A representation and comparison of three cubic macro-elements

TL;DR

This paper introduces a unified basis function representation for three types of cubic splines over triangulations, highlighting their differences and demonstrating their applicability in numerical approximation methods.

Contribution

It provides a new unified geometric and Bernstein--Bézier based framework for three cubic spline types, enhancing their versatility in numerical analysis.

Findings

Unified basis functions facilitate comparison of spline types.

Numerical experiments demonstrate practical applicability.

Differences in complexity, polynomial reproduction, and smoothness are characterized.

Abstract

The paper is concerned with three types of cubic splines over a triangulation that are characterized by three degrees of freedom associated with each vertex of the triangulation. The splines differ in computational complexity, polynomial reproduction properties, and smoothness. With the aim to make them a versatile tool for numerical analysis, a unified representation in terms of locally supported basis functions is established. The construction of these functions is based on geometric concepts and is expressed in the Bernstein--B\'ezier form. They are readily applicable in a range of standard approximation methods, which is demonstrated by a number of numerical experiments.