  Schwarz methods use a decomposition of the computational domain into
subdomains and need to put boundary conditions on the subdomain boundaries. In
domain truncation one restricts the unbounded domain to a bounded computational
domain and also needs to put boundary conditions on the computational domain
boundaries. It turns out to be fruitful to think of the domain decomposition in
Schwarz methods as truncation of the domain onto subdomains. The first truly
optimal Schwarz method that converges in a finite number of steps was proposed
in 1994 and used precisely transparent boundary conditions as transmission
conditions between subdomains. Approximating these transparent boundary
conditions for fast convergence of Schwarz methods led to the development of
optimized Schwarz methods -- a name that has become common for Schwarz methods
based on domain truncation. Compared to classical Schwarz methods which use
simple Dirichlet transmission conditions and have been successfully used in a
wide range of applications, optimized Schwarz methods are much less well
understood, mainly due to their more sophisticated transmission conditions.
This present situation is the motivation for our survey: to give a
comprehensive review and precise exploration of convergence behaviors of
optimized Schwarz methods based on Fourier analysis taking into account the
original boundary conditions, many subdomain decompositions and layered media.
The transmission conditions we study include the lowest order absorbing
conditions (Robin), and also more advanced perfectly matched layers (PML), both
developed first for domain truncation.
