An efficient finite-difference method for solving the isotropic Helmholtz equation relies on a discretization scheme and an appropriate solver. Accordingly, we have adopted an average-derivative optimal scheme that has two advantages: (1) it can be applied to unequal directional sampling intervals and (2) it requires less than four grid points of sampling per wavelength. Direct methods are not of interest for industry-sized problems due to the high memory requirements; Krylov subspace methods such as the biconjugate gradient stabilized method and the flexible generalized minimal residual method that combine a multigrid-based preconditioner are better alternatives. However, standard geometric multigrid algorithms fail to converge when there exist unequal directional sampling intervals; this is called anisotropic grids in terms of the multigrid. We first review our previous research on 2D anisotropic grids: the semicoarsening strategy, line-wise relaxation operator, and matrix-dependent interpolation were used to modify the standard V-cycle multigrid algorithms, resulting in convergence. Although directly extending to the 3D case by substituting line relaxation for plane relaxation deteriorates the convergence rate considerably, we then find that a multilevel generalized minimal residual preconditioner-combined semicoarsening strategy is more suitable for anisotropic grids and the convergence rate is faster in the 2D and 3D cases. The results of the numerical experiments indicate that the standard geometric multigrid does not work for anisotropic grids, whereas our method demonstrates a faster convergence rate than the previous method.

中文翻译：

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各向异性多级预处理器，用于求解方向采样间隔不相等的Helmholtz方程

用于求解各向同性亥姆霍兹方程的有效有限差分方法依赖于离散化方案和适当的求解器。因此，我们采用了具有两个优点的平均导数最优方案：（1）可以将其应用于不相等的方向采样间隔；（2）每个波长需要少于四个网格点的采样。由于内存需求量大，直接方法对于行业规模的问题不感兴趣。结合基于多网格的预处理器的双共轭梯度稳定方法和灵活的广义最小残差方法等Krylov子空间方法是更好的选择。但是，当存在不相等的方向采样间隔时，标准的几何多重网格算法无法收敛。就多重网格而言，这称为各向异性网格。我们首先回顾一下我们以前在2D各向异性网格上的研究：使用半粗化策略，逐行弛豫算子和依赖于矩阵的插值来修改标准V周期多网格算法，从而实现收敛。尽管用线松弛代替平面松弛直接扩展到3D情况会大大降低收敛速度，但是我们发现，多层广义最小残差预处理器组合半粗化策略更适合各向异性网格，并且在2D和2D中收敛速度更快。 3D案例​​。数值实验结果表明，标准几何多重网格不适用于各向异性网格，而我们的方法显示出比以前的方法更快的收敛速度。