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Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods


Authors: Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, Debora Sesana and Hendrik Speleers
Journal: Math. Comp. 86 (2017), 1343-1373
MSC (2010): Primary 15A18, 15B05, 41A15, 15A69, 65N30
DOI: https://doi.org/10.1090/mcom/3143
Published electronically: August 3, 2016
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Abstract: A linear full elliptic second-order Partial Differential Equation (PDE), defined on a $ d$-dimensional domain $ \Omega $, is approximated by the isogeometric Galerkin method based on uniform tensor-product B-splines of degrees $ (p_1,\ldots ,p_d)$. The considered approximation process leads to a $ d$-level stiffness matrix, banded in a multilevel sense. This matrix is close to a $ d$-level Toeplitz structure if the PDE coefficients are constant and the physical domain $ \Omega $ is the hypercube $ (0,1)^d$ without using any geometry map. In such a simplified case, a detailed spectral analysis of the stiffness matrices has already been carried out in a previous work. In this paper, we complete the picture by considering non-constant PDE coefficients and an arbitrary domain $ \Omega $, parameterized with a non-trivial geometry map. We compute and study the spectral symbol of the related stiffness matrices. This symbol describes the asymptotic eigenvalue distribution when the fineness parameters tend to zero (so that the matrix-size tends to infinity). The mathematical tool used for computing the symbol is the theory of Generalized Locally Toeplitz (GLT) sequences.


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Additional Information

Carlo Garoni
Affiliation: Department of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133 Rome, Italy – and – Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, Italy
Email: garoni@mat.uniroma2.it, carlo.garoni@uninsubria.it

Carla Manni
Affiliation: Department of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133 Rome, Italy
Email: manni@mat.uniroma2.it

Stefano Serra-Capizzano
Affiliation: Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, Italy – and – Department of Information Technology, Division of Scientific Computing, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden
Email: stefano.serrac@uninsubria.it, stefano.serra@it.uu.se

Debora Sesana
Affiliation: Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, Italy
Email: debora.sesana@uninsubria.it

Hendrik Speleers
Affiliation: Department of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133 Rome, Italy
Email: speleers@mat.uniroma2.it

DOI: https://doi.org/10.1090/mcom/3143
Keywords: Spectral distribution, symbol, Galerkin method, B-splines, isogeometric analysis
Received by editor(s): February 9, 2015
Received by editor(s) in revised form: October 10, 2015
Published electronically: August 3, 2016
Additional Notes: This work was partially supported by INdAM-GNCS Gruppo Nazionale per il Calcolo Scientifico, by the MIUR ‘Futuro in Ricerca 2013’ Programme through the project DREAMS, and by Donation KAW 2013.0341 from the Knut & Alice Wallenberg Foundation in collaboration with the Royal Swedish Academy of Sciences, supporting Swedish research in mathematics.
Article copyright: © Copyright 2016 American Mathematical Society

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