Iterative Substructuring Methods for Elliptic Problems and Related Algorithms

大系统数值方法的发展是流体力学、弹性力学和其他连续介质力学核心问题有效代码发展的核心。这种代码中的许多其他任务相对容易并行化。因此，随着并行和分布式计算系统的出现，代数系统求解器的重要性日益增加，这些系统具有大量快速处理器，每个处理器都具有相对较大的内存。改进方法的发展，加上更强大的计算机系统，将使以相当高的分辨率相对容易地进行三维模拟成为可能。工作将继续发展迭代子结构和其他领域分解方法，以解决日益困难的椭圆问题。区域分解算法是一种迭代方法，通常是共轭梯度型的，用于在偏微分方程被有限元、有限差分或谱方法离散时出现的大型线性或非线性代数方程组的并行解。在每个迭代步骤中，近似地求解局部问题，这些局部问题表示原始问题对潜在的大量子区域的限制。子区域可以分配给并行计算机的各个处理器，形成问题的整个域的分解。此外，粗子问题的包含大大提高了前置条件的效率。在本研究中，将数学分析与算法的设计和数值测试相结合，特别强调了单元和其他高阶有限元方法的研究，以及砂浆有限元等非一致性方法的研究，以及与扭曲子区域或单元相关的困难。

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.