Stable Simplex Spline Bases for C 3 Quintics on the Powell–Sabin 12-Split

For the space of C 3 quintics on the Powell–Sabin 12-split of a triangle, we determine explicitly the six symmetric simplex spline bases that reduce to a B-spline basis on each edge and have a positive partition of unity, a Marsden identity that splits into real linear factors, and an intuitive doma...

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Bibliographic Details
Published in:Constructive approximation 2017-01, Vol.45 (1), p.1-32
Main Authors: Lyche, Tom, Muntingh, Georg
Format: Article
Language:English
Subjects:
Conditioning
Splines
Online Access:Get full text
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Summary:For the space of C 3 quintics on the Powell–Sabin 12-split of a triangle, we determine explicitly the six symmetric simplex spline bases that reduce to a B-spline basis on each edge and have a positive partition of unity, a Marsden identity that splits into real linear factors, and an intuitive domain mesh. The bases are stable in the L ∞ norm with a condition number independent of the geometry and have a well-conditioned Lagrange interpolant at the domain points and a quasi-interpolant with local approximation order 6. We show an h 2 bound for the distance between the control points and the values of a spline at the corresponding domain points. For one of these bases, we derive C 0 , C 1 , C 2 , and C 3 conditions on the control points of two splines on adjacent macrotriangles.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-016-9332-8