  Domain decomposition (DD) methods are a natural way to take advantage of
parallel computers when solving large scale linear systems. Their scalability
depends on the design of the coarse space used in the two-level method. The
analysis of adaptive coarse spaces we present here is quite general since it
applies to symmetric and non symmetric problems, to symmetric preconditioners
such the additive Schwarz method (ASM) and to the non-symmetric preconditioner
restricted additive Schwarz (RAS), as well as to exact or inexact subdomain
solves. The coarse space is built by solving generalized eigenvalues in the
subdomains and applying a well-chosen operator to the selected eigenvectors.
