SIAM Review, Volume 66, Issue 1, Page 125-146, February 2024.
We analyze the spectrum of the operator $\Delta^{-1} [\nabla \cdot (K\nabla u)]$ subject to homogeneous Dirichlet or Neumann boundary conditions, where $\Delta$ denotes the Laplacian and $K=K(x,y)$ is a symmetric tensor. Our main result shows that this spectrum can be derived from the spectral decomposition $K=Q \Lambda Q^T$, where $Q=Q(x,y)$ is an orthogonal matrix and $\Lambda=\Lambda(x,y)$ is a diagonal matrix. More precisely, provided that $K$ is continuous, the spectrum equals the convex hull of the ranges of the diagonal function entries of $\Lambda$. The domain involved is assumed to be bounded and Lipschitz. In addition to studying operators defined on infinite-dimensional Sobolev spaces, we also report on recent results concerning their discretized finite-dimensional counterparts. More specifically, even though $\Delta^{-1} [\nabla \cdot (K\nabla u)]$ is not compact, it turns out that every point in the spectrum of this operator can, to an arbitrary accuracy, be approximated by eigenvalues of the associated generalized algebraic eigenvalue problems arising from discretizations. Our theoretical investigations are illuminated by numerical experiments. The results presented in this paper extend previous analyses which have addressed elliptic differential operators with scalar coefficient functions. Our investigation is motivated by both preconditioning issues (efficient numerical computations) and the need to further develop the spectral theory of second order PDEs (core analysis).

中文翻译：

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二阶自伴微分算子广义谱的简单公式

SIAM Review，第 66 卷，第 1 期，第 125-146 页，2024 年 2 月。
我们分析算子 $\Delta^{-1} [\nabla \cdot (K\nabla u)]$ 的谱，服从齐次狄利克雷或诺依曼边界条件，其中 $\Delta$ 表示拉普拉斯算子，$K=K(x,y)$ 是对称张量。我们的主要结果表明，该谱可以从谱分解 $K=Q \Lambda Q^T$ 中导出，其中 $Q=Q(x,y)$ 是正交矩阵，$\Lambda=\Lambda(x, y)$ 是对角矩阵。更准确地说，假设 $K$ 是连续的，则谱等于 $\Lambda$ 对角函数项范围的凸包。所涉及的域被假定为有界且 Lipschitz。除了研究无限维 Sobolev 空间上定义的算子之外，我们还报告了有关其离散有限维对应算子的最新结果。更具体地说，即使 $\Delta^{-1} [\nabla \cdot (K\nabla u)]$ 并不紧凑，但事实证明，该算子的频谱中的每个点都可以以任意精度表示为由离散化产生的相关广义代数特征值问题的特征值来近似。我们的理论研究通过数值实验得到阐明。本文提出的结果扩展了先前的分析，这些分析已经解决了具有标量系数函数的椭圆微分算子。我们的研究是出于预处理问题（高效数值计算）和进一步发展二阶偏微分方程谱理论（核心分析）的需要。