Iterative Substructuring Methods for Elliptic Problems

大型代数系统的数值方法的发展是计算流体力学、弹性力学和其他连续介质力学核心问题的有效程序开发的核心。这类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，代数系统解算器的重要性正在增加，这些系统具有大量的快速处理器，每个处理器都具有相对较大的内存。迭代子结构和其他区域分解算法的一个非常理想的特征是它们尊重现代并行和分布式计算系统的存储层次结构，这对于接近峰值浮点性能是必不可少的。改进方法的发展，加上功能更强大的计算机系统，使得以相当高的分辨率相对容易地进行三维模拟成为可能。这项工作现在得到了高质量的软件系统的支持，例如Argonne的PETSC库，它将促进代码开发以及访问各种并行和分布式计算机系统。对于日益困难的偏微分方程，将继续开发迭代子结构和其他区域分解方法。区域分解算法是一种迭代方法，通常是预条件共轭梯度类型的迭代方法，用于并行求解大型线性或非线性代数方程组，这些代数方程组是在偏微分方程组被有限元、有限差分或谱方法离散时产生的。在每个迭代步骤中，表示原始问题对潜在的大量子区域的限制的局部问题被近似地解决。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，包含一个粗略的问题通常会大大提高预处理器的效率。这项研究将把数学分析与算法的设计和数值测试结合起来。将特别强调谱单元和其他高阶有限元方法的研究，以及非协调方法，如Morar和Nedelec有限元方法。后者是为麦克斯韦方程发展起来的。还将重点讨论弹性有限元近似中经常出现的非常病态的问题。还将为亥姆霍兹方程和其他时间调和模型开发迭代方法和生成代码，例如麦克斯韦方程。这些问题对迭代求解器的开发提出了非常现实的挑战，并且在许多工程应用中也具有重要意义。

The development of numerical methods for large algebraic systems is central in the development of efficient codes for computational 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. A very desirable feature of iterative substructuring and other domain decomposition algorithms is that they respect the memory hierarchy of modern parallel and distributed computing systems, which is essential for approaching peak floating point performance. The development of improved methods is, together with more powerful computer systems, making it possible to carry out simulations in three dimensions, with quite high resolution, relatively easily. This work is now supported by high quality software systems, such as Argonne's PETSC library, which will facilitate code development as well as the access to a variety of parallel and distributed computer systems. Work willcontinue in developing iterative substructuring and other domain decomposition methods for increasingly difficult partial differential equations. Domain decomposition algorithms are iterative methods, often of preconditioned 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 proble m often substantially increases the efficiency of the preconditioner. This study will combine mathematical analysis with the design and numerical testing of algorithms. A special emphasis will be placed on the study of spectral elements and other high order finite element methods, as well as on nonconforming methods such as the mortar and Nedelec finite element methods. The latter have been developed for Maxwell's equation. There will also be a focus on the often very ill-conditioned problems which arise in finite element approximations of elasticity. Iterative methods and production codes will also be developed for Helmholtz's equation and other time-harmonic models arising, e.g., from Maxwell's equations. These problems pose very real challenges for the development of iterative solvers and are also of great importance in a number of engineering applications.