Multidirectionnal sweeping preconditioners with non-overlapping checkerboard domain decomposition for Helmholtz problems

TL;DR

This paper introduces a family of generalized sweeping preconditioners for Helmholtz problems with checkerboard domain decompositions, enabling efficient iterative solutions through multi-directional sweeping and GMRES flexibility.

Contribution

It develops a flexible approach to apply existing sweeping preconditioners to checkerboard partitions with multiple directions, improving scalability and convergence.

Findings

Effective multi-directional sweeping preconditioners demonstrated

Accelerated convergence with GMRES in complex 2D Helmholtz problems

Comparative results show improved performance over traditional methods

Abstract

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.