Approximating surface areas by interpolations on triangulations

TL;DR

This paper investigates how different interpolation methods on triangulations can approximate surface areas, proving convergence under various conditions and extending classical results.

Contribution

It provides an alternative proof for Lagrange interpolation convergence and shows Crouzeix--Raviart interpolation converges without geometric restrictions.

Findings

Lagrange interpolation converges under maximum angle condition.

Crouzeix--Raviart interpolation converges without geometric constraints.

Extends classical surface area approximation results.

Abstract

We consider surface area approximations by Lagrange and Crouzeix--Raviart interpolations on triangulations. For Lagrange interpolation, we give an alternative proof for Young's classical result that claims the areas of inscribed polygonal surfaces converge to the area of the original surface under the maximum angle condition on the triangulation. For Crouzeix--Raviart interpolation we show that the approximated surface areas converge to the area of the original surface without any geometric conditions on the triangulation.