SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2054-2074, January 2021.
We consider radial complex scaling/perfectly matched layer methods for scalar resonance problems in homogeneous exterior domains. We introduce a new abstract framework to analyze the convergence of domain truncations and discretizations. Our theory requires rather minimal assumptions on the scaling profile and includes affine, smooth, and unbounded profiles. We report a swift technique to analyze the convergence of domain truncations and a more technical one for approximations through simultaneous truncation and discretization. Our established results include convergence rates of eigenvalues and eigenfunctions. The framework introduced is based on the following ideas: to interpret the domain truncation as Galerkin approximation, to apply theory on holomorphic Fredholm operator eigenvalue approximation theory to a linear eigenvalue problem, to employ the notion of weak T-coercivity and T-compatible approximations, to construct a suitable T-operator as multiplication operator, to smooth its symbol, and to apply the discrete commutator technique.

中文翻译：

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径向复标度方法分析：标量共振问题

SIAM 数值分析杂志，第 59 卷，第 4 期，第 2054-2074 页，2021 年 1 月。
我们考虑径向复杂缩放/完美匹配层方法来解决均匀外部域中的标量共振问题。我们引入了一个新的抽象框架来分析域截断和离散化的收敛。我们的理论对缩放轮廓需要相当少的假设，包括仿射、平滑和无界轮廓。我们报告了一种快速技术来分析域截断的收敛性，以及一种通过同时截断和离散化进行近似的更具技术性的技术。我们建立的结果包括特征值和特征函数的收敛速度。引入的框架基于以下思想：将域截断解释为Galerkin近似，将全纯Fredholm算子特征值逼近理论应用于线性特征值问题，