  This paper introduces a class of approximate transparent boundary conditions
for the solution of Helmholtz-type resonance and scattering problems on
unbounded domains. The computational domain is assumed to be a polygon. A
detailed description of two variants of the Hardy space infinite element method
which relays on the pole condition is given. The method can treat
waveguide-type inhomogeneities in the domain with non-compact support. The
results of the Hardy space infinite element method are compared to a perfectly
matched layer method. Numerical experiments indicate that the approximation
error of the Hardy space decays exponentially in the number of Hardy space
modes.
