Schwarz preconditioner with $H_k$-GenEO coarse space for the indefinite Helmholtz problem

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.