In the present article, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div(−a(x)∇·), with a continuous and positive over $\bar{\mathrm{\Omega }}$, Ω being an open and bounded subset of ${\mathbb{R}}^d$, d≥1. For the numerical approximation, we consider the classical ${\mathbb{P}}_k$ Finite Elements, in the case of Friedrichs–Keller triangulations, leading, as usual, to sequences of matrices of increasing size. The new results concern the spectral analysis of the resulting matrix‐sequences in the direction of the global distribution in the Weyl sense, with a concise overview on localization, clustering, extremal eigenvalues, and asymptotic conditioning. We study in detail the case of constant coefficients on Ω=(0,1)² and we give a brief account in the more involved case of variable coefficients and more general domains. Tools are drawn from the Toeplitz technology and from the rather new theory of Generalized Locally Toeplitz sequences. Numerical results are shown for a practical evidence of the theoretical findings.

中文翻译：

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广义局域Toeplitz技术在Friedrichs-Keller三角剖分的情况下Pk有限元矩阵的谱分析

在本文章中，我们考虑一类椭圆偏微分方程狄利克雷边界条件和在操作者是DIV（ -一个（X）·∇），具有一个连续和正上方$\stackrel{〜}{\mathrm{\Omega }}$，Ω是的一个开放有界子集 ${\mathbb{[R}}^d$，d ≥1。对于数值逼近，我们考虑经典${\mathbb{P}}_{ķ}$对于Friedrichs-Keller三角剖分，有限元通常会导致尺寸递增的矩阵序列。新结果涉及在Weyl意义上对矩阵序列在全局分布方向上的频谱分析，并简要介绍了定位，聚类，极值特征值和渐近条件。我们详细研究了Ω=（0,1）²上的常数系数的情况，并简要介绍了可变系数和更一般域的更复杂情况。工具是从Toeplitz技术和广义本地Toeplitz序列的新理论中汲取的。数值结果显示了理论发现的实用证据。