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

In this paper, we issue an error analysis for integration over discrete surfaces using the surface parametrization presented in Praetorius and Stenger (Arch Numer Softw 1(1):2022, 2022) as well as prove why even-degree polynomials utilized for approximating both the smooth surface and the integrand 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.