BIT Numerical Mathematics

, Volume 42, Issue 4, pp 779–795

Overlapping Schwarz Waveform Relaxation for the Heat Equation in N Dimensions

  • Martin J. Gander
  • Hongkai Zhao

DOI: 10.1023/A:1021900403785

Cite this article as:
Gander, M.J. & Zhao, H. BIT Numerical Mathematics (2002) 42: 779. doi:10.1023/A:1021900403785


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.

Domain decompositionwaveform relaxationSchwarz method for parabolic problemssuperlinear convergence

Copyright information

© Swets & Zeitlinger 2002

Authors and Affiliations

  • Martin J. Gander
    • 1
  • Hongkai Zhao
    • 2
  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Department of MathematicsUniversity of CaliforniaIrvineUSA