Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866914676223246336 |
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| author | Gopalakrishnan, Jay Neunteufel, Michael Schöberl, Joachim Wardetzky, Max |
| author_facet | Gopalakrishnan, Jay Neunteufel, Michael Schöberl, Joachim Wardetzky, Max |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_09343 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics Gopalakrishnan, Jay Neunteufel, Michael Schöberl, Joachim Wardetzky, Max Numerical Analysis 65N30 (Primary) 53A70, 83C27 (Secondary) 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. |
| title | Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics |
| topic | Numerical Analysis 65N30 (Primary) 53A70, 83C27 (Secondary) |
| url | https://arxiv.org/abs/2206.09343 |