Regge metrics with enhanced trace

TL;DR

This paper develops advanced Regge finite element metrics with improved trace properties, enabling better mathematical modeling for applications like general relativity and elasticity using high-order polynomials and refined meshes.

Contribution

It introduces new variants of Regge metrics with enhanced trace properties, including surjectivity to finite element spaces, applicable on refined meshes and high-order polynomials.

Findings

Metrics based on high-order polynomials constructed on refined meshes.

Enhanced trace operator is surjective to a finite element space of continuous functions.

Potential applications in general relativity, elasticity, and conformal geometry.

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.