Time Domain Maxwell Equations Solved with Schwarz Waveform Relaxation Methods

It is very natural to solve time dependent problems with Domain Decomposition Methods by using an implicit scheme for the time variable and then applying a classical iterative domain decomposition method at each time step. This is however not what the Schwarz Waveform Relaxation (SWR) methods do. The SWR methods are a combination of the Schwarz Domain Decomposition methods, see Schwarz [1870], and the Waveform Relaxation algorithm, see Lelarasmee et al. [1982]. Combined, one obtains a new method which decomposes the domain into subdomains on which time dependent problems are solved. Iterations are then introduced, where communication between subdomains is done at artificial interfaces along the whole time window. This new approach has been introduced by Bjørhus [1995] for hyperbolic problems with Dirichlet boundary conditions and was analyzed for the heat equation by Gander and Stuart [1998]. Giladi and Keller [2002] analyzed this same approach applied to the advection diffusion equation with constant coefficients. For the wave equation and SWR see Gander and Halpern [2001] in which they treat the one-dimensional case with overlapping subdomains and for the n-dimensional case Gander and Halpern [2005], again with overlap. In this paper, we analyze for the first time the SWR algorithm applied to the time domain Maxwell equations.