  We examine the use of a two-level deflation preconditioner combined with
GMRES to locally solve the subdomain systems arising from applying domain
decomposition methods to Helmholtz problems. Our results show that the direct
solution method can be replaced with an iterative approach. This will be
particularly important when solving large 3D high-frequency problems as
subdomain problems can be too large for direct inversion or otherwise become
inefficient. We additionally show that, even with a relatively low tolerance,
inexact solution of the subdomain systems does not lead to a drastic increase
in the number of outer iterations. As a result, it is promising that a
combination of a two-level domain decomposition preconditioner with inexact
subdomain solves could provide more economical and memory efficient numerical
solutions to large-scale Helmholtz problems.
