Advances in Computational Mathematics

, Volume 40, Issue 2, pp 459–490 | Cite as

Strongly stable bases for adaptively refined multilevel spline spaces

  • Carlotta Giannelli
  • Bert Jüttler
  • Hendrik SpeleersEmail author


The problem of constructing a normalized hierarchical basis for adaptively refined spline spaces is addressed. Multilevel representations are defined in terms of a hierarchy of basis functions, reflecting different levels of refinement. When the hierarchical model is constructed by considering an underlying sequence of bases \(\{\Gamma ^{\ell }\}_{\ell =0,\ldots ,N-1}\) with properties analogous to classical tensor-product B-splines, we can define a set of locally supported basis functions that form a partition of unity and possess the property of coefficient preservation, i.e., they preserve the coefficients of functions represented with respect to one of the bases \(\Gamma ^{\ell }\). Our construction relies on a certain truncation procedure, which eliminates the contributions of functions from finer levels in the hierarchy to coarser level ones. Consequently, the support of the original basis functions defined on coarse grids is possibly reduced according to finer levels in the hierarchy. This truncation mechanism not only decreases the overlapping of basis supports, but it also guarantees strong stability of the construction. In addition to presenting the theory for the general framework, we apply it to hierarchically refined tensor-product spline spaces, under certain reasonable assumptions on the given knot configuration.


Hierarchical splines Truncated hierarchical basis Partition of unity Local refinement Stability 

Mathematics Subject Classifications (2010)

41A15 65D07 65D17 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Carlotta Giannelli
    • 1
  • Bert Jüttler
    • 1
  • Hendrik Speleers
    • 2
    Email author
  1. 1.Institute of Applied GeometryJohannes Kepler UniversityLinzAustria
  2. 2.Department of Computer ScienceKatholieke Universiteit LeuvenLeuvenBelgium

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