  Over the last two decades, classical Schwarz methods have been extended to
systems of hyperbolic partial differential equations, and it was observed that
the classical Schwarz method can be convergent even without overlap in certain
cases. This is in strong contrast to the behavior of classical Schwarz methods
applied to elliptic problems, for which overlap is essential for convergence.
Over the last decade, optimized Schwarz methods have been developed for
elliptic partial differential equations. These methods use more effective
transmission conditions between subdomains, and are also convergent without
overlap for elliptic problems. We show here why the classical Schwarz method
applied to the hyperbolic problem converges without overlap for Maxwell's
equations. The reason is that the method is equivalent to a simple optimized
Schwarz method for an equivalent elliptic problem. Using this link, we show how
to develop more efficient Schwarz methods than the classical ones for the
Maxwell's equations. We illustrate our findings with numerical results.
