A method of estimating all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in T. Gergelits, K.A. Mardal, B.F. Nielsen, Z. Strako\v{s}: Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator, SIAM Journal on Numerical Analysis 57(3) (2019), 1369-1394. Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite element method, preconditioned by the inverse of a matrix of the same operator with different data. Our results hold for mixed Dirichlet and Robin or periodic boundary conditions applied to the original and preconditioning problems. The bounds are two-sided, guaranteed, easily accessible, and depend solely on the material data.

中文翻译：

---

由有限元方法求解的预处理扩散和弹性问题的所有特征值的保证两侧边界

最近在 T. Gergelits, KA Mardal, BF Nielsen, Z. Strako\v{s}: Laplacian preconditioning of elliptic PDEs: Localization of the离散化算子的特征值，SIAM 数值分析期刊 57(3) (2019)，1369-1394。受本文的启发，我们提供了一种稍微不同的方法，在某些方向上扩展了先前的结果。也就是说，我们提供了具有张量数据的一般扩散或弹性算子的所有渐增有序特征值的边界，用符合要求的有限元方法离散，由具有不同数据的相同算子的矩阵的逆进行预处理。我们的结果适用于混合 Dirichlet 和 Robin 或应用于原始和预处理问题的周期性边界条件。边界是双向的、有保证的、易于访问的，并且完全取决于材料数据。