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Mathematical Problems in Engineering
Volume 2015, Article ID 612862, 11 pages
http://dx.doi.org/10.1155/2015/612862
Research Article

A Convergence Study of Multisubdomain Schwarz Waveform Relaxation for a Class of Nonlinear Problems

School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China

Received 17 March 2015; Revised 16 June 2015; Accepted 7 July 2015

Academic Editor: Kyandoghere Kyamakya

Copyright © 2015 Liping Zhang and Shu-Lin Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Schwarz waveform relaxation (SWR) is a new type of domain decomposition methods, which is suited for solving time-dependent PDEs in parallel manner. The number of subdomains, namely, , has a significant influence on the convergence rate. For the representative nonlinear problem , convergence behavior of the algorithm in the two-subdomain case is well-understood. However, for the multisubdomain case (i.e., ), the existing results can only predict convergence when . Therefore, there is a gap between and . In this paper, we try to finish this gap. Precisely, for a specified subdomain number , we find that there exists a quantity such that convergence of the algorithm on unbounded time domains is guaranteed if . The quantity depends on and we present concise formula to calculate it. We show that the analysis is useful to study more complicated PDEs. Numerical results are provided to support the theoretical predictions.