  In this paper we are concerned with restricted additive Schwarz with local
impedance transformation conditions for a family of Helmholtz problems in two
dimensions. These problems are discretized by the finite element method with
conforming nodal finite elements. We design and analyze a new adaptive coarse
space for this kind of restricted additive Schwarz method. This coarse space is
spanned by some eigenvalue functions of local generalized eigenvalue problems,
which are defined by weighted positive semi-definite bilinear forms on
subspaces consisting of local discrete Helmholtz-harmonic functions from
impedance boundary data. We proved that a two-level hybrid Schwarz
preconditioner with the proposed coarse space possesses uniformly convergence
independent of the mesh size, the subdomain size and the wave numbers under
suitable assumptions. We also introduce an economic coarse space to avoid
solving generalized eigenvalue problems. Numerical experiments confirm the
theoretical results.
