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

Evan S. Gawlik, Jack McKee

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.

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

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.

Paper Structure

This paper contains 12 sections, 13 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Computing connection forms.
  3. Contributions.
  4. Notation
  5. Sobolev Spaces on Manifolds with Regge Metrics
  6. Quantifying Quasi-Isometry
  7. Trace and Inverse Inequalities
  8. Riemannian Shape-Regularity
  9. Error Analysis of the Hodge-Laplace Problem on Manifolds with Regge Metrics
  10. Calculation of Connection Forms as a Hodge-Laplace Problem
  11. Error Analysis of the Connection Form Problem
  12. Numerical Example and Benchmarking

Key Result

Lemma 1

Let $V$ be a vector space and $(V,\|\cdot\|_{g_1})$, $(V,\|\cdot\|_{g_2})$ be two reflexive Banach space structures on $V$. By abuse of notation, also use the same symbols for the induced operator norms on $V^*$. Then: and

Figures (2)

  • Figure 1: Graphs of relative error $\|du_h - \star\alpha\|_{L^2(M,g)}/\|\alpha\|_{L^2(M,g)}$ vs. the parameter $h$ which controls the diameter of mesh elements.
  • Figure 2: Plots of $E$ = relative error of $du_h$ measured in the $g$ metric, ndof = total degrees of freedom (including those used in constructing the Regge metric).

Theorems & Definitions (29)

  • Remark 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Lemma 2
  • Remark 2
  • proof
  • Theorem 2: $H^1$ Trace Inequality for Riemannian Simplices
  • ...and 19 more