GitHub

Whitney forms

This page documents the lowest-order Whitney basis/reconstruction layer.

Mathematical objects

  • Whitney 0-forms: piecewise-linear scalar basis on vertices.
  • Whitney 1-forms: edge-based basis on triangles.
  • Whitney 2-forms: face-based constant area-density basis.

Degrees and cochain spaces

  • degree-0: c0::Vector on vertices,
  • degree-1: c1::Vector on oriented edges,
  • degree-2: c2::Vector on oriented faces.

Orientation convention

  • Global edge orientation follows canonical edge ordering (i < j) from build_topology.
  • Face-local Whitney-1 ordering is (1→2), (2→3), (3→1) in face orientation.
  • Local/global sign transfer uses topo.face_edge_signs.

Storage convention

  • Reconstructed 0-form: facewise affine scalar.
  • Reconstructed 1-form: tangent-vector representation (α^#) evaluated facewise.
  • Reconstructed 2-form: facewise constant density (c2[f] / area[f]).

API

  • whitney0_basis_local, whitney1_basis_local, whitney2_basis_local
  • eval_whitney0_local, eval_whitney1_local, eval_whitney2_local
  • reconstruct_0form_face, reconstruct_1form_face, reconstruct_2form_face
  • reconstruct_0form, reconstruct_1form, reconstruct_2form

Minimal example

using FrontIntrinsicOps, StaticArrays

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

c1 = interpolate_1form(x -> SVector{3,Float64}(-x[2], x[1], 0.0), mesh, geom)
vface = reconstruct_1form(c1, mesh, geom; representation=:facewise_tangent)
println(norm(vface[1]))

Limitations and non-goals

  • Lowest-order only.
  • Facewise reconstruction API (not full global point-location/evaluation).
  • No higher-order Whitney-like families in this round.