We consider piecewise polynomial discontinuous Galerkin discretizations of boundary integral reformulations of the heat equation. The resulting linear systems are dense and block-lower triangular and hence can be solved by block forward elimination. For the fast evaluation of the history part, the matrix is subdivided into a family of sub-matrices according to the temporal separation. Separated blocks are approximated by Chebyshev interpolation of the heat kernel in time. For the spatial variable, we propose an adaptive cross approximation (ACA) framework to obtain a data-sparse approximation of the entire matrix. We analyse how the ACA tolerance must be adjusted to the temporal separation and present numerical results for a benchmark problem to confirm the theoretical estimates.

翻译：本文研究热方程边界积分重构的分段多项式间断伽勒金离散化方法。所得线性系统具有稠密块下三角结构，因此可通过块前向消元法求解。为实现历史部分的快速计算，矩阵根据时间分离度被划分为一系列子矩阵。分离块通过热核在时间维度上的切比雪夫插值进行近似。针对空间变量，我们提出自适应交叉逼近（ACA）框架以获得全矩阵的数据稀疏近似。我们分析了ACA容差如何随时间分离度进行调整，并通过基准问题的数值结果验证了理论估计。