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On Quasi-Interpolation Operators in Spline Spaces

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Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 114))

Abstract

We propose the construction of a class of L 2 stable quasi-interpolation operators onto the space of splines on tensor-product meshes, in any space dimension. The estimate we propose is robust with respect to knot repetition and to knot “vicinity” (up to p + 1 knots), so it applies to the most general scenario in which the B-spline functions are known to be well defined.

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Notes

  1. 1.

    This constant depends on the polynomial degree p, since the number of B-spline basis functions acting on a single mesh element is p + 1.

  2. 2.

    For each Q we consider the representation Q = I 1 × … × I d .

  3. 3.

    Without loss of generality, we denote by C 2 the constant in Remark 1 for each coordinate direction, and thus, \(\frac{\vert \tilde{Q}\vert } {\vert \mathop{\mathrm{supp}}\nolimits \beta \vert }\leqslant C_{2}^{d}\), for all \(Q \in \mathcal{Q}\) such that Q ⊂ suppβ.

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Acknowledgements

Annalisa Buffa and Giancarlo Sangalli were partially supported by the European Research Council through the FP7 ERC Consolidator Grant n.616563 HIGEOM, and by the Italian MIUR through the PRIN “Metodologie innovative nella modellistica differenziale numerica”. Eduardo M. Garau was partially supported by CONICET through grant PIP 112-2011-0100742, by Universidad Nacional del Litoral through grants CAI+D 500 201101 00029 LI, 501 201101 00476 LI, by Agencia Nacional de Promoción Científica y Tecnológica, through grants PICT-2012-2590 and PICT-2014-2522 (Argentina). Carlotta Giannelli was supported by the project DREAMS (MIUR “Futuro in Ricerca” RBFR13FBI3) and by the Gruppo Nazionale per il Calcolo Scientifico (GNCS) of the Istituto Nazionale di Alta Matematica (INdAM). This support is gratefully acknowledged.

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Authors and Affiliations

  1. Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” (CNR), Pavia, Italy

    Annalisa Buffa, Eduardo M. Garau & Giancarlo Sangalli

  2. Instituto de Matemática Aplicada del Litoral (CONICET-UNL) and Facultad de Ingeniería Química (UNL), Santa Fe, Argentina

    Eduardo M. Garau

  3. Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Firenze, Italy

    Carlotta Giannelli

  4. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Pavia, Italy

    Giancarlo Sangalli

Authors
  1. Annalisa Buffa
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  2. Eduardo M. Garau
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  3. Carlotta Giannelli
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  4. Giancarlo Sangalli
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Corresponding author

Correspondence to Annalisa Buffa .

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Editors and Affiliations

  1. Department of Mathematics & Statistics, University of Strathclyde, Glasgow, United Kingdom

    Gabriel R. Barrenechea

  2. Istituto di Matematica Applicata e Tecnologie Informatiche - Pavia, Consiglio Nazionale delle Ricerche, Pavia, Italy

    Franco Brezzi

  3. Department of Mathematics, University of Leicester, Leicester, United Kingdom

    Andrea Cangiani

  4. Department of Mathematics, National Technical University of Athens, Zografou, Greece

    Emmanuil H. Georgoulis

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Buffa, A., Garau, E.M., Giannelli, C., Sangalli, G. (2016). On Quasi-Interpolation Operators in Spline Spaces. In: Barrenechea, G., Brezzi, F., Cangiani, A., Georgoulis, E. (eds) Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 114. Springer, Cham. https://doi.org/10.1007/978-3-319-41640-3_3

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