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

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 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 $$L_\infty $$ 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.