  This paper is concerned with resolvent estimates on the real axis for the
Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet
boundary conditions when the obstacle is trapping. There are two resolvent
estimates for this situation currently in the literature: (i) in the case of
elliptic trapping the general "worst case" bound of exponential growth applies,
and examples show that this growth can be realised through some sequence of
wavenumbers, (ii) in the prototypical case of hyperbolic trapping where the
Helmholtz equation is posed in the exterior of two strictly convex obstacles
(or several obstacles with additional constraints) the nontrapping resolvent
estimate holds with a logarithmic loss.
  This paper proves the first resolvent estimate for parabolic trapping by
obstacles, studying a class of obstacles the prototypical example of which is
the exterior of two squares (in 2-d), or two cubes (in 3-d), whose sides are
parallel. We show, via developments of the vector-field/multiplier argument of
Morawetz and the first application of this methodology to trapping
configurations, that a resolvent estimate holds with a polynomial loss over the
nontrapping estimate. We use this bound, along with the other trapping
resolvent estimates, to prove results about integral-equation formulations of
the boundary value problem in the case of trapping. Feeding these bounds into
existing frameworks for analysing finite and boundary element methods, we
obtain the first wavenumber-explicit proofs of convergence for numerical
methods for solving the Helmholtz equation in the exterior of a trapping
obstacle.
