The polynomial spline functions defined on triangulations are tools widely used in many different fields, both theoretical and applied [1]. It is well known that C^r-regularity of a spline on a given triangulation is obtained if all derivatives up to order 2r at the vertices of the triangles, in which case the degree must be greater than or equal to 4r+1 [2].  As in practice it is essential to use splines of the lowest degree for a given class, different finite elements obtained by subdividing every triangle have been introduced and analysed in the literature, among them the Clough-Tocher (3-CT), Powell-Sabin (6-PS) and Morgan-Scott (MS-) splits [3, 4, 5], so that C^2 smoothness results, for minimum degrees 6, 5 and 5, respectively. The construction of C^2-continuous quartic splines on a triangulation endowed with a mixed split consisting of macro-triangles with PS-6 or Modified Morgan-Scott (MMS-10) refinements is addressed. Indeed, in [5, 6] it is proved that under a certain geometrical conditions between macro-triangles and edge split points, the space of almost C^2-continuous splines introduced in [7] becomes a subspace of the space of C^2-continuous functions. Joining the opposite vertices of every two triangles sharing an edge gives, in general, a mixed-type triangulation in the above sense. This procedure may result in a PS-6 refinement or an MS-split, from which an MMS-10 split is easily obtained.

For the mixed-type sub-triangulation, the construction of a basis of B-spline-like functions will be provided to establish a suitable representation of the C^2-continuous functions of the space.

References

[1] M. J. Lai, L.L. Schumaker. Spline Functions on Triangulations, CUP, Cambridge, 2007.

[2] A. Ženı́šek. A general theorem on triangular finite C (m) -elements, Rev Fr Automat Infor Analyse numérique, 8(R2) (1974) 119–127.
[3] R. W. Clough, J. L. Tocher. Finite element stiffness matrices for analysis of plates in bending, Proc. Conference on Matrix Methods in Structural Mechanics, Wright-Paterson A. F. B., Ohio, 515–545, 1985.
[4] M. Powell, M. Sabin. Piecewise quadratic approximations on triangles, ACM Trans Math Softw 3 (1977) 316–325.
[5] D. Sbibih, A. Serghini, A. Tijini. C 1 quadratic and C 2 quartic macro-elements on a modified Morgan-Scott triangulation, Mediterr J Math, 10 (2013) 1273–1292.
[6] D. Barrera, S. Eddargani, M. J. Ibáñez, A. Lamnii, A geometric characterization of Powell-Sabin triangulations allowing the construction of C 2 quartic splines,Computers & Mathematics with Applications 100 (2021) 30–40.
[7] J. Grošelj, M. Krajnc. Quartic splines on Powell-Sabin triangulations, Comput Aided Geom Design, 49 (2016) 1–16.