Theory and Applications of Multigrid and Domain Decomposition Methods

“多重网格和区域分解方法的理论和应用”项目有三个研究领域。第一个领域涉及加性多重网格收敛理论，该理论有效地分析了多重网格算法的收缩数相对于小于完全椭圆正则性的边值问题的光顺步数的渐近行为。以Richardson松弛为光滑器的V-循环PI算法的开创性结果将推广到一般光滑器、非协调有限元、混合格式和F-循环算法。第二个领域涉及应力强度因子和奇异解的多重网格法。这些方法可以利用边值问题在不具有完全椭圆正则性的区域中的解的形式。利用这种方法，我们得到了简单网格上的简单有限元在具有凹角或裂纹的二维区域上的边值问题的通常的准最优收敛速度。该项目将研究技术上更具挑战性的界面问题和三维问题。第三个领域是有限元撕裂和互连(FETI)方法的分析，这是一种非重叠区域分解方法，其中必须对Schur补矩阵的(伪)逆进行预条件处理。该项目这一部分的目标是在加性Schwarz预条件子的框架内对FETI方法及其一些变体进行分析，并研究子域之间全局通信的新机制。本项目所分析的方法是求解偏微分方程组的有效算法。这种方程在科学和工程中极其重要，因为它们是大多数物理现象的控制方程。部分研究涉及到应力强度因子的快速计算，这是断裂力学中的基本指标。本项目中研究的FETI方法已经在使用多达1000台处理器的并行超级计算机上实现了大规模工程问题。因此，该项目取得的进展将对许多科学和技术领域产生影响，如航空航天工程、裂缝预测和流体流动问题。

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.