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On the improved convergence of lifted distributional Gauss curvature from Regge elements

Jay Gopalakrishnan, Michael Neunteufel, Joachim Schöberl, Max Wardetzky

Abstract

Although Regge finite element functions are not continuous, useful generalizations of nonlinear derivatives like the curvature, can be defined using them. This paper is devoted to studying the convergence of the finite element lifting of a generalized (distributional) Gauss curvature defined using a metric tensor approximation in the Regge finite element space. Specifically, we investigate the interplay between the polynomial degree of the curvature lifting by Lagrange elements and the degree of the metric tensor in the Regge finite element space. Previously, a superconvergence result, where convergence rate of one order higher than expected, was obtained when the approximate metric is the canonical Regge interpolant of the exact metric. In this work, we show that an even higher order can be obtained if the degree of the curvature lifting is reduced by one polynomial degree and if at least linear Regge elements are used. These improved convergence rates are confirmed by numerical examples.

On the improved convergence of lifted distributional Gauss curvature from Regge elements

Abstract

Although Regge finite element functions are not continuous, useful generalizations of nonlinear derivatives like the curvature, can be defined using them. This paper is devoted to studying the convergence of the finite element lifting of a generalized (distributional) Gauss curvature defined using a metric tensor approximation in the Regge finite element space. Specifically, we investigate the interplay between the polynomial degree of the curvature lifting by Lagrange elements and the degree of the metric tensor in the Regge finite element space. Previously, a superconvergence result, where convergence rate of one order higher than expected, was obtained when the approximate metric is the canonical Regge interpolant of the exact metric. In this work, we show that an even higher order can be obtained if the degree of the curvature lifting is reduced by one polynomial degree and if at least linear Regge elements are used. These improved convergence rates are confirmed by numerical examples.
Paper Structure (15 sections, 8 theorems, 79 equations, 2 figures)

This paper contains 15 sections, 8 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Regge metric
  4. Differential geometry
  5. Finite element spaces
  6. Lifted distributional Gauss curvature
  7. Distributional covariant differential operators
  8. Error analysis
  9. Statement of main theorem
  10. Basic estimates
  11. Analysis of distributional rotrot operator
  12. Proof of Theorem \ref{['thm:improved_rate_Hm1']}
  13. Analysis of the lifting of pure Gauss curvature
  14. Proof of Corollary \ref{['cor:improved_conv_dens_curvature']}
  15. Numerical examples

Key Result

Lemma 2.5

Let $\sigma\in\mathrm{Reg}(\mathscr{T})$ and $u\in \mathcal{V}(\mathscr{T})$. There holds

Figures (2)

  • Figure 1: Convergence of lifted densitized Gauss curvature with respect to number of degrees of freedom (ndof) in different norms for Regge elements ${g_h}\in\mathrm{Reg}_h^k$ of order $k=1,2,3$.
  • Figure 2: Convergence of pure lifted Gauss curvature with respect to number of degrees of freedom (ndof) in different norms for Regge elements ${g_h}\in\mathrm{Reg}_h^k$ of order $k=1,2,3$.

Theorems & Definitions (20)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • Corollary 3.2
  • Remark 3.3: Convergence of pure Gauss curvature
  • Remark 3.4: Optimal convergence
  • ...and 10 more