  We consider GMRES applied to discretisations of the high-frequency Helmholtz
equation with strong trapping; recall that in this situation the problem is
exponentially ill-conditioned through an increasing sequence of frequencies.
Under certain assumptions about the distribution of the eigenvalues, we prove
upper bounds on how the number of GMRES iterations grows with the frequency.
Our main focus is on boundary-integral-equation formulations of the exterior
Dirichlet and Neumann obstacle problems in 2- and 3-d; for these problems, we
investigate numerically the sharpness (in terms of dependence on frequency) of
both our bounds and various quantities entering our bounds. This paper is
therefore the first comprehensive study of the frequency-dependence of the
number of GMRES iterations for Helmholtz boundary-integral equations under
trapping.
