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

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.

References in corpus (7)

• Finite element exterior calculus: from Hodge theory to numerical stability
• Finite elements for symmetric tensors in three dimension
• Discrete structures in gravity
• Complexes from complexes
• Avoiding Membrane Locking with Regge Interpolation
• Finite element approximation of the Levi-Civita connection and its curvature in two dimensions
• Tullio Regge's legacy: Regge calculus and discrete gravity

Cited by in corpus (4)

• Finite element approximation of the Levi-Civita connection and its curvature in two dimensions
• Exact formulas for the approximation of connections and curvature
• Blow-up Whitney forms, shadow forms, and Poisson processes
• On the improved convergence of lifted distributional Gauss curvature from Regge elements