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Spectral analysis of matrices in Galerkin methods based on generalized B-splines with high smoothness

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Abstract

We present a first step towards the spectral analysis of matrices arising from IgA Galerkin methods based on hyperbolic and trigonometric GB-splines. Second order differential problems with constant coefficients are considered and discretized by means of sequences of both nested and non-nested spline spaces. We prove that there always exists an asymptotic eigenvalue distribution which can be compactly described by a symbol, just like in the polynomial case. There is a complete similarity between the symbol expressions in the hyperbolic, trigonometric and polynomial cases. This results in similar spectral features of the corresponding matrices. We also analyze the IgA discretization based on trigonometric GB-splines for the eigenvalue problem related to the univariate Laplace operator. We prove that, for non-nested spaces, the phase parameter can be exploited to improve the spectral approximation with respect to the polynomial case. As part of the analysis, we derive several Fourier properties of cardinal GB-splines.

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Notes

  1. The functions \(N_{i,1}^{U,V}\) may also depend on p because of the definition of \(\widetilde{U}_i, \widetilde{V}_i\), but we omit the parameter p in order to avoid a heavier notation. Moreover, in the most interesting cases (polynomial, hyperbolic and trigonometric GB-splines) the functions \( N_{i, 1}^{U,V}\) are independent of p.

  2. For polynomial B-splines, a degree p in Galerkin approximation corresponds to a degree \(2p+1\) in collocation, see [11, Remark 3.2]. We have followed the same rule here.

  3. A different ordering has been used in [14] in the bivariate setting.

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Acknowledgments

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 the ‘Uncovering Excellence’ Programme of the University of Rome ‘Tor Vergata’ through the project DEXTEROUS.

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  1. Department of Mathematics, University of Turin, Via Carlo Alberto 10, 10123, Turin, Italy

    Fabio Roman

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

    Carla Manni & Hendrik Speleers

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  1. Fabio Roman
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  2. Carla Manni
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Correspondence to Hendrik Speleers.

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Roman, F., Manni, C. & Speleers, H. Spectral analysis of matrices in Galerkin methods based on generalized B-splines with high smoothness. Numer. Math. 135, 169–216 (2017). https://doi.org/10.1007/s00211-016-0796-z

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  • DOI: https://doi.org/10.1007/s00211-016-0796-z

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