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.