High-Order Approximation of Gaussian Curvature with Regge Finite Elements

Evan S. Gawlik

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arXiv · 1905.07004

High-Order Approximation of Gaussian Curvature with Regge Finite Elements

Evan S. Gawlik

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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 $k \ge -1$.

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Evan S. Gawlik. 2019-05-16. High-Order Approximation of Gaussian Curvature with Regge Finite Elements. https://arxiv.org/abs/1905.07004

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