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

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.