# High-Order Approximation of Gaussian Curvature with Regge Finite Elements

## TL;DR

This paper introduces a high-order method for approximating Gaussian curvature on triangulated surfaces using Regge finite elements, providing error estimates and a generalization of classical angle defect techniques.

## Contribution

It develops a novel high-order approximation framework for Gaussian curvature based on Regge finite elements, extending classical methods and providing rigorous error analysis.

## Key findings

- Error estimates in Sobolev norms for high-order approximations
- Relation between angle defect linearization and div-div operator discretization
- Generalization of angle defect to higher order methods

## Abstract

A widely used approximation of the Gaussian curvature on a triangulated surface is the angle defect, which measures the deviation between $2\pi$ and the sum of the angles between neighboring edges emanating from a common vertex. We show that the linearization of the angle defect about an arbitrary piecewise constant Regge metric is related to the classical Hellan-Herrmann-Johnson finite element discretization of the div-div operator. Integrating this relation leads to an integral formula for the angle defect which is well-suited for analysis and generalizes naturally to higher order. We prove error estimates for these high-order approximations of the Gaussian curvature in $H^k$-Sobolev norms of integer order $k \ge -1$.