  We propose a pure source transfer domain decomposition method (PSTDDM) for
solving the truncated perfectly matched layer (PML) approximation in bounded
domain of Helmholtz scattering problem. The method is a modification of the
STDDM proposed by [Z. Chen and X. Xiang, SIAM J. Numer. Anal., 51 (2013), pp.
2331--2356]. After decomposing the domain into $N$ non-overlapping layers, the
STDDM is composed of two series steps of sources transfers and wave expansions,
where $N-1$ truncated PML problems on two adjacent layers and $N-2$ truncated
half-space PML problems are solved successively. While the PSTDDM consists
merely of two parallel source transfer steps in two opposite directions, and in
each step $N-1$ truncated PML problems on two adjacent layers are solved
successively. One benefit of such a modification is that the truncated PML
problems on two adjacent layers can be further solved by the PSTDDM along
directions parallel to the layers. And therefore, we obtain a block-wise PSTDDM
on the decomposition composed of $N^2$ squares, which reduces the size of
subdomain problems and is more suitable for large-scale problems. Convergences
of both the layer-wise PSTDDM and the block-wise PSTDDM are proved for the case
of constant wave number. Numerical examples are included to show that the
PSTDDM gives good approximations to the discrete Helmholtz equations with
constant wave numbers and can be used as an efficient preconditioner in the
preconditioned GMRES method for solving the discrete Helmholtz equations with
constant and heterogeneous wave numbers.
