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.

该项目的研究涉及多格里德方法的理论和应用。 目标之一是将研究者的添加剂多移民理论推广，该理论可以处理对不合格方法的V周期和F周期算法的融合，以解决诸如各种问题，非对称问题，非对称问题和无限问题的问题，以及不确定的问题，以及不适用的问题，以及不适合新的既定技术，例如新的iCtaritization extiqualization extiqualization intar firation firation Elemente Elemente Elemente Methite方法和脱节方法。 另一个目标是将PI的奇异解决方案和应力强度因子的多族方法扩展到更复杂的问题和三个维度。 这种新型的多人方法可以恢复简单网格上简单有限元的最佳收敛速率，即使在存在非平滑几何形状，边界条件的突然变化或部分微分方程的系数中的突然变化引起的强烈奇异性。 它还可以充分利用超融合现象，外推技术和并行实现。多族式方法可以为大型方程式生成快速解决方案。 多机溶液的误差与最小可能的错误相媲美，同时，多式方法的计算成本与未知数的数量成正比。 因此，多机方法具有最佳的复杂性，并且它们（单独或与其他方法结合在一起）是用于大规模科学计算的强大引擎。 该项目的结果可以提供有关多格里德方法可靠性的重要问题的答案，并为开发新算法提供指南。 它们还将为材料科学，断裂力学，流体流和电磁作挑战性问题的许多具有挑战性的问题生成有用的计算工具。