Construction of a $C^2$ class finite element based on the Clough-Tocher subdivision

TL;DR

This paper introduces a new $C^2$ finite element method based on Clough-Tocher subdivision, utilizing derivatives up to second order at vertices and edges, resulting in a globally smooth, piecewise polynomial interpolant.

Contribution

The paper presents a novel $C^2$ finite element construction with specific derivative conditions and polynomial degree, enhancing smoothness and local support properties.

Findings

Constructed a globally $C^2$ interpolant of degree ≤ 5

Uses derivatives up to second order at vertices and edges

Achieves local support with smoothness across the mesh

Abstract

In this paper, we construct a $C^2$ finite element based on the Clough-Tocher subdivision. We use derivatives order up to two at the vertices and cross boundary derivatives order up to two along the exterior edges of the triangle. The centroid of the triangle is just evaluated. The interpolant used is globally $C^2,$ has local support, is piecewise polynomial of degree less or equal to 5.