Theory and Applications of Multigrid

多重网格理论与应用

• 批准号: 0311790
• 负责人: Susanne Brenner
• 金额: $ 11.26万
• 依托单位: University South Carolina Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-09-01 至 2007-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0311790&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 ofsuperconvergence 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.

本项目主要研究多重网格方法的理论和应用。 其中一个目标是推广研究者的加法多重网格理论，它可以处理收敛的V-循环和F-循环算法的迭代方法，更困难的问题，如各向异性问题，非对称问题和不确定的问题，和新的离散技术，如砂浆有限元方法和间断伽辽金方法。 另一个目标是扩展PI的多重网格法奇异的解决方案和应力强度因子更复杂的问题和三维。 这种新的多重网格方法可以恢复简单的有限元在简单的网格上的最佳收敛速度，即使在存在由非光滑的几何形状，边界条件的突变，或跳跃的偏微分方程的系数引起的强奇异性。 多重网格方法还可以充分利用超收敛现象、外推技术和并行实现。多重网格方法可以快速求解大型方程组。 多重网格方法的误差与最小可能误差相当，同时多重网格方法的计算成本与未知数的数量成正比。 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方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产生有用的计算工具。