Shifted Laplacian RAS Solvers for the Helmholtz Equation

where Ω is a bounded polygonal region in <, and the ∂ΩD, ∂ΩN and ∂ΩS correspond to subsets of ∂Ω where the Dirichlet, Neumann and Sommerfeld boundary conditions are imposed. The main purpose of this paper is to introduce novel two-level overlapping Schwarz methods for solving the Helmholtz equation. Among the most effective parallel two-level domain decomposition solvers for the Helmholtz equation on general unstructured meshes, we mention the FETI-H method introduced by Farhat et al. [2000], and the WRAS-H-RC method introduced by Kimn and Sarkis [2007]. FETI-H type preconditioners belong to the class of nonoverlapping domain decomposition methods. FETI-H methods can be viewed as a modification of the original FETI method introduced by Farhat et al. [1994]. The local solvers in FETI-H are based on Sommerfeld boundary conditions, see Deprés [1991], while the coarse problem is based on plane waves. WRAS-H-RC type preconditioners belong to the class of overlapping Schwarz methods. They can be viewed as a miscellaneous of several methods to enhance the effectiveness of the solver for Helmholtz problems. The first ingredient of WRAS-H-RC preconditioners is the use of Sommerfeld boundary conditions for the local solvers on overlapping subdomains. This idea is