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Regge metrics with enhanced trace

Snorre H. Christiansen, Ting Lin

Abstract

We introduce variants of Regge finite element metrics with enhanced properties of the trace. In particular the trace operator is surjective to a finite element space of continuous functions. Multiplying these scalar functions by the identity tensor brings one back to the finite element space of metrics. The metrics can be based on high order polynomials and be constructed on refinements, such as the Clough-Tocher or Worsey-Farin splits. Potential applications to general relativity, incompressible elasticity and conformal geometry are sketched.

Regge metrics with enhanced trace

Abstract

We introduce variants of Regge finite element metrics with enhanced properties of the trace. In particular the trace operator is surjective to a finite element space of continuous functions. Multiplying these scalar functions by the identity tensor brings one back to the finite element space of metrics. The metrics can be based on high order polynomials and be constructed on refinements, such as the Clough-Tocher or Worsey-Farin splits. Potential applications to general relativity, incompressible elasticity and conformal geometry are sketched.
Paper Structure (16 sections, 23 theorems, 54 equations, 4 figures)

This paper contains 16 sections, 23 theorems, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Notations
  4. Continuous strain complexes with enhanced trace
  5. Goals and applications
  6. Regge elements and complexes: old and new
  7. Trace enhanced Regge elements in dimension two
  8. Trace enhanced elements
  9. Complexes built on trace enhanced elements
  10. Trace enhanced Regge complexes on the Hsieh--Clough--Tocher split
  11. Trace enhanced Regge elements in three dimensions
  12. Construction on general meshes
  13. Trace enhanced element space on Worsey-Farin split
  14. Regarding cohomologies
  15. Cohomology of high order Regge complexes in dimension two
  16. ...and 1 more sections

Key Result

Proposition 1.1

The map $\sigma \to (\mathop{\mathrm{tr}}\nolimits \sigma) \mathbb{I}$ sends $\underline{\underline{\mathrm{H}}}_{\mathop{\mathrm{tr}}\nolimits}(\Omega)$ to $\mathrm{H}^1(\Omega) \mathbb{I}$ (by definition) and the latter space is included in $\underline{\underline{\mathrm{H}}}_{\mathop{\mathrm{tr}}

Figures (4)

  • Figure 1: Regge complex \ref{['eq:regge-2d-k']} with $k=1$ (starting with $\mathrm{Lag}_2 \otimes \mathbb{R}^2$)
  • Figure 2: The trace enhanced Regge complex \ref{['eq:discrete-2D']} with $k=3$.
  • Figure 3: trace enhanced Regge complex \ref{['eq:dicsrete-2D-low']} on CT split
  • Figure 4: Reduced trace enhanced Regge complex based on $\widetilde{\Sigma}(T)$ on CT split.

Theorems & Definitions (47)

  • Proposition 1.1
  • Proposition 1.2
  • proof
  • Remark 1.1
  • Proposition 1.3
  • proof
  • Remark 1.2: Hellinger-Reissner
  • Theorem 2.1
  • Proposition 2.2
  • Remark 2.1
  • ...and 37 more