Double Source Transfer Domain Decomposition Method For Helmholtz Problems

We propose and study a double source transfer domain decomposition method (Double STDDM) for solving the truncated perfectly matched layer approximation in the bounded domain of Helmholtz problems. Based on the decomposition of the domain into non-overlapping layers and instead of transferring the source along one direction in STDDM [Z. Chen and X. Xiang, 2013], Double STDDM transfers the source in each layer along two directions, which can capture of the reflection information for heterogenous media. Double STDDM is an iterative scheme, and in each iteration, it first transfers the source from down to up and produces the Up wave (the wave propagating from down to up), and then transfers the source from up to down and produces the Down wave (the wave propagating from up to down). The output of Double STDDM is the summation of the Up and Down waves that are produced during the iteration. By using the fundamental solution of the PML equation, the convergence of Double STDDM is proved for the case of a constant wavenumber. Numerical examples are included to show the efficient performance of using Double STDDM as a preconditioner both for the problems with constant and heterogenous wavenumbers. For problems with a low velocity contrast, the number of iterations is independent of the wavenumber and mesh size, whereas for problems with a high velocity contrast, double STDDM performs much better than STDDM.