Analysis of Curvature Approximations via Covariant Curl and incompatibility for Regge metrics

Authors

Jay Gopalakrishnan, Portland State UniversityFollow
Michael Neunteufel, Institute for Analysis and Scientific Computing
Joachim Schöberl, Institute for Analysis and Scientific Computing
Max Wardetzky, Institute of Numerical and Applied Mathematics

Published In

JOURNAL OF COMPUTATIONAL MATHEMATICS

Document Type

Article

Publication Date

9-2023

Subjects

Differential geometry -- Gauss curvature

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.

Rights

License: CC-BY 4.0 Copyrights: The authors retain unrestricted copyrights and publishing rights

Locate the Document

10.5802/smai-jcm.98

DOI

10.5802/smai-jcm.98

Persistent Identifier

https://archives.pdx.edu/ds/psu/42684

Citation Details

Gopalakrishnan, J., Neunteufel, M., Schöberl, J., & Wardetzky, M. (2023). Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics. The SMAI Journal of computational mathematics, 9, 151-195.