  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.
