Commuting projections

This page describes the canonical lowest-order FEEC interpolation operators.

Mathematical objects and DOFs

• Π0: scalar 0-form interpolation by vertex evaluation.
• Π1: 1-form interpolation by oriented edge integrals.
• Π2: 2-form interpolation by oriented face integrals.

Commuting identities

For suitable smooth fields, the layer enforces the canonical discrete identities:

• Π1(df) ≈ d0 Π0(f)
• Π2(dα) ≈ d1 Π1(α)

projection_commutator_01 and projection_commutator_12 expose residual vectors for these checks.

Orientation and storage convention

• Edge DOFs follow canonical edge orientation (i<j).
• Face DOFs follow mesh face orientation.
• The returned cochains are dense Julia vectors (Vector{T}) in the package's global simplex ordering.

API

• interpolate_0form, interpolate_1form, interpolate_2form
• aliases: Π0, Π1, Π2
• helpers: interpolate_exact_gradient, interpolate_exact_flux_density
• diagnostics: projection_commutator_01, projection_commutator_12, verify_commuting_projection

Minimal example

using FrontIntrinsicOps, LinearAlgebra

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

f = x -> x[1]^2 - 0.3x[2] + 0.1
r = projection_commutator_01(f, mesh, geom, dec)
println(norm(r))

Limitations and non-goals

• Canonical interpolators only (no smoothed bounded projections yet).
• Lowest-order simplicial setting only.
• Π2 currently targets surface meshes (no curve 2-cells).