Abstract
Schwarz waveform relaxation methods are Schwarz methods applied to evolution problems. Like for steady problems, they are based on an overlapping domain decomposition of the spatial domain, and an iteration which only requires subdomain solutions, now in space-time, to get better and better approximations of the global, monodomain solution. Fourier analysis has been used to study the convergence of both Schwarz and Schwarz waveform relaxation methods. We show here that their convergence is however quite different: for steady problems of diffusive type, Schwarz methods converge linearly, which is also well predicted by Fourier analysis. For a time dependent heat equation however, the Schwarz waveform relaxation algorithm first has a rapid convergence phase, followed by a slow down, and eventually convergence increases again to become superlinear, none of which is predicted by classical Fourier analysis. Introducing a new Fourier analysis combined with kernel estimates, we can explain this behavior for the heat equation. We then generalize our approach to the case of advection reaction diffusion problems. We illustrate all our results with numerical experiments.
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