  GenEO (`Generalised Eigenvalue problems on the Overlap') is a method from the
family of spectral coarse spaces that can efficiently rely on local eigensolves
in order to build a robust parallel domain decomposition preconditioner for
elliptic PDEs. When used as a preconditioner in a conjugate gradient, this
method is extremely efficient in the positive-definite case, yielding an
iteration count completely independent of the number of subdomains and
heterogeneity. In a previous work this theory was extended to the cased of
convection--diffusion--reaction problems, which may be non-self-adjoint and
indefinite, and whose discretisations are solved with preconditioned GMRES. The
GenEO coarse space was then defined here using a generalised eigenvalue problem
based on a self-adjoint and positive definite subproblem. The resulting method,
called $\Delta$-GenEO becomes robust with respect to the variation of the
coefficient of the diffusion term in the operator and depends only very mildly
on variations of the other coefficients. However, the iteration number
estimates get worse as the non-self-adjointness and indefiniteness of the
operator increases, which is often the case for the high frequency Helmholtz
problems. In this work, we will improve on this aspect by introducing a new
version, called $H_k$-GenEO, which uses a generalised eigenvalue problem based
directly on the indefinite operator which will lead to a robust method with
respect to the increase in the wave-number. We provide theoretical estimates
showing the dependence of the size of the coarse space on the wave-number.
