In this paper we examine the use of low-rank approximations for the handling of radiation boundary conditions in a transient heat equation given a cavity radiation setting. The finite element discretization that arises from cavity radiation is well known to be dense, which poses difficulties for efficiency and scalability of solvers. Here we consider a special treatment of the cavity radiation discretization using a block low-rank approximation combined with hierarchical matrices. We provide an overview of the methodology and discusses techniques that can be used to improve efficiency within the framework of hierarchical matrices, including the usage of the approximate cross approximation (ACA) method. We provide a number of numerical results that demonstrate the accuracy and efficiency of the approach in practical problems, and demonstrate significant speedup and memory reduction compared to the more conventional "dense matrix" approach.

翻译：本文研究了在瞬态热传导方程中，针对腔体辐射条件下边界条件处理过程中低秩逼近方法的应用。众所周知，腔体辐射引起的有限元离散矩阵是稠密的，这对求解器的效率和可扩展性构成了挑战。本文采用一种将块低秩逼近与层次化矩阵相结合的特殊处理方法处理腔体辐射离散问题。我们概述了该方法论，并讨论了可在层次化矩阵框架内提升效率的技术，包括使用近似交叉逼近方法。我们提供了若干数值结果，展示了该方法在实际问题中的准确性和效率，并证明了相较于传统的"稠密矩阵"方法，该方法在显著加速计算和减少内存占用方面的优势。