Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 612862, 11 pages

http://dx.doi.org/10.1155/2015/612862

## 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.

#### 1. Introduction

Let be a bounded spatial domain of interest. We are interested in the Schwarz waveform relaxation (SWR) algorithm applied to compute solution of the initial-boundary value problem (IBVP):where denotes a function which in general depends in a nonlinear manner on . This is a fundamental model for analyzing the convergence properties of the SWR algorithm and some important results are revisited as follows.

Gander [1] studied the SWR algorithm on bounded and unbounded time intervals in the two-subdomain case. Particularly, the author proved linear convergence of the algorithm on unbounded time intervals, if the derivative of can be bounded from above by a constant , which satisfies (other related or similar studies can be found in [2–4]). In the case of subdomains with , Gander and Stuart [5] analyzed the convergence behavior of the SWR algorithm for the linear heat equation on unbounded time intervals. It was shown that the convergence rate depends on and deteriorates as increases. For IBVP (1) with and , the work in [6] can be generalized to obtain a similar convergence result in the case of . In summary, in the multisubdomain case, the convergence behavior of the SWR algorithm for (1) on unbounded time domains is well-understood, when . For and , however, we know nothing up to now.

In this paper, we try to finish this gap. After a brief description of the multisubdomain SWR algorithm in Section 2, we perform a convergence analysis for the multisubdomain SWR algorithm in Section 3. For given , we present concise formula to calculate the allowed upper bound of , namely, , which guarantees convergence of the algorithm on unbounded time domains. We show that the analysis for (1) can be used to study the multisubdomain domain decomposition methods [7, 8] for more complicated PDEs: . Section 4 provides numerical results to support the theoretical prediction and we finish this paper by giving some concluding remarks in Section 5.

#### 2. The Schwarz Waveform Relaxation Algorithm

For the initial-boundary value problem (IBVP) (1), we decompose the whole space domain into subdomains: , where , , , and for . We assume that so that all the subdomains overlap but domains which are not adjacent do not overlap, as shown in Figure 1. Then, the -subdomain SWR algorithm for IBVP (1) can be written aswhere and for all . Let () be the error function at the th iteration. Then, we havewhere we have used the remainder term in Taylor’s expansion for some function which lies between and . In (3), and for all . Following in this section, we define , , and .