  This paper explores a family of generalized sweeping preconditionners for
Helmholtz problems with non-overlapping checkerboard partition of the
computational domain. The domain decomposition procedure relies on high-order
transmission conditions and cross-point treatments, which cannot scale without
an efficient preconditioning technique when the number of subdomains increases.
With the proposed approach, existing sweeping preconditioners, such as the
symmetric Gauss-Seidel and parallel double sweep preconditioners, can be
applied to checkerboard partitions with different sweeping directions (e.g.
horizontal and diagonal). Several directions can be combined thanks to the
flexible version of GMRES, allowing for the rapid transfer of information in
the different zones of the computational domain, then accelerating the
convergence of the final iterative solution procedure. Several two-dimensional
finite element results are proposed to study and to compare the sweeping
preconditioners, and to illustrate the performance on cases of increasing
complexity.
