Simplified convergence proof in B\'ezier finite elements on D-dimensional simplex

TL;DR

This paper presents a simplified convergence proof for Bézier finite elements on D-dimensional simplices, utilizing the Stone-Weierstrass theorem to derive error estimates involving exponential function approximation.

Contribution

It introduces a new, simplified proof of convergence for Bézier polynomials on arbitrary dimensional simplices using topological methods.

Findings

Convergence of Bézier polynomials is established on D-dimensional simplices.

Error estimates relate approximation accuracy to exponential function approximation.

The proof simplifies previous approaches by leveraging the Stone-Weierstrass theorem.

Abstract

By using a general formalism, we expose a simplified proof of the convergence of the B\'ezier polynomials attached to a continuous function defined in arbitrary dimensional simplex. We obtain an error estimate that contains the error in approximating by exponential functions. Our new proof is based on the topological Stone-Weierstrass theorem.