  The aim of this paper is to develop stable and accurate numerical schemes for
boundary integral formulations of the heat equation with Dirichlet boundary
conditions. The accuracy of Galerkin discretisations for the resulting boundary
integral formulations depends mainly on the choice of discretisation space. We
develop a-priori error analysis utilising a proof technique that involves norm
equivalences in hierarchical wavelet subspace decompositions. We apply this to
a full tensor product discretisation, showing improvements over existing
results, particularly for discretisation spaces having low polynomial degrees.
We then use the norm equivalences to show that an anisotropic sparse grid
discretisation yields even higher convergence rates. Finally, a simple adaptive
scheme is proposed to suggest an optimal shape for the sparse grid index sets.
