## Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods

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- by Carlo Garoni, Carla Manni, Stefano Serra-Capizzano, Debora Sesana and Hendrik Speleers PDF
- Math. Comp.
**86**(2017), 1343-1373 Request permission

## 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.## References

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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
- MR Author ID: 1021672
- 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
- MR Author ID: 119310
- 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
- MR Author ID: 332436
- 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
- MR Author ID: 855861
- Email: debora.sesana@uninsubria.it
**Hendrik Speleers**- Affiliation: Department of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133 Rome, Italy
- MR Author ID: 778374
- Email: speleers@mat.uniroma2.it
- 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.
- © Copyright 2016 American Mathematical Society
- Journal: Math. Comp.
**86**(2017), 1343-1373 - MSC (2010): Primary 15A18, 15B05, 41A15, 15A69, 65N30
- DOI: https://doi.org/10.1090/mcom/3143
- MathSciNet review: 3614020