A representation and comparison of three cubic macro-elements
For researchers in numerical analysis, this work provides a practical tool for using cubic macro-elements, but the contribution is incremental as it focuses on representation rather than new theory or performance gains.
The paper presents a unified representation for three types of cubic macro-element splines using locally supported basis functions in Bernstein-Bézier form, enabling their use in standard approximation methods. Numerical experiments demonstrate their applicability.
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--Bézier form. They are readily applicable in a range of standard approximation methods, which is demonstrated by a number of numerical experiments.