Overlapping Schwarz Waveform Relaxation for the Heat Equation in N Dimensions
 Martin J. Gander,
 Hongkai Zhao
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Abstract
We analyze overlapping Schwarz waveform relaxation for the heat equation in n spatial dimensions. We prove linear convergence of the algorithm on unbounded time intervals and superlinear convergence on bounded time intervals. In both cases the convergence rates are shown to depend on the size of the overlap. The linear convergence result depends also on the number of subdomains because it is limited by the classical steady state result of overlapping Schwarz for elliptic problems. However the superlinear convergence result is independent of the number of subdomains. Thus overlapping Schwarz waveform relaxation does not need a coarse space for robust convergence independent of the number of subdomains, if the algorithm is in the superlinear convergence regime. Numerical experiments confirm our analysis. We also briefly describe how our results can be extended to more general parabolic problems.
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 Title
 Overlapping Schwarz Waveform Relaxation for the Heat Equation in N Dimensions
 Journal

BIT Numerical Mathematics
Volume 42, Issue 4 , pp 779795
 Cover Date
 20021201
 DOI
 10.1023/A:1021900403785
 Print ISSN
 00063835
 Online ISSN
 15729125
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Domain decomposition
 waveform relaxation
 Schwarz method for parabolic problems
 superlinear convergence
 Industry Sectors
 Authors

 Martin J. Gander ^{(1)}
 Hongkai Zhao ^{(2)}
 Author Affiliations

 1. Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 2K6, Canada
 2. Department of Mathematics, University of California, Irvine, CA, 926973875, USA