High-Order Approximation of Gaussian Curvature with Regge Finite Elements
Provides a theoretical foundation and higher-order generalization for Gaussian curvature approximation on triangulated surfaces, benefiting computational geometry and numerical analysis.
The paper shows that the angle defect approximation of Gaussian curvature on triangulated surfaces is related to the Hellan-Herrmann-Johnson finite element discretization, and generalizes this to higher-order approximations with proven error estimates in Sobolev norms.
A widely used approximation of the Gaussian curvature on a triangulated surface is the angle defect, which measures the deviation between $2π$ 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 $k \ge -1$.