We introduce an $hp$-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and $hp$-version a priori error analysis of the proposed method, by deriving suitable $hp$-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.

翻译：本文提出了一种用于线性玻尔兹曼输运问题的$hp$版本不连续伽辽金有限元方法（DGFEM）。该方法的核心特征在于，在提供任意阶收敛速率的同时，其实现形式几乎与标准多群离散纵标法相同，这意味着可在现有软件中以高精度高效地并行计算解。该方法统一了积分-微分方程的空间、角度和能量域的离散化，并自然地融入局部网格与各计算域内局部多项式阶数的变化。此外，该方法可处理通用多面体单元，从而实现对复杂空间几何问题的高效离散。我们通过推导合适的$hp$逼近估计及新型inf-sup界，研究了所提方法的稳定性及$hp$版本先验误差分析。数值实验展示了该方法在多能及单能问题中的性能表现。