Rahla, Ryma Imene; Serra-Capizzano, Stefano; Tablino-Possio, Cristina Spectral analysis of \(\mathbb{P}_k\) finite element matrices in the case of Friedrichs-Keller triangulations via generalized locally Toeplitz technology. (English) Zbl 07250717 Numer. Linear Algebra Appl. 27, No. 4, e2302, 28 p. (2020). Summary: In the present article, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions and where the operator is \(\operatorname{div}(-a(x)\nabla\cdot)\), with \(a\) continuous and positive over \(\overline{\Omega}\), \(\Omega\) being an open and bounded subset of \(\mathbb{R}^d\), \(d\geq 1\). 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 \(\Omega =(0,1)^2\) 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. Cited in 3 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 15A18 Eigenvalues, singular values, and eigenvectors 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 15A12 Conditioning of matrices 65F10 Iterative numerical methods for linear systems Keywords:finite element approximations; matrix-sequences; spectral analysis PDFBibTeX XMLCite \textit{R. I. Rahla} et al., Numer. Linear Algebra Appl. 27, No. 4, e2302, 28 p. (2020; Zbl 07250717) Full Text: DOI