Maximally smooth cubic spline quasi-interpolants on arbitrary triangulations

We investigate the construction of C2 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 C2 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.