We consider sweeping domain decomposition preconditioners to solve the Helmholtz equation in the case of stripwise domain decomposition with or without overlaps. We unify their derivation and convergence studies by expressing them as Jacobi, Gauss-Seidel, and Symmetric Gauss-Seidel methods for different numbering of the unknowns. The proposed framework enables theoretical comparisons between the double sweep methods in [Nataf and Nier (1997), Vion and Geuzaine (2018)] and those in [Stolk (2013, 2017), Vion and Geuzaine (2014)]. Additionally, it facilitates the introduction of a new sweeping algorithm. We provide numerical test cases to assess the validity of the theoretical studies.

翻译：我们考虑采用扫描型区域分解预处理器来求解亥姆霍兹方程，处理带状区域分解（可重叠或非重叠）的情况。通过将这些方法表述为对不同未知量编号的雅可比、高斯-赛德尔以及对称高斯-赛德尔方法，我们统一了其推导过程与收敛性研究。所提出的框架能够实现对[Nataf and Nier (1997), Vion and Geuzaine (2018)]中的双扫描方法与[Stolk (2013, 2017), Vion and Geuzaine (2014)]中的方法进行理论比较。此外，该框架还有助于引入一种新的扫描算法。我们提供数值测试案例以评估理论研究的有效性。