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 Chen and Xiang (SIAM J Numer Anal 51:2331–2356, 2013). 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.

我们提出了一种纯源转移域分解方法（PSTDDM），用于求解亥姆霍兹散射问题的有界域中的截断的完美匹配层（PML）近似。该方法是Chen和Xiang（SIAM J Numer Anal 51：2331–2356，2013）提出的STDDM的修改。在将域分解为N个非重叠层之后，STDDM由源传输和波扩展的两个系列步骤组成，其中\（N-1 \）截断了两个相邻层上的PML问题，而\（N-2 \）被截断了相继解决了半空间PML问题。而PSTDDM仅由两个相反方向上的两个并行源传输步骤组成，每个步骤中的\（N-1 \）连续解决了两个相邻层上截短的PML问题。这种修改的一个好处是，PSTDDM可以沿着平行于两层的方向进一步解决两个相邻层上的PML截断问题。因此，我们得到由\（N ^ 2 \）组成的分解的逐块PSTDDM平方，减少了子域问题的大小，更适合大规模问题。对于恒定波数的情况，证明了逐层PSTDDM和逐块PSTDDM的收敛。数值算例表明，PSTDDM对具有恒定波数的离散亥姆霍兹方程具有良好的近似性，可以用作预处理GMRES方法中求解具有恒定波数和异构波数的离散Helmholtz方程的有效预处理器。