In this work, we propose and analyze two two-level hybrid Schwarz preconditioners for solving the Helmholtz equation with high wave number in two and three dimensions. Both preconditioners are defined over a set of overlapping subdomains, with each preconditioner formed by a global coarse solver and one local solver on each subdomain. The global coarse solver is based on the localized orthogonal decomposition (LOD) technique, which was proposed in [30,31] originally for the discretization schemes for elliptic multiscale problems with heterogeneous and highly oscillating coefficients and Helmholtz problems with high wave number to eliminate the pollution effect. The local subproblems are Helmholtz problems in subdomains with homogeneous boundary conditions (the first preconditioner) or impedance boundary conditions (the second preconditioner). Both preconditioners are shown to be optimal under some reasonable conditions, that is, a uniform upper bound of the preconditioned operator norm and a uniform lower bound of the field of values are established in terms of all the key parameters, such as the fine mesh size, the coarse mesh size, the subdomain size and the wave numbers. It is the first time to show that the LOD solver can be a very effective coarse solver when it is used appropriately in the Schwarz method with multiple overlapping subdomains. Numerical experiments are presented to confirm the optimality and efficiency of the two proposed preconditioners.

翻译：本文提出并分析了两种用于求解二维和三维高波数亥姆霍兹方程的两层混合施瓦茨预条件子。两种预条件子均定义在一组重叠子域上，每个预条件子由一个全局粗求解器和各子域上的局部求解器构成。全局粗求解器基于局部正交分解（LOD）技术，该技术最初在文献[30,31]中针对具有异质高振荡系数的椭圆多尺度问题及高波数亥姆霍兹问题的离散化方案提出，以消除污染效应。局部子问题为具有齐次边界条件（第一种预条件子）或阻抗边界条件（第二种预条件子）的子域亥姆霍兹问题。在合理条件下，两种预条件子均被证明具有最优性，即预条件算子范数的统一上界和数值域的统一下界均能以所有关键参数（如细网格尺寸、粗网格尺寸、子域尺寸及波数）表示。本研究首次证明，当LOD求解器在多重叠子域的施瓦茨方法中得到恰当应用时，可成为非常有效的粗求解器。数值实验验证了所提两种预条件子的最优性与计算效率。