Shifted Laplacian RAS Solvers 2 for the Helmholtz Equation 3

The main purpose of this paper is to introduce novel two-level overlapping 15 Schwarz methods for solving the Helmholtz equation. Among the most effective par16 allel two-level domain decomposition solvers for the Helmholtz equation on general 17 unstructured meshes, we mention the FETI-H method introduced by Farhat et al. [5], 18 and the WRAS-H-RC method introduced by Kimn and Sarkis [10]. FETI-H type pre19 conditioners belong to the class of nonoverlapping domain decomposition methods. 20 FETI-H methods can be viewed as a modification of the original FETI method in21 troduced by Farhat et al. [6]. The local solvers in FETI-H are based on Sommerfeld 22 boundary conditions, see [3], while the coarse problem is based on plane waves. 23 WRAS-H-RC type preconditioners belong to the class of overlapping Schwarz 24 methods. They can be viewed as a miscellaneous of several methods to enhance the 25 effectiveness of the solver for Helmholtz problems. The first ingredient of WRAS26 H-RC preconditioners is the use of Sommerfeld boundary conditions for the local 27 solvers on overlapping subdomains. This idea is similar to what was done in FETI28 H, however, now for the overlapping case. This idea can be found for instance in the 29 work of Cai et al. [2] and Kimn [8]. The second ingredient is the use of the Weighted 30 Restricted Additive Schwarz (WRAS) method introduced by Cai and Sarkis [1] in 31 order to average the local overlapping solutions. The third ingredient is the use of 32