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The discretization of the double-layer potential integral equation for theinterior Dirichlet Laplace problem in a domain with smooth boundary results ina linear system that has a bounded condition number. Thus, the number ofiterations 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 iscomplicated, the number of Krylov iterations can be quite large---to the extendthat 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 newclass of preconditioners based on an FMM-based spatial decomposition of thedouble-layer operator. We present an experimental study in which we compare thedifferent approaches and we discuss the merits and shortcomings of ourapproach. Our method can be easily extended to other second-kind integralequations with non-oscillatory kernels in two and three dimensions.

On preconditioners for the Laplace double-layer in 2D