A parsimonious approach to $C^2$ cubic splines on arbitrary triangulations: Reduced macro-elements on the cubic Wang-Shi split

TL;DR

This paper introduces a simplified method for constructing $C^2$ cubic spline subspaces on arbitrary triangulations refined by the cubic Wang-Shi split, using macro-elements and Hermite degrees of freedom to reduce complexity.

Contribution

It develops a new macro-element approach for $C^2$ cubic splines on Wang-Shi refined triangulations, with explicit local bases and reduced degrees of freedom.

Findings

Subspaces contain cubic polynomials with fewer degrees of freedom.

Dimension can be as small as six times the number of vertices.

Explicit local basis functions are provided for practical implementation.

Abstract

We present a general method to obtain interesting subspaces of the $C^2$ cubic spline space defined on the cubic Wang-Shi refinement of a given arbitrary triangulation $\mathcal{T}$. These subspaces are characterized by specific Hermite degrees of freedom associated with only the vertices and edges of $\mathcal{T}$, or even only the vertices of $\mathcal{T}$. Each subspace still contains cubic polynomials while saving a consistent number of degrees of freedom compared with the full space. The dimension of the considered subspaces can be as small as six times the number of vertices of $\mathcal{T}$. The method fits in the setting of macro-elements: any function of such a subspace can be constructed on each triangle of $\mathcal{T}$ separately by specifying the necessary Hermite degrees of freedom. The explicit local representation in terms of a local simplex spline basis is also provided. This simplex spline basis intrinsically takes care of the complex geometry of the Wang-Shi split, making it transparent to the user.