Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics
Gopalakrishnan, Jay1 ; Neunteufel, Michael2 ; Schöberl, Joachim2 ; Wardetzky, Max3
1 Portland State University, PO Box 751, Portland OR 97207, USA
2 Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstr. 8-10, 1040 Wien, Austria
3 Institute of Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany
The SMAI Journal of computational mathematics, Tome 9 (2023), pp. 151-195

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.

Publié le :
DOI : 10.5802/smai-jcm.98
Classification : 65N30, 53A70, 83C27
Keywords: Gauss curvature, Regge calculus, finite element method, differential geometry

Gopalakrishnan, Jay 1 ; Neunteufel, Michael 2 ; Schöberl, Joachim 2 ; Wardetzky, Max 3

1 Portland State University, PO Box 751, Portland OR 97207, USA
2 Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstr. 8-10, 1040 Wien, Austria
3 Institute of Numerical and Applied Mathematics, University of Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany 
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Gopalakrishnan, Jay; Neunteufel, Michael; Schöberl, Joachim; Wardetzky, Max. Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics. The SMAI Journal of computational mathematics, Tome 9 (2023), pp. 151-195. doi: 10.5802/smai-jcm.98

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