We study overlapping Schwarz methods for the Helmholtz equation posed in any dimension with large, real wavenumber and smooth variable wave speed. The radiation condition is approximated by a Cartesian perfectly-matched layer (PML). The domain-decomposition subdomains are overlapping hyperrectangles with Cartesian PMLs at their boundaries. The overlaps of the subdomains and the widths of the PMLs are all taken to be independent of the wavenumber. For both parallel (i.e., additive) and sequential (i.e., multiplicative) methods, we show that after a specified number of iterations -- depending on the behaviour of the geometric-optic rays -- the error is smooth and smaller than any negative power of the wavenumber. For the parallel method, the specified number of iterations is less than the maximum number of subdomains, counted with their multiplicity, that a geometric-optic ray can intersect. These results, which are illustrated by numerical experiments, are the first wavenumber-explicit results about convergence of overlapping Schwarz methods for the Helmholtz equation, and the first wavenumber-explicit results about convergence of any domain-decomposition method for the Helmholtz equation with a non-trivial scatterer (here a variable wave speed).

翻译：我们研究了任意维度中具有大实波数和光滑变波速的Helmholtz方程的重叠Schwarz方法。辐射条件通过笛卡尔完美匹配层近似。区域分解子区域为重叠的超矩形，其边界处设有笛卡尔PML。子区域的重叠区域和PML宽度均设为与波数无关。对于并行方法与序贯方法，我们证明：在特定迭代次数后——该次数取决于几何光学射线的行为——误差变得光滑且小于波数的任意负幂次。对于并行方法，特定迭代次数小于几何光学射线可能穿过的最大子区域数量（计及子区域重叠次数）。数值实验验证了这些结果。这是首次关于Helmholtz方程重叠Schwarz方法收敛性的波数显式结果，也是首次关于具有非平凡散射体（此处为变波速）Helmholtz方程的区域分解方法收敛性的波数显式结果。