  We examine the use of the Dirichlet-to-Neumann coarse space within an
additive Schwarz method to solve the Helmholtz equation in 2D. In particular,
we focus on the selection of how many eigenfunctions should go into the coarse
space. We find that wave number independent convergence of a preconditioned
iterative method can be achieved in certain special cases with an appropriate
and novel choice of threshold in the selection criteria. However, this property
is lost in a more general setting, including the heterogeneous problem.
Nonetheless, the approach converges in a small number of iterations for the
homogeneous problem even for relatively large wave numbers and is robust to the
number of subdomains used.
