This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e., the orthogonal projector on the associated eigenspace. This allows us to deal with degenerate (multiple) eigenvalues within the framework. All the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues; we show how this assumption can be numerically checked. Our bounds are guaranteed and converge with the same speed as the exact errors. They can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available, which is presented in two particular cases: the Laplace eigenvalue problem discretized with conforming finite elements, and a Schrodinger operator with periodic boundary conditions of the form $−∆ + V$ discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems.

中文翻译：

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特征值和特征向量的保证 $\textit {a后验}$ 边界：多重性和簇

本文提出了具有紧解算的二阶自伴随椭圆线性算子的特征值簇的符合数值近似的后验误差估计。给定一组特征值，我们估计特征值总和的误差，以及通过密度矩阵表示的特征向量的误差，即相关特征空间上的正交投影仪。这使我们能够在框架内处理退化（多个）特征值。所有边界在唯一假设下是有效的，即集群与周围越来越小的特征值分开；我们展示了如何在数值上检查这个假设。我们的边界是有保证的，并且以与精确误差相同的速度收敛。一旦残差的对偶范数的估计可用，它们就可以变成完全可计算的边界，这在两种特殊情况下呈现：使用一致的有限元离散的拉普拉斯特征值问题，以及具有周期性边界条件的薛定谔算子用平面波离散化的形式 $−∆ + V$。对于这两种情况，提供了一组测试问题的数值说明。