Construction of 2D explicit cubic quasi-interpolating splines in Bernstein-B\'ezier form

TL;DR

This paper presents a method for constructing smooth cubic quasi-interpolating splines in 2D using Bernstein-Bézier coefficients, ensuring high polynomial reproduction and global properties, validated by numerical tests.

Contribution

It introduces a new approach for building $C^{1}$ cubic quasi-interpolants on a 2D mesh directly via Bernstein-Bézier coefficients, emphasizing polynomial reproduction and global properties.

Findings

Numerical tests confirm the approximation capabilities.

The method achieves high polynomial reproduction.

Constructs $C^{1}$ cubic quasi-interpolants on 2D meshes.

Abstract

In this paper, the construction of $C^{1}$ cubic quasi-interpolants on a three-direction mesh of $\RR^{2}$ is addressed. The quasi-interpolating splines are defined by directly setting their Bernstein-B\'{e}zier coefficients relative to each triangle from point and gradient values in order to reproduce the polynomials of the highest possible degree. Moreover, additional global properties are required. Finally, we provide some numerical tests confirming the approximation properties.