Intrinsic Finite Element Error Analysis on Manifolds with Regge Metrics, with Applications to Calculating Connection Forms

TL;DR

This paper develops an intrinsic finite element error analysis framework for PDEs on manifolds with Regge metrics, enabling accurate computation of connection forms without relying on preferred coordinates or embeddings.

Contribution

It introduces an intrinsic approach to finite element exterior calculus on manifolds with approximate metrics, facilitating computation of connection forms in a coordinate-free manner.

Findings

Error bounds for finite element approximations on Regge metrics

Implementation of a method to compute approximate Levi-Civita connection forms

Demonstration of the approach on a disc with an approximate metric

Abstract

We present some aspects of the theory of finite element exterior calculus as applied to partial differential equations on manifolds, especially manifolds endowed with an approximate metric called a Regge metric. Our treatment is intrinsic, avoiding wherever possible the use of preferred coordinates or a preferred embedding into an ambient space, which presents some challenges but also conceptual and possibly computational advantages. As an application, we analyze and implement a method for computing an approximate Levi-Civita connection form for a disc whose metric is itself approximate.