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Completeness Characterization of Type-I Box Splines

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Geometric Challenges in Isogeometric Analysis

Part of the book series: Springer INdAM Series ((SINDAMS,volume 49))

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Abstract

We present a completeness characterization of box splines on three-directional triangulations, also called type-I box spline spaces, based on edge-contact smoothness properties. For any given type-I box spline, of specific maximum degree and order of global smoothness, our results allow to identify the local linear subspace of polynomials spanned by the box spline translates. We use the global super-smoothness properties of box splines as well as the additional super-smoothness conditions at edges to characterize the spline space spanned by the box spline translates. Subsequently, we prove the completeness of this space with respect to the local polynomial space induced by the box spline translates. The completeness property allows the construction of hierarchical spaces spanned by the translates of box splines for any polynomial degree on multilevel type-I grids. We provide a basis for these hierarchical box spline spaces under explicit geometric conditions of the domain.

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Notes

  1. 1.

    Observe that \(\ker A^{(i)} = \hat{\mathbb {S}}^{\boldsymbol{d}}(M,\mathscr {V}_{\boldsymbol{n}})\).

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Acknowledgements

We would like to thank the anonymous referees for carefully reading our initial submission providing many helpful comments which led to an improved presentation of the results in this paper. N. Villamizar and A. Mantzaflaris wish to thank the support by The Alliance Hubert Curien Programme 2021, Project number: 515492678. N. Villamizar was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) New Investigator Award EP/V012835/1.

Author information

Authors and Affiliations

  1. Swansea University, Swansea, United Kingdom

    Nelly Villamizar

  2. Inria Sophia Antipolis–Méditerranée, Université Côte d’Azur, Nice, France

    Angelos Mantzaflaris

  3. Johannes Kepler University Linz & RICAM, Linz, Austria

    Bert Jüttler

Authors
  1. Nelly Villamizar
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  2. Angelos Mantzaflaris
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  3. Bert Jüttler
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Corresponding author

Correspondence to Nelly Villamizar .

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Editors and Affiliations

  1. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Carla Manni

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Hendrik Speleers

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Villamizar, N., Mantzaflaris, A., Jüttler, B. (2022). Completeness Characterization of Type-I Box Splines. In: Manni, C., Speleers, H. (eds) Geometric Challenges in Isogeometric Analysis. Springer INdAM Series, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-92313-6_12

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