Interpolation with C2 quartic macro-elements based on 10-splits

Abstract

In this paper interpolation methods for the construction of quartic splines on triangulations refined with 10-splits are proposed. After examining the C2 macro-structure on a single triangle in terms of the Bernstein–Bézier representation, three methods that can be applied on general triangulations are developed. The methods make use of Hermite interpolation data prescribed at the vertices and (optionally) on the edges of the triangulation. The first two approaches lead to splines that are C2 continuous on triangles and C1 continuous across the interior edges of the triangulation. The third method gives rise to splines with overall C2 continuity, which is an exceptionally high order of smoothness for splines of degree four, but comes at the cost of solving a global system of linear equations. The derived results are accompanied with a few numerical examples that show an interesting behavior of splines in dependence of interpolated data.