  In this article we consider the iterative solution of the linear system of
equations arising from the discretisation of the poly-energetic linear
Boltzmann transport equation using a discontinuous Galerkin finite element
approximation in space, angle, and energy. In particular, we develop
preconditioned Richardson iterations which may be understood as generalisations
of source iteration in the mono-energetic setting, and derive computable a
posteriori bounds for the solver error incurred due to inexact linear algebra,
measured in a relevant problem-specific norm. We prove that the convergence of
the resulting schemes and a posteriori solver error estimates are independent
of the discretisation parameters. We also discuss how the poly-energetic
Richardson iteration may be employed as a preconditioner for the generalised
minimal residual (GMRES) method. Furthermore, we show that standard
implementations of GMRES based on minimising the Euclidean norm of the residual
vector can be utilized to yield computable a posteriori solver error estimates
at each iteration, through judicious selections of left- and
right-preconditioners for the original linear system. The effectiveness of
poly-energetic source iteration and preconditioned GMRES, as well as their
respective a posteriori solver error estimates, is demonstrated through
numerical examples arising in the modelling of photon transport.
