差异或光谱方法。 在每个迭代步骤中，局部问题代表 将原始问题限制在潜在的大量 子区域近似求解。 分区域，可分配给 并行计算机的单个处理器，形成了整个 问题的领域。 此外，包含一个粗糙的子问题， 基本上增加了预处理器的效率。 本研究将数学分析与设计、 算法的数值测试，特别强调的是研究 单元和其他高阶有限元方法， 对砂浆有限元等方法进行了研究，并对难点进行了探讨 其与扭曲的子区域或元素有关。

The development of numerical methods for large systems is central in the development of efficient codes for fluid dynamics, elasticity, and other core problems of continuum mechanics. Many other tasks in such codes parallelize relatively easily. The importance of the algebraic system solvers is therefore increasing with the appearance of parallel and distributed computing systems, with a substantial number of fast processors, each with relatively large memory. The development of improved methods will, together with more powerful computer systems, make it possible to carry out simulations in three dimensions, with quite high resolution, relatively easily. Work will continue in developing iterative substructuring and other domain decomposition methods for increasingly difficult elliptic problems. Domain decomposition algorithms are iterative methods, often of conjugate gradient type, for the parallel solution of the large linear, or nonlinear, systems of algebraic equations that arise when partial differential equations are discretized by finite elements, finite differences, or spectral methods. In each iteration step, local problems representing the restriction of the original problem to a potentially large number of subregions are solved approximately. The subregions, which can be allocated to individual processors of a parallel computer, form a decomposition of the entire domain of the problem. In addition, the inclusion of a coarse subproblem substantially increases the efficiency of the preconditioner. In this study, which combines mathematical analysis with the design and numerical testing of algorithms, a special emphasis is on the study of elements and other high order finite element methods, on nonconforming methods such as the mortar finite elements, and on the difficulties which are related to distorted subregions or elements.