发表机构
University College London; University of Bath; Sorbonne Université, Université Paris Cité, CNRS, INRIA, Laboratoire Jacques-Louis Lions（伦敦大学学院; 巴斯大学; 索邦大学，巴黎西岱大学，法国国家科学研究中心，法国国家信息与自动化研究所，雅克-路易·利翁斯实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对高频亥姆霍兹方程，提出带预渐近分段多项式粗空间的两级混合Schwarz预条件子，证明其收敛迭代次数最多随logk增长，数值实验验证理论且表明结果尖锐。
AI 中文摘要
我们研究用于高频亥姆霍兹方程（波数为$k$）有限元离散化的两级混合Schwarz区域分解预条件子。迄今为止，所有针对这些预条件子的$k$-显式理论均考虑渐近 regime 下的粗空间，即Galerkin解是$k$-一致拟最优的。我们考虑细空间和粗空间分别由宽度为$h$和$H_c$的网格上次数为$p$的分段多项式构成的情形。在熟知的网格阈值$(H_c k)^{2p} \rho(k)$足够小的条件下，我们证明了预渐近 regime 下粗空间的相关结果，其中$\rho(k)$是解算子的范数（自由空间中亥姆霍兹方程的$\rho(k)\backsim k$）。我们证明，在该 regime 下，使用此预条件子的不动点迭代的收敛迭代次数最多随$\boldsymbol{\text{log}\boldsymbol{k}}$增长。我们给出数值实验以验证该理论，且实验表明我们的理论结果是尖锐的：当$(H_c k)^{2p} \rho(k) \boldsymbol{\text{远大于}\boldsymbol{1}}$时，这些预条件子收敛所需的迭代次数随$k$快速增长。
英文摘要
We consider two-level hybrid Schwarz domain-decomposition preconditioners for finite element discretizations of the high-frequency Helmholtz equation (with wavenumber $k$). To date, all existing $k$-explicit theory for these preconditioners considers coarse spaces in the $\textit{asymptotic regime}$, i.e., where the Galerkin solution is $k$-uniformly quasi-optimal. We consider the situation where the fine and coarse spaces consist, respectively, of piecewise polynomials of degree $p$ on meshes of width $h$ and $H_c$. We prove results for the coarse space in the $\textit{preasymptotic regime}$, under the familiar mesh threshold that $(H_c k)^{2p} ρ(k)$ is sufficiently small, where $ρ(k)$ is the norm of the solution operator (such that $ρ(k)\sim k$ for the Helmholtz equation in free space). We prove that, in this regime, the fixed-point iteration with this preconditioner converges in a number of iterations that grows at most like $\log k$. We give numerical experiments demonstrating this theory. Furthermore, these experiments indicate that our theoretical results are sharp, in that the number of iterations required for these preconditioners to converge with $(H_c k)^{2p} ρ(k) \gg 1$ grows rapidly with $k$.
