A note on the rate of convergence of integration schemes for closed surfaces

TL;DR

This paper analyzes the convergence rates of surface integration schemes, explaining why even-degree polynomials perform better than odd-degree ones, supported by numerical examples and a proposed improved method.

Contribution

It provides an error analysis for surface integration schemes, explains the higher convergence of even-degree polynomials, and introduces a potential approach to address existing issues.

Findings

Even-degree polynomials have higher convergence rates than odd-degree polynomials.

Numerical examples support the theoretical error analysis.

A new approach is proposed to overcome limitations of the original scheme.

Abstract

In this paper, we issue an error analysis for integration over discrete surfaces using the surface parametrization presented in [PS22] as well as prove why even-degree polynomials exhibit a higher convergence rate than odd-degree polynomials. Additionally, we provide some numerical examples that illustrate our findings and propose a potential approach that overcomes the problems associated with the original one.