  The discretization of the double-layer potential integral equation for the
interior Dirichlet Laplace problem in a domain with smooth boundary results in
a linear system that has a bounded condition number. Thus, the number of
iterations required for the convergence of a Krylov method is, asymptotically,
independent of the discretization size $N$. Using the Fast Multipole Method
(FMM) to accelerate the matrix-vector products, we obtain an optimal
$\mathcal{O}(N)$ solver. In practice, however, when the geometry is
complicated, the number of Krylov iterations can be quite large---to the extend
that necessitates the use of preconditioning.
  We summarize the different methodologies that have appeared in the literature
(single-grid, multigrid, approximate sparse inverses) and we propose a new
class of preconditioners based on an FMM-based spatial decomposition of the
double-layer operator. We present an experimental study in which we compare the
different approaches and we discuss the merits and shortcomings of our
approach. Our method can be easily extended to other second-kind integral
equations with non-oscillatory kernels in two and three dimensions.
