Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jay Gopalakrishnan, Michael Neunteufel, Joachim Schöberl, Max Wardetzky

TL;DR

The paper develops a Regge-element based framework to approximate intrinsic Gauss curvature and the Levi-Civita connection on 2D manifolds, proving superconvergence for curvature and connection when analyzed through covariant curl and incompatibility. It introduces distributional covariant operators on piecewise smooth metrics, extends them to Regge metrics via a glued smooth structure, and couples them with Regge interpolation to obtain higher-order convergence than standard estimates. A rigorous error-analysis establishes bounds for covariant curl, incompatibility, curvature, and the connection form, supported by detailed numerical experiments that confirm the predicted rates. The work advances finite element exterior calculus for Regge metrics and suggests broader applicability to discretizations of intrinsic geometric quantities on manifolds.

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.

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

TL;DR

The paper develops a Regge-element based framework to approximate intrinsic Gauss curvature and the Levi-Civita connection on 2D manifolds, proving superconvergence for curvature and connection when analyzed through covariant curl and incompatibility. It introduces distributional covariant operators on piecewise smooth metrics, extends them to Regge metrics via a glued smooth structure, and couples them with Regge interpolation to obtain higher-order convergence than standard estimates. A rigorous error-analysis establishes bounds for covariant curl, incompatibility, curvature, and the connection form, supported by detailed numerical experiments that confirm the predicted rates. The work advances finite element exterior calculus for Regge metrics and suggests broader applicability to discretizations of intrinsic geometric quantities on manifolds.

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.
Paper Structure (33 sections, 31 theorems, 222 equations, 10 figures)

This paper contains 33 sections, 31 theorems, 222 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Spaces on the manifold.
  4. Curvature.
  5. Approximate metric.
  6. Tangents and normals on element boundaries.
  7. Finite element spaces.
  8. Integrals over the manifold's triangulation.
  9. Curvature approximation
  10. A finite element curvature approximation
  11. Implementation issues
  12. A model case for further analysis
  13. Covariant derivatives using the nonsmooth metric
  14. Covariant curl and incompatibility for smooth metric
  15. Covariant curl in the Regge metric
  16. ...and 18 more sections

Key Result

Lemma 3.2

The geodesic curvature along each edge of an element boundary $\partial T$ is given by

Figures (10)

  • Figure 1: Illustration of the vertex jump defined in \ref{['eq:vertexxjump']}.
  • Figure 2: Angle between frames on different elements.
  • Figure 3: Left: Embedded surface, color indicates to $z$-component. Right: Exact Gauss curvature as graph over the domain $\varOmega$.
  • Figure 4: Left: Domain with Dirichlet and Neumann boundaries. Middle and right: perturbed unstructured triangular mesh grids.
  • Figure 5: Convergence of Gauss curvature with respect to number of degrees of freedom (ndof) in different norms for Regge elements ${g}\in\mathcal{R}_h^k$ of order $k=0,1,2$.
  • ...and 5 more figures

Theorems & Definitions (68)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • proof
  • Definition 4.2
  • ...and 58 more