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An electrical engineering perspective on naturality in computational physics

P. Robert Kotiuga, Valtteri Lahtinen

TL;DR

This paper derives formulas involving simplices and barycentric coordinates, defining discrete vector fields and a discrete Lie derivative as a result of a discrete analogue of Cartan’s magic formula, which is the main mathematical result of the paper.

Abstract

We look at computational physics from an electrical engineering perspective and suggest that several concepts of mathematics, not so well-established in computational physics literature, present themselves as opportunities in the field. We discuss elliptic complexes and highlight the category theoretical background and its role as a unifying language between algebraic topology, differential geometry, and modelling software design. In particular, the ubiquitous concept of naturality is central. Natural differential operators have functorial analogues on the cochains of triangulated manifolds. In order to establish this correspondence, we derive formulas involving simplices and barycentric coordinates, defining discrete vector fields and a discrete Lie derivative as a result of a discrete analogue of Cartan's magic formula. This theorem is the main mathematical result of the paper.

An electrical engineering perspective on naturality in computational physics

TL;DR

This paper derives formulas involving simplices and barycentric coordinates, defining discrete vector fields and a discrete Lie derivative as a result of a discrete analogue of Cartan’s magic formula, which is the main mathematical result of the paper.

Abstract

We look at computational physics from an electrical engineering perspective and suggest that several concepts of mathematics, not so well-established in computational physics literature, present themselves as opportunities in the field. We discuss elliptic complexes and highlight the category theoretical background and its role as a unifying language between algebraic topology, differential geometry, and modelling software design. In particular, the ubiquitous concept of naturality is central. Natural differential operators have functorial analogues on the cochains of triangulated manifolds. In order to establish this correspondence, we derive formulas involving simplices and barycentric coordinates, defining discrete vector fields and a discrete Lie derivative as a result of a discrete analogue of Cartan's magic formula. This theorem is the main mathematical result of the paper.

Paper Structure

This paper contains 31 sections, 3 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Methodology and objectives
  4. The structure of the article
  5. Mathematical structures for discrete solvers
  6. (Co)chain complexes.
  7. Elliptic complexes
  8. Naturality
  9. Naturality associated with differentiable structure
  10. Naturality associated with geometric structure
  11. Homology and cohomology
  12. An electrical engineering interlude
  13. Correspondence between cochains and differential forms
  14. The path to discrete: Finite element method
  15. Weak formulation
  16. ...and 16 more sections

Key Result

Lemma 1

where and $\frac{\partial}{\partial \mu_i} \mathbin{\raisebox{\depth}{[2]{$\lrcorner$}}} {\rm d} \mu_j$ is subject to

Theorems & Definitions (5)

  • Remark
  • Lemma 1: Discrete contraction 1
  • Lemma 2: Discrete contraction 2
  • Definition 1
  • Theorem 1: Discrete Cartan's magic formula