Maximally smooth cubic spline quasi-interpolants on arbitrary triangulations

We investigate the construction of $C^2$ cubic spline quasi-interpolants on a given arbitrary triangulation $T$ to approximate a sufficiently smooth function f. The proposed quasi-interpolants are locally represented in terms of a simplex spline basis defined on the cubic Wang–Shi refinement of the triangulation. This basis behaves like a B-spline basis within each triangle of $T$ and like a Bernstein basis for imposing smoothness across the edges of $T$. Any element of the cubic Wang–Shi spline space can be uniquely identified by considering a local Hermite interpolation problem on every triangle of $T$. Different $C^2$ cubic spline quasi-interpolants are then obtained by feeding different sets of Hermite data to this Hermite interpolation problem, possibly reconstructed via local polynomial approximation. All the proposed quasi-interpolants reproduce cubic polynomials and their performance is illustrated with various numerical examples.