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.

翻译：我们针对二维三角曲面 $\Gamma \subset \mathbb{R}^3$ 上拉普拉斯超奇异方程的边界元离散线性系统提出一种预条件方法。允许 $\Gamma$ 属于一大类称为多边形多屏的几何结构（可非流形）。在引入一种新的简单协调伽辽金离散格式后，我们分析了一种基于有限元方法思想的子结构区域分解预条件器。将曲面 $\Gamma$ 细分为非重叠区域，通过求解每个子块上的超奇异方程并附加粗空间校正得到预条件算子。证明了预条件系统条件数以 $H/h$（粗细网格尺寸比）的多对数增长，数值实验表明该估计是紧的。因此，即使在使用大量子域时，该区域分解算法也能显著加速迭代求解器。