A domain decomposition method of Schwarz waveform relaxation type for singularly perturbed nonlinear parabolic problems

Abstract

We introduce a domain decomposition method of discrete Schwarz waveform relaxation (DSWR) type for a singularly perturbed nonlinear parabolic problem. The method utilizes Shishkin transition parameter for a space–time decomposition of the computational domain. In each subdomain, the problem is discretized using the central differencing and backward difference schemes on a uniform mesh in space and time directions, respectively. Further, the exchange of information between the subdomains is done through the Dirichlet data that leads to optimal convergence. We analyse the convergence of the developed method and show that the method converges very fast for small perturbation parameter and provides uniformly convergent approximations to the solution of the nonlinear problem. Finally, with some numerical experiments, we illustrate our theoretical results.

Keywords:

• Domain decomposition
• nonlinear parabolic problems
• Schwarz waveform relaxation methods
• singularly perturbed
• robust convergence

AMS Subject Classifications:

• 65M55
• 65M06
• 65M15
• 65M12

Acknowledgments

The authors greatly acknowledge the valuable comments and suggestions of the anonymous referees which improved the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the Science and Engineering Research Board (SERB) under Project No. MTR/2017/001036.