Optimal-order nonnested multigrid methods for solving finite element equations. I. On quasi-uniform meshes
Optimal-order nonnested multigrid methods for solving finite element equations. I. On quasi-uniform meshes
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DOI：
10.1090/s0025-5718-1990-1035947-0
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发表时间：
1990-09
影响因子：
2
通讯作者：
Shangyou Zhang
中科院分区：
文献类型：
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作者：
Shangyou Zhang
我们证明了多重网格方法即使在多重网格不嵌套的情况下也能以最优的计算顺序进行计算。当粗网格不是细网格的子网格时，粗级校正通常不具有a(-，-)投影性质，并且确实放大了某些分量的迭代误差。尽管如此，误差的低频分量仍然可以通过粗级校正来捕获。由于通过精细平滑可以有效地抑制(放大的)高频误差，所以仍能保持标准多重网格法的最优工作顺序。然而，与嵌套网格的情况不同，具有一次光顺的非嵌套多重网格法无论是V循环方法还是W循环方法，通常都不收敛。数值结果表明，非嵌套多重网格法的收敛速度不一定比嵌套多重网格法差。由于非嵌套多重网格方法接受相当任意相关的网格，因此我们可以将自适应精化和多重网格算法的效率结合起来。
We prove that the multigrid method works with optimal computational order even when the multiple meshes are not nested. When a coarse mesh is not a submesh of the finer one, the coarse-level correction usually does not have the a(-, -) projection property and does amplify the iterative error in some components. Nevertheless, the low-frequency components of the error can still be caught by the coarse-level correction. Since the (amplified) highfrequency errors will be damped out by the fine-level smoothing efficiently, the optimal work order of the standard multigrid method can still be maintained. However, unlike the case of nested meshes, a nonnested multigrid method with one smoothing does not converge in general, no matter whether it is a V-cycle or a W-cycle method. It is shown numerically that the convergence rates of nonnested multigrid methods are not necessarily worse than those of nested ones. Since nonnested multigrid methods accept quite arbitrarily related meshes, we may then combine the efficiencies of adaptive refinements and of multigrid algorithms.