A Unified Framework for Double Sweep Methods for the Helmholtz Equation

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 as Jacobi, Gauss-Seidel or Symmetric Gauss-Seidel for different numbering of the unknowns. This enables the theoretical comparisons of the double sweep methods in [Nataf and Nier (1997), Vion and Geuzaine (2018)] with that of [Stolk (2013, 2017), Vion and Geuzaine (2014)]. It also makes possible the introduction of two new sweeping algorithms. We provide numerical test cases that assess the validity of the theoretical studies.