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Exterior Algebra Extensions

This page documents low-order practical operators added for DEC-form workflows.

Mathematical object

Implemented additions:

  • discrete wedge product (wedge, wedge0k, wedge11),
  • interior product / contraction (interior_product),
  • Lie derivative (lie_derivative) via Cartan formula:
    • L_X α = i_X dα + d(i_X α).

Cochain space / degree

Surface support

  • 0∧0 -> 0
  • 0∧1, 1∧0 -> 1
  • 0∧2, 2∧0 -> 2
  • 1∧1 -> 2
  • contraction:
    • i_X : Ω¹ -> Ω⁰,
    • i_X : Ω² -> Ω¹
  • Lie derivative:
    • Ω⁰ -> Ω⁰,
    • Ω¹ -> Ω¹,
    • Ω² -> Ω²

Curve support

  • wedge: 0∧0, 0∧1, 1∧0
  • contraction/Lie:
    • i_X : Ω¹ -> Ω⁰ with X as tangent-speed field,
    • L_X on Ω⁰ and Ω¹ via Cartan formula in 1D.

Orientation convention

  • Wedge products are orientation-dependent through face normals and edge orientation.
  • wedge11(α,β) is antisymmetric (α∧β = -β∧α) under this orientation convention.

Storage convention

  • Forms are stored as vectors in vertex/edge/face ordering.
  • Vector field X for contraction/Lie derivative is currently represented as per-face tangent vectors (representation=:face_vector).
  • On closed curves (nV == nE), pass degree=0 or degree=1 for interior_product / lie_derivative to disambiguate cochain degree.

API

wedge
wedge0k
wedge11
interior_product
lie_derivative
cartan_lie_derivative

Minimal example

using FrontIntrinsicOps
using LinearAlgebra

mesh = generate_icosphere(1.0, 1)
geom = compute_geometry(mesh)
dec = build_dec(mesh, geom)

u = [p[1] for p in mesh.points]
v = [p[2] for p in mesh.points]
α = dec.d0 * u
β = dec.d0 * v

w = wedge11(α, β, mesh, geom, dec)
X = [0.2 .* geom.face_normals[fi] for fi in 1:length(mesh.faces)]
Lα = lie_derivative(X, α, mesh, geom, dec)

(antisymmetry=norm(w + wedge11(β, α, mesh, geom, dec)), lie_norm=norm(Lα))

Limitations and non-goals

  • This is intentionally low-order and practical; no symbolic/all-degree abstraction layer.
  • Interior/Lie currently use practical fixed representations:
    • surface: representation=:face_vector,
    • curve: representation=:tangent_speed.
  • The implementation is designed for PDE operators and diagnostics, not for exact algebraic identities on arbitrary high-order elements.