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.

翻译：本文研究了基于间断伽辽金有限元方法对多能线性玻尔兹曼输运方程进行空间、角度和能量离散化后所产生线性方程组的迭代求解问题。我们重点发展了预条件理查森迭代方法，该方法可视为单能情形下源迭代的推广，并推导了因非精确线性代数求解产生的、以问题相关范数衡量的可计算后验求解器误差界。我们证明了所得迭代格式的收敛性及后验求解器误差估计与离散化参数无关。同时讨论了如何将多能理查森迭代作为广义最小残差法的预条件子。此外，通过合理选择原线性系统的左、右预条件子，基于残差向量欧几里得范数最小化的标准GMRES实现可在每次迭代中提供可计算的后验求解器误差估计。最后，通过光子输运建模的数值算例，验证了多能源迭代与预条件GMRES方法及其对应后验求解器误差估计的有效性。