  This paper analyzes the Krylov convergence rate of a Helmholtz problem
preconditioned with Multigrid. The multigrid method is applied to the Helmholtz
problem formulated on a complex contour and uses GMRES as a smoother substitute
at each level. A one-dimensional model is analyzed both in a continuous and
discrete way. It is shown that the Krylov convergence rate of the continuous
problem is independent of the wave number. The discrete problem, however, can
deviate significantly from this bound due to a pitchfork in the spectrum. It is
further shown in numerical experiments that the convergence rate of the Krylov
method approaches the continuous bound as the grid distance $h$ gets small.

Numerical Analysis