Volume 57, Number 4, July-August 2023
|2397 - 2426
|18 July 2023
Fully discrete Schwarz waveform relaxation analysis for the heat equation on a finite spatial domain
Department of Mathematics & Statistics, Memorial University of Newfoundland, St. John’s Newfoundland A1C 5S7, Canada
* Corresponding author: firstname.lastname@example.org
Accepted: 28 April 2023
Schwarz waveform relaxation methods provide space-time parallelism for the solution of time dependent partial differential equations. The algorithms are differentiated by the choice of the transmission conditions enforced at the introduced space-time boundaries. Early results considered the theoretical analysis of these algorithms in the continuous and semi-discrete (in space) settings for various families of linear partial differential equations. Later, fully discrete results were obtained under the simplifying assumption of an infinite spatial domain. In this paper, we provide a first analysis of a fully discrete classical Schwarz Waveform algorithm for the one-dimensional heat equation on an arbitrary but finite number of bounded subdomains. The θ-method is chosen as the time integrator. Convergence results are given in both the infinity norm and two norm, with an explicit contraction given in the case of a uniform partitioning. The results are compared to the numerics and to the earlier theoretical results.
Mathematics Subject Classification: 65M55 / 65M12 / 65M15 / 65Y05 / 35K05
Key words: Heat equation / Domain decomposition / Schwarz waveform relaxation
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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