Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics

TL;DR

This paper demonstrates that certain Regge finite element approximations of Riemannian metrics can achieve higher convergence rates for curvature and connection computations by leveraging covariant curl and incompatibility operators, supported by theoretical analysis and numerical validation.

Contribution

It introduces a novel analysis framework using covariant curl and incompatibility operators to achieve superconvergence in Regge metric approximations.

Findings

Higher convergence rates for curvature and connection approximations.

Superconvergence results based on properties of the Regge space interpolant.

Numerical experiments confirm theoretical error estimates.

Abstract

The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and such approximations can be used to compute approximations to the Gauss curvature and the Levi-Civita connection of the manifold. It is shown that certain Regge approximations yield curvature and connection approximations that converge at a higher rate than previously known. The analysis is based on covariant (distributional) curl and incompatibility operators which can be applied to piecewise smooth matrix fields whose tangential-tangential component is continuous across element interfaces. Using the properties of the canonical interpolant of the Regge space, we obtain superconvergence of approximations of these covariant operators. Numerical experiments further illustrate the results from the error analysis.