Iterative Substructuring Methods for Elliptic Problems and Related Algorithms

大型系统数值方法的开发是流体动力学、弹性和连续介质力学其他核心问题的高效代码开发的核心。 此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，代数系统求解器的重要性日益增加，该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。 改进方法的发展将与更强大的计算机系统一起，使得能够相对容易地以相当高的分辨率进行三维模拟。 我们将继续针对日益困难的椭圆问题开发迭代子结构和其他域分解方法。 域分解算法是迭代方法，通常是共轭梯度类型，用于并行求解大型线性或非线性代数方程组，这些代数方程组是通过有限元、有限差分或谱方法离散偏微分方程时出现的。 在每个迭代步骤中，近似解决表示原始问题对潜在大量子区域的限制的局部问题。 可以分配给并行计算机的各个处理器的子区域形成问题的整个域的分解。 此外，包含粗略子问题大大提高了预处理器的效率。 本研究将数学分析与算法设计和数值测试相结合，特别强调单元和其他高阶有限元方法、砂浆有限元等非一致性方法以及与扭曲子区域或单元相关的困难的研究。

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.