Stable simplex spline bases for $C^3$ quintics on the Powell-Sabin 12-split

TL;DR

This paper explicitly constructs and analyzes six symmetric simplex spline bases for $C^3$ quintic splines on the Powell-Sabin 12-split, ensuring stability, positive partition of unity, and high-quality approximation properties.

Contribution

The paper introduces six explicit symmetric simplex spline bases with desirable stability, partition of unity, and approximation features for $C^3$ quintics on Powell-Sabin 12-split.

Findings

Bases are stable in $L_ abla_ ext{infty}$ norm with geometry-independent condition number.

A well-conditioned Lagrange interpolant at domain points.

A quasi-interpolant with local approximation order 6.

Abstract

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, 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 $L_\infty$ norm with a condition number independent of the geometry, 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.