A numerical scheme is presented for solving the Helmholtz equation with Dirichlet or Neumann boundary conditions on piecewise smooth open curves, where the curves may have corners and multiple junctions. Existing integral equation methods for smooth open curves rely on analyzing the exact singularities of the density at endpoints for associated integral operators, explicitly extracting these singularities from the densities in the formulation, and using global quadrature to discretize the boundary integral equation. Extending these methods to handle curves with corners and multiple junctions is challenging because the singularity analysis becomes much more complex, and constructing high-order quadrature for discretizing layer potentials with singular and hypersingular kernels and singular densities is nontrivial. The proposed scheme is built upon the following two observations. First, the single-layer potential operator and the normal derivative of the double-layer potential operator serve as effective preconditioners for each other locally. Second, the recursively compressed inverse preconditioning (RCIP) method can be extended to address "implicit" second-kind integral equations. The scheme is high-order, adaptive, and capable of handling corners and multiple junctions without prior knowledge of the density singularity. It is also compatible with fast algorithms, such as the fast multipole method. The performance of the scheme is illustrated with several numerical examples.

翻译：本文提出了一种数值格式，用于求解分段光滑开曲线上具有Dirichlet或Neumann边界条件的Helmholtz方程，其中曲线可能包含角点和多重连接点。针对光滑开曲线的现有积分方程方法依赖于分析关联积分算子在端点处密度的精确奇异性，在公式中显式地从密度中提取这些奇异性，并使用全局求积法离散边界积分方程。将这些方法扩展到处理具有角点和多重连接点的曲线具有挑战性，因为奇异性分析变得复杂得多，并且为离散具有奇异及超奇异核与奇异密度的层势构造高阶求积法并非易事。所提出的方案基于以下两个观察构建。首先，单层势算子和双层势算子的法向导数在局部互为有效的预条件子。其次，递归压缩逆预条件（RCIP）方法可以扩展到处理“隐式”第二类积分方程。该方案是高阶、自适应的，能够在无需预先知道密度奇异性的情况下处理角点和多重连接点。它也与快速算法（如快速多极子方法）兼容。通过若干数值算例展示了该方案的性能。