Iterative Pure Source Transfer Domain Decomposition Methods for Helmholtz Equations in Heterogeneous Media

Abstract

We extend the pure source transfer domain decomposition method (PSTDDM) to solve the perfectly matched layer approximation of Helmholtz scattering problems in heterogeneous media. We first propose some new source transfer operators, and then introduce the layer-wise and block-wise PSTDDMs based on these operators. In particular, it is proved that the solution obtained by the layer-wise PSTDDM in $\mathbb{R}^2$ coincides with the exact solution to the heterogeneous Helmholtz problem in the computational domain. Second, we propose the iterative layer-wise and blockwise PSTDDMs, which are designed by simply iterating the PSTDDM alternatively over two staggered decompositions of the computational domain. Finally, extensive numerical tests in two and three dimensions show that, as the preconditioner for the GMRES method, the iterative PSTDDMs are more robust and efficient than PSTDDMs for solving heterogeneous Helmholtz problems.