Rigorous quantitative analysis of multigrid, I: constant coefficients two-level cycle with L 2 -norm
Rigorous quantitative analysis of multigrid, I: constant coefficients two-level cycle with L 2 -norm
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DOI：
10.1137/0731087
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发表时间：
1994-12
影响因子：
2.9
通讯作者：
A. Brandt
中科院分区：
文献类型：
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作者：
A. Brandt
任何多重网格循环的精确数值收敛因子都可以通过局部模式（傅立叶）分析来预测。对于在均匀网格离散的一般域中具有分段平滑系数的一般线性椭圆偏微分方程（PDE）系统，证明了在小网格尺寸的限制下，只要在边界处和边界附近对循环进行适当的处理，确实可以获得这些预测因子。事实证明，这种处理所花费的额外计算机工作量可以忽略不计。除了模态分析之外，还引入了粗网格近似（CGA）条件，这对于多重网格算法的正常工作来说是必要且充分的。本部分研究常数系数方程的一个周期内$L_2$收敛性。在续集中[Brandt，多重网格的严格定量分析，II：扩展和实际意义，手稿]扩展被讨论到许多循环（渐近收敛），到具有任意循环类型的更多级别......
Exact numerical convergence factors for any multigrid cycle can be predicted by local mode (Fourier) analysis. For general linear elliptic partial differential equation (PDE) systems with piecewise smooth coefficients in general domains discretized by uniform grids, it is proved that, in the limit of small meshsizes, these predicted factors are indeed obtained, provided the cycle is supplemented with a proper processing at and near the boundaries. That processing, it is proved, costs negligible extra computer work. Apart from mode analysis, a coarse grid approximation (CGA) condition is introduced which is both necessary and sufficient for the multigrid algorithm to work properly. The present part studies the $L_2 $ convergence in one cycle for equations with constant coefficients. In the sequel [Brandt, Rigorous quantitative analysis of multigrid, II: Extensions and practical implications, manuscript] extensions are discussed to many cycles (asymptotic convergence), to more levels with arbitrary cycle ty...