使用源误差估计器可靠估计特征空间误差
Reliable eigenspace error estimation using source error estimators
AI总结：
本文提出了一种利用源误差估计器计算特征空间与离散化间隙的方法，应用于FOSLS和DPG离散化，展示了自适应算法在处理特征簇时的改进效果。
摘要：
我们介绍了一个框架，用于将源问题的误差估计器重新利用以计算特征空间与其离散化之间的间隙估计器。感兴趣的包括无界非自共轭线性算子有限簇特征值的特征空间。首先研究了此类算子有理函数的特征空间和特征值。在假设解析近似在算子范数下收敛和源问题误差估计器的全局可靠性假设下，我们证明了特征空间近似间隙可以被一个全局可靠且可计算的误差估计器所限制。此外还包括了将理论框架应用于一阶系统最小二乘（FOSLS）离散化和不连续Petrov-Galerkin（DPG）离散化，两者都给出了新的误差间隙估计器。使用新估计器的自共轭模型问题和渗漏非自共轭波导本征问题的数值实验显示，使用新估计器的自适应算法生成的细化模式瞄准整个簇而不是单个本征函数。
英文摘要：
We introduce a framework for repurposing error estimators for source problems to compute an estimator for the gap between eigenspaces and their discretizations. Of interest are eigenspaces of finite clusters of eigenvalues of unbounded nonselfadjoint linear operators with compact resolvent. Eigenspaces and eigenvalues of rational functions of such operators are studied as a first step. Under an assumption of convergence of resolvent approximations in the operator norm and an assumption on global reliability of source problem error estimators, we show that the gap in eigenspace approximations can be bounded by a globally reliable and computable error estimator. Also included are applications of the theoretical framework to first-order system least squares (FOSLS) discretizations and discontinuous Petrov-Galerkin (DPG) discretizations, both yielding new estimators for the error gap. Numerical experiments with a selfadjoint model problem and with a leaky nonselfadjoint waveguide eigenproblem show that adaptive algorithms using the new estimators give refinement patterns that target the cluster as a whole instead of individual eigenfunctions.
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