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

Abstract EN

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, N , has a significant influence on the convergence rate.

For the representative nonlinear problem ∂ t u = ∂ x x u + f ( u ) , convergence behavior of the algorithm in the two-subdomain case is well-understood.

However, for the multisubdomain case (i.e., N ≥ 3 ), the existing results can only predict convergence when f ′ ( u ) ≤ 0 ( ∀ u ∈ R ) .

Therefore, there is a gap between N ≥ 3 and f ′ ( u ) > 0 .

In this paper, we try to finish this gap.

Precisely, for a specified subdomain number N , we find that there exists a quantity d max such that convergence of the algorithm on unbounded time domains is guaranteed if f ′ ( u ) ≤ d max ( ∀ u ∈ R ) .

The quantity d max depends on N 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.