Theory and Applications of Multigrid

多重网格理论与应用

• 批准号: 0738028
• 负责人: Susanne Brenner
• 金额: $ 0.58万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-10-15 至 2009-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0738028&HistoricalAwards=false
• 关键词: Theory; Applications; Multigrid

The research in this project is on the theory and applications of multigrid methods. One of the goals is to generalize the investigator's additive multigrid theory, which can handle the convergence of V-cycle and F-cycle algorithms for nonconforming methods, to more difficult problems such as anisotropic problems, nonsymmetric problems and indefinite problems, and to new discretization techniques such as mortar finite element methods and discontinuous Galerkin methods. Another goal is to extend the PI's multigrid method for singular solutions and stress intensity factors to more complicated problems and to three dimensions. This new multigrid approach can recover the optimal convergence rates of simple finite elements on simple grids, even in the presence of strong singularities caused by nonsmooth geometries, abrupt changes in boundary conditions, or jumps in the coefficients of partial differential equations. It can also take full advantage of superconvergence phenomena, extrapolation techniques, and parallel implementations.Multigrid methods can produce fast solutions to large systems of equations. The errors of multigrid solutions are comparable to the smallest possible errors and at the same time the computational cost of multigrid methods is proportional to the number of unknowns. Therefore multigrid methods have optimal complexity, and they (either on their own or combined with other methods) are powerful engines for large scale scientific computations. The results of this project can provide answers to the important question of the reliability of multigrid methods and provide guidelines for the development of new algorithms. They will also generate useful computational tools for many challenging problems in material science, fracture mechanics, fluid flow and electromagnetism.

本课题的研究内容是多重网格法的理论与应用。其中一个目标是将研究者的加性多重网格理论推广到更困难的问题，如各向异性问题、非对称问题和不定问题，以及新的离散化技术，如Morar有限元方法和间断Galerkin方法。另一个目标是将PI的奇异解和应力强度因子多重网格法推广到更复杂的问题和三维问题。这种新的多重网格方法可以在简单网格上恢复简单有限元的最优收敛速度，即使在由于几何形状不光滑、边界条件突变或偏微分方程组系数跳跃引起的强奇异性存在的情况下也是如此。它还可以充分利用超收敛现象、外推技术和并行实现。多重网格法可以快速求解大型方程组。多重网格法的误差与最小误差相当，同时多重网格法的计算量与未知数成正比。因此，多重网格方法具有最佳的复杂性，它们(无论是单独使用还是与其他方法结合使用)都是大规模科学计算的强大引擎。该项目的结果可以为多重网格法的可靠性这一重要问题提供答案，并为新算法的开发提供指导。它们还将为材料科学、断裂力学、流体流动和电磁学中的许多具有挑战性的问题生成有用的计算工具。