Analysis of a Two-level Schwarz Method with Coarse Spaces Based on Local Dirichlet-to-Neumann Maps
Analysis of a Two-level Schwarz Method with Coarse Spaces Based on Local Dirichlet-to-Neumann Maps
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DOI：
10.2478/cmam-2012-0027
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
V. Dolean;F. Nataf;Robert Scheichl;N. Spillane
中科院分区：
文献类型：
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作者：
V. Dolean;F. Nataf;Robert Scheichl;N. Spillane
摘要为了得到可伸缩的区域分解方法，粗网格校正是一个关键因素。对于光滑问题，这种两层方法的理论和实践已经很成熟，但对于系数变化复杂和对比度较高的问题，情况并非如此。在先前的研究中，两位作者利用子域DTN映射的低频模式引入了一种适用于高度异质系数的粗空间。在这项工作中，我们给出了一个具有这种粗空间的两级重叠加性Schwarz方法的严格分析，它为每个子域需要增加多少个模来获得常系数情形的收敛速度提供了一个自动判据。我们的方法适合于并行实现，并通过对一些具有高度异构性的具有挑战性的自动划分问题的数值例子来验证其有效性。
Abstract Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. For smooth problems, the theory and practice of such two-level methods is well established, but this is not the case for problems with complicated variation and high contrasts in the coefficients. In a previous study, two of the authors introduced a coarse space adapted to highly heterogeneous coefficients using the low frequency modes of the subdomain DtN maps. In this work, we present a rigorous analysis of a two-level overlapping additive Schwarz method with this coarse space, which provides an automatic criterion for the number of modes that need to be added per subdomain to obtain a convergence rate of the order of the constant coefficient case. Our method is suitable for parallel implementation and its efficiency is demonstrated by numerical examples on some challenging problems with high heterogeneities for automatic partitionings.