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On the spectrum of stiffness matrices arising from isogeometric analysis

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Abstract

We study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems. Motivated by the applicative interest in the fast solution of the related linear systems, we are looking for a spectral characterization of the involved matrices. In particular, we investigate non-singularity, conditioning (extremal behavior), spectral distribution in the Weyl sense, as well as clustering of the eigenvalues to a certain (compact) subset of \(\mathbb C\). All the analysis is related to the notion of symbol in the Toeplitz setting and is carried out both for the cases of 1D and 2D problems.

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Notes

  1. The essential range of \(f\) coincides exactly with the range of \(f\) whenever \(f\) is continuous. In this paper we will only deal with continuous functions \(f\).

  2. If \(\beta \ne 0\) then \(\mathrm{Im}\,\frac{1}{n}A_n^{[2]}\) is irreducible and \(\sigma \left( \mathrm{Im}\,\frac{1}{n}A_n^{[2]}\right) \subset \left( -\frac{11|\beta |}{12n},\frac{11|\beta |}{12n}\right) \). In (76) we have included the endpoints \(\pm \frac{11|\beta |}{12n}\) to cover the case \(\beta =0\).

  3. In this way, \(A_{n_1,n_2}^{[p_1,p_2]}\) is really a sequence of matrices, since only \(n_1\) is a free parameter. The relation \(n_2=\nu n_1\) must be kept in mind while reading this section. We point out that this request could be replaced by even milder conditions, but at the price of heavier notations.

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Authors and Affiliations

  1. Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 , Como, Italy

    Carlo Garoni & Stefano Serra-Capizzano

  2. Department of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica, 00133 , Rome, Italy

    Carla Manni & Francesca Pelosi

  3. Department of Computer Science, University of Leuven, Celestijnenlaan 200A, 3001 , Heverlee (Leuven), Belgium

    Hendrik Speleers

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  1. Carlo Garoni
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Correspondence to Hendrik Speleers.

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Garoni, C., Manni, C., Pelosi, F. et al. On the spectrum of stiffness matrices arising from isogeometric analysis. Numer. Math. 127, 751–799 (2014). https://doi.org/10.1007/s00211-013-0600-2

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