Construction of [Formula omitted] Cubic Splines on Arbitrary Triangulations

• Published in: Foundations of computational mathematics 2022-10, Vol.22 (5), p.1309
• Main Authors: Lyche, Tom, Manni, Carla, Speleers, Hendrik
• Format: Article
• Language: English
• Subjects: Methods; Spline theory; Triangulation
• ISSN: 1615-3375
• DOI: 10.1007/s10208-022-09553-z

In this paper, we address the problem of constructing [Formula omitted] cubic spline functions on a given arbitrary triangulation [Formula omitted]. To this end, we endow every triangle of [Formula omitted] with a Wang-Shi macro-structure. The [Formula omitted] cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in this space can be locally built on each of the macro-triangles independently via Hermite interpolation. We provide a simplex spline basis for the space of [Formula omitted] cubics defined on a single macro-triangle which behaves like a Bernstein/B-spline basis over the triangle. The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for [Formula omitted] joins across the edges of neighboring triangles, and they enjoy a Marsden-like identity. Also, there is a single control net to facilitate control and early visualization of a spline function over the macro-triangle. Thanks to these properties, the complex geometry of the Wang-Shi macro-structure is transparent to the user. Stable global bases for the full space of [Formula omitted] cubics on the Wang-Shi refined triangulation [Formula omitted] are deduced from the local simplex spline basis by extending the concept of minimal determining sets.