Theory and Applications of Multigrid

本课题主要研究多网格方法的理论与应用。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、非对称问题和不确定问题，以及新的离散化技术，如砂浆有限元方法和不连续伽辽金方法。另一个目标是将PI的奇异解和应力强度因子多重网格方法扩展到更复杂的问题和三维空间。这种新的多重网格方法可以恢复简单网格上简单有限元的最优收敛速度，即使在存在由非光滑几何、边界条件突变或偏微分方程系数跳跃引起的强奇点的情况下。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。多网格解的误差与最小可能误差相当，同时多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优的复杂性，它们（单独使用或与其他方法结合使用）是大规模科学计算的强大引擎。该项目的研究结果可以回答多网格方法可靠性这一重要问题，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提供有用的计算工具。

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.