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Spectral analysis of P k Finite Element matrices in the case of Friedrichs–Keller triangulations via Generalized Locally Toeplitz technology
Author(s) -
Rahla Ryma Imene,
SerraCapizzano Stefano,
TablinoPossio Cristina
Publication year - 2020
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2302
Subject(s) - toeplitz matrix , mathematics , eigenvalues and eigenvectors , bounded function , dirichlet distribution , operator (biology) , matrix (chemical analysis) , dirichlet boundary condition , boundary (topology) , finite element method , partial differential equation , elliptic operator , pure mathematics , mathematical analysis , boundary value problem , physics , thermodynamics , biochemistry , chemistry , materials science , repressor , quantum mechanics , transcription factor , composite material , gene
Summary 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 overΩ ‾ , Ω being an open and bounded subset ofR d , d ≥1. For the numerical approximation, we consider the classicalP kFinite 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.