  We present a preconditioning method for the linear systems arising from the
boundary element discretization of the Laplace hypersingular equation on a
$2$-dimensional triangulated surface $\Gamma$ in $\mathbb{R}^3$. We allow
$\Gamma$ to belong to a large class of geometries that we call polygonal
multiscreens, which can be non-manifold. After introducing a new, simple
conforming Galerkin discretization, we analyze a substructuring
domain-decomposition preconditioner based on ideas originally developed for the
Finite Element Method. The surface $\Gamma$ is subdivided into non-overlapping
regions, and the application of the preconditioner is obtained via the solution
of the hypersingular equation on each patch, plus a coarse subspace correction.
We prove that the condition number of the preconditioned linear system grows
poly-logarithmically with $H/h$, the ratio of the coarse mesh and fine mesh
size, and our numerical results indicate that this bound is sharp. This
domain-decomposition algorithm therefore guarantees significant speedups for
iterative solvers, even when a large number of subdomains is used.
