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子来求解Helmholtz方程，其中涉及带或不带重叠的带状区域分解。通过将不同未知数编号方式下的Jacobi、Gauss-Seidel及对称Gauss-Seidel方法统一表述，我们实现了这类预处理子的推导过程与收敛性研究的统一。该框架不仅能够对[Nataf and Nier (1997), Vion and Geuzaine (2018)]与[Stolk (2013, 2017), Vion and Geuzaine (2014)]中的双重扫描方法进行理论对比分析，还便于引入新型扫描算法。我们通过数值算例验证了理论研究的有效性。