Improvements to the Theoretical Estimates of the Schwarz Preconditioner with Δ-GenEO Coarse Space for the Indefinite Helmholtz Problem

GenEO (`Generalised Eigenvalue problems on the Overlap') is a method from thefamily of spectral coarse spaces that can efficiently rely on local eigensolvesin order to build a robust parallel domain decomposition preconditioner forelliptic PDEs. When used as a preconditioner in a conjugate gradient, thismethod is extremely efficient in the positive-definite case, yielding aniteration count completely independent of the number of subdomains andheterogeneity. In a previous work this theory was extended to the cased ofconvection–diffusion–reaction problems, which may be non-self-adjoint andindefinite, and whose discretisations are solved with preconditioned GMRES. TheGenEO coarse space was then defined here using a generalised eigenvalue problembased on a self-adjoint and positive definite subproblem. The resulting method,called Δ-GenEO becomes robust with respect to the variation of thecoefficient of the diffusion term in the operator and depends only very mildlyon variations of the other coefficients. However, the iteration numberestimates get worse as the non-self-adjointness and indefiniteness of theoperator increases, which is often the case for the high frequency Helmholtzproblems. In this work, we will improve on this aspect by introducing a newversion, called H_k-GenEO, which uses a generalised eigenvalue problem baseddirectly on the indefinite operator which will lead to a robust method withrespect to the increase in the wave-number. We provide theoretical estimatesshowing the dependence of the size of the coarse space on the wave-number.