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Spectral analysis and spectral symbol of matrices in isogeometric collocation methods


Authors: Marco Donatelli, Carlo Garoni, Carla Manni, Stefano Serra-Capizzano and Hendrik Speleers
Journal: Math. Comp. 85 (2016), 1639-1680
MSC (2010): Primary 15A18, 15B05, 41A15, 15A69, 65N35
DOI: https://doi.org/10.1090/mcom/3027
Published electronically: October 15, 2015
MathSciNet review: 3471103
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Abstract: We consider a linear full elliptic second order partial differential equation in a $ d$-dimensional domain, $ d\ge 1$, approximated by isogeometric collocation methods based on uniform (tensor-product) B-splines of degrees $ \boldsymbol {p}:=(p_1,\ldots ,p_d)$, $ p_j\ge 2$, $ j=1,\ldots ,d$. We give a construction of the inherently non-symmetric matrices arising from this approximation technique and we perform an analysis of their spectral properties. In particular, we find and study the associated (spectral) symbol, that is, the function describing their asymptotic spectral distribution (in the Weyl sense) when the matrix-size tends to infinity or, equivalently, the fineness parameters tend to zero. The symbol is a non-negative function with a unique zero of order two at $ \boldsymbol {\theta }=\mathbf {0}$ (where $ \boldsymbol {\theta }:=(\theta _1,\ldots ,\theta _d)$ are the Fourier variables), but with infinitely many `numerical zeros' for large $ \Vert\boldsymbol {p}\Vert _\infty $. Indeed, the symbol converges exponentially to zero with respect to $ p_j$ at all the points $ \boldsymbol {\theta }$ such that $ \theta _j=\pi $. In other words, if $ p_j$ is large, all the points $ \boldsymbol {\theta }$ with $ \theta _j=\pi $ behave numerically like a zero of the symbol. The presence of the zero of order two at $ \boldsymbol {\theta }=\mathbf {0}$ is expected because it is intrinsic in any local approximation method, such as finite differences and finite elements, of second order differential operators. However, the `numerical zeros' lead to the surprising fact that, for large $ \Vert\boldsymbol {p}\Vert _\infty $, there is a subspace of high frequencies where the collocation matrices are ill-conditioned. This non-canonical feature is responsible for the slowdown, with respect to $ \boldsymbol {p}$, of standard iterative methods. On the other hand, this knowledge and the knowledge of other properties of the symbol can be exploited to construct iterative solvers with convergence properties independent of the fineness parameters and of the degrees $ \boldsymbol {p}$.


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Additional Information

Marco Donatelli
Affiliation: Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, Italy
Email: marco.donatelli@uninsubria.it

Carlo Garoni
Affiliation: Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, Italy
Email: carlo.garoni@uninsubria.it

Carla Manni
Affiliation: Department of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica, 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; Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden
Email: stefano.serrac@uninsubria.it, stefano.serra@it.uu.se

Hendrik Speleers
Affiliation: Department of Mathematics, University of Rome ‘Tor Vergata’, Via della Ricerca Scientifica, 00133 Rome, Italy; Department of Computer Science, University of Leuven, Celestijnenlaan 200A, 3001 Heverlee (Leuven), Belgium
Email: speleers@mat.uniroma2.it, hendrik.speleers@cs.kuleuven.be

DOI: https://doi.org/10.1090/mcom/3027
Keywords: Spectral distribution, symbol, collocation method, B-splines, isogeometric analysis
Received by editor(s): April 21, 2014
Received by editor(s) in revised form: December 6, 2014
Published electronically: October 15, 2015
Additional Notes: This work was partially supported by INdAM-GNCS Gruppo Nazionale per il Calcolo Scientifico, by MIUR - PRIN 2012 N. 2012MTE38N, by the MIUR ‘Futuro in Ricerca 2013’ Programme through the project DREAMS, by the Research Foundation Flanders, and by the Program ‘Becoming the Number One – Sweden (2014)’ of the Knut and Alice Wallenberg Foundation.
Article copyright: © Copyright 2015 American Mathematical Society