We consider one-level additive Schwarz preconditioners for the Helmholtz equation (with increasing wavenumber $k$), discretized using fixed-order nodal conforming finite elements on a family of simplicial fine meshes with diameter $h$, chosen to maintain accuracy as $k$ increases. The preconditioners combine independent local solves (with impedance boundary conditions) on overlapping subdomains of diameter $H$ and overlap $\delta$, and prolongation/restriction operators defined using a partition of unity, this formulation was previously proposed in [J.H. Kimn and M. Sarkis, Comp. Meth. Appl. Mech. Engrg. 196, 1507-1514, 2007]. In numerical experiments (with $\delta \sim H$) we observe robust (i.e. $k-$independent) GMRES convergence as $k$ increases, both with $H$ fixed, and with $H$ decreasing moderately as $k$ increases. This provides a highly-parallel, $k-$robust one-level domain-decomposition method. We provide supporting theory for this observation by studying the preconditioner applied to a range of absorptive problems, $k^2\mapsto k^2+ \mathrm{i} \varepsilon$, with absorption parameter $\varepsilon$, including the "pure Helmholtz" case ($\varepsilon = 0$). Working in the Helmholtz "energy" inner product, we prove a robust upper bound on the norm of the preconditioned matrix, valid for all $\varepsilon, \delta$. Under additional conditions on $\varepsilon$ and $\delta$, we also prove a strictly-positive lower bound on the distance of the field of values of the preconditioned matrix from the origin. Using these results, combined with previous results of [M.J. Gander, I.G. Graham and E.A. Spence, Numer. Math. 131(3), 567-614, 2015] we obtain theoretical support for the observed robustness of the preconditioner for the pure Helmholtz problem with increasing wavenumber $k$.

中文翻译：

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具有吸收的亥姆霍兹方程的局部阻抗条件域分解

我们考虑 Helmholtz 方程的一级加法 Schwarz 预处理器（随着波数 $k$ 的增加），在直径 $h$ 的简单细网格族上使用固定阶节点符合有限元进行离散化，选择以保持精度为 $k $ 增加。预处理器将直径为 $H$ 和重叠 $\delta$ 的重叠子域上的独立局部求解（具有阻抗边界条件）与使用统一划分定义的扩展/限制算子相结合，该公式先前在 [JH Kimn 和 M . Sarkis, Comp. 冰毒。应用程序 机械。工程师 196, 1507-1514, 2007]。在数值实验中（使用 $\delta \sim H$），我们观察到稳健的（即 $k-$independent）GMRES 收敛随着 $k$ 的增加而增加，$H$ 固定，$H$ 作为 $k$ 适度减少增加。这提供了一种高度并行的、$k-$robust 的一级域分解方法。我们通过研究应用于一系列吸收问题的预处理器 $k^2\mapsto k^2+ \mathrm{i} \varepsilon$，以及吸收参数 $\varepsilon$，包括“纯亥姆霍兹”案例（$\varepsilon = 0$）。在亥姆霍兹“能量”内积中工作，我们证明了预处理矩阵范数的稳健上限，对所有 $\varepsilon, \delta$ 都有效。在 $\varepsilon$ 和 $\delta$ 的附加条件下，我们还证明了预处理矩阵的值域与原点的距离的严格正下界。使用这些结果，结合 [MJ Gander, IG Graham 和 EA Spence, Numer. 数学。131(3), 567-614,