Theory and Applications of Multigrid and Domain Decomposition Methods

多重网格和域分解方法的理论与应用

• 批准号: 0074246
• 负责人: Susanne Brenner
• 金额: $ 9.85万
• 依托单位: University of South Carolina at Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-01 至 2004-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0074246&HistoricalAwards=false
• 关键词: Theory; Applications; Multigrid; Domain; Decomposition

There are three areas of research in the project ``Theory and Applications of Multigrid and Domain Decomposition Methods''. The first area concerns the additive multigrid convergence theory, which is effective in analyzing the asymptotic behavior of the contraction numbers of multigrid algorithms with respect to the number of smoothing steps for boundary value problems with less than full elliptic regularity. The seminal result obtained by the PI for V-cycle algorithm with Richardson relaxation as the smoother will be extended to general smoothers, nonconforming finite elements, the mixed formulation and the F-cycle algorithm. The second area involves multigrid methods for stress intensity factors and singular solutions. These are methods that can take advantage of the form of the solution of the boundary value problem in the regions where it does not have full elliptic regularity. Using this approach, usual quasi-optimal convergence rates have been obtained for simple finite elements on simple grids for boundary value problems on two-dimensional domains with reentrant corners or cracks. The technically more challenging interface problems and three dimensional problems will be investigated in this project. The third area is the analysis of the Finite Element Tearing and Interconnecting (FETI) method, a nonoverlapping domain decomposition method in which the (pseudo) inverse of the Schur complement matrix has to be preconditioned. The goal of this part of the project is to carry out an analysis of the FETI method and some of its variants within the framework of additive Schwarz preconditioners, and to investigate new mechanisms for the global communication among the subdomains. The methods analyzed in this project are efficient algorithms for the numerical solution of partial differential equations. Such equations are extremely important in science and engineering since they are the governing equations for most physical phenomena. Part of the research involves the fast computation of stress intensity factors, which are essential indicators in fracture mechanics. The FETI method to be studied in this project has already been implemented for large scale engineering problems using parallel supercomputers with up to a thousand processors. The advances resulting from this project will therefore have an impact on many areas of science and technology, such as aerospace engineering, fracture prediction and fluid flow problems.

该项目中有三个研究领域``Multigrid和域分解方法的理论和应用''。 第一个领域涉及添加剂多族收敛理论，该理论有效地分析了跨越算法的收缩数量相对于针对边界价值问题的平滑步骤的数量，而椭圆时期的规律性小。 PI通过Richardson弛豫获得的V-Cycle算法获得的开创性结果将扩展到普通Smoother，不合格的有限元，混合配方和F-Cycle算法。 第二个区域涉及用于应力强度因子和奇异溶液的多机方法。 这些方法可以利用没有完全椭圆规则性的区域中边界值问题的解决方案的形式。 使用这种方法，对于简单网格上的简单有限元素，已经获得了通常的准最佳收敛速率，以解决具有重进入角或裂缝的二维域上的边界价值问题。 在该项目中将研究技术更具挑战性的界面问题和三维问题。 第三个区域是对有限元撕裂和互连方法（FETI）方法的分析，这是一种非重叠域分解方法，其中（伪）必须对Schur补体矩阵的（伪）进行预处理。 该项目的这一部分的目的是对feti方法进行分析及其在添加剂Schwarz先验者框架内的某些变体，并研究子域之间全球通信的新机制。 该项目中分析的方法是偏微分方程数值解的有效算法。 这些方程在科学和工程中非常重要，因为它们是大多数物理现象的管理方程式。研究的一部分涉及应力强度因子的快速计算，这是断裂力学中的重要指标。 在该项目中要研究的FETI方法已经针对具有多达一千个处理器的平行超级计算机实施了大规模工程问题。 因此，该项目的进步将对许多科学技术领域产生影响，例如航空工程，断裂预测和流体流动问题。