Convergence Analysis of Two-Level Space-Time Additive Schwarz Method for Parabolic Equations
Convergence Analysis of Two-Level Space-Time Additive Schwarz Method for Parabolic Equations
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DOI：
10.1137/140993776
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发表时间：
2015-12
期刊：
影响因子：
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通讯作者：
Shishun Li;X. Cai
中科院分区：
文献类型：
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作者：
Shishun Li;X. Cai
加性Schwarz （additive Schwarz， AS）方法最初是为预处理椭圆型问题而设计的，最近该方法被成功地推广为求解抛物型问题的耦合时空预处理方法。然而，现有的加性Schwarz方法理论并不能直接应用于时空离散问题。本文建立了两能级时空加性Schwarz预条件的最优收敛理论。我们获得了AS预条件算子的谱的下界和上界，以及它们对空间和时间上的细/粗网格大小、子域数量和窗口大小（即耦合时间步数）的依赖关系。一些数值实验报告证实了这一理论。
The additive Schwarz (AS) method was originally designed for preconditioning elliptic problems, and recently the method was extended successfully as a coupled space-time preconditioner for parabolic problems. However, the existing theory for the additive Schwarz method doesn't apply directly to the space-time discretized problems. In this paper, we develop an optimal convergence theory for the two-level space-time additive Schwarz preconditioner. We obtain lower and upper bounds for the spectrum of the AS preconditioned operator and their dependency on the fine/coarse mesh sizes in space and time, the number of subdomains, and the window size (i.e., the number of coupled time steps). Some numerical experiments are reported to confirm the theory.