Abstract
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.
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