  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.
