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We present a preconditioning method for the linear systems arising from theboundary 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 polygonalmultiscreens, which can be non-manifold. After introducing a new, simpleconforming Galerkin discretization, we analyze a substructuringdomain-decomposition preconditioner based on ideas originally developed for theFinite Element Method. The surface $\Gamma$ is subdivided into non-overlappingregions, and the application of the preconditioner is obtained via the solutionof the hypersingular equation on each patch, plus a coarse subspace correction.We prove that the condition number of the preconditioned linear system growspoly-logarithmically with $H/h$, the ratio of the coarse mesh and fine meshsize, and our numerical results indicate that this bound is sharp. Thisdomain-decomposition algorithm therefore guarantees significant speedups foriterative solvers, even when a large number of subdomains is used.

Boundary Element Methods for the Laplace Hypersingular Integral Equation on Multiscreens: a two-level Substructuring Preconditioner