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Hodge Decomposition of 1-Forms

Overview

The Hodge–Helmholtz decomposition (or Hodge decomposition) states that every smooth 1-form $\alpha$ on a compact Riemannian manifold can be uniquely decomposed as:

\[\alpha = d\phi + \delta\psi + h\]

where:

  • \[d\phi = \nabla_\Gamma \phi\]

    is the exact component (gradient of a potential $\phi$),

  • \[\delta\psi = \star d \star \psi\]

    is the coexact component (codifferential of a potential $\psi$),

  • \[h\]

    is the harmonic component ($dh = 0$, $\delta h = 0$).

The dimension of the harmonic space equals the first Betti number $b_1 = 2g$, where $g$ is the genus of the surface:

SurfaceGenus $g$$b_1$Harmonic forms
Sphere00None (trivial harmonic space)
Torus12Two independent harmonic 1-forms
Genus-$g$ surface$g$$2g$$2g$ independent harmonic 1-forms

Discrete Hodge decomposition

Step 1: Exact component

Solve the scalar Poisson equation for $\phi$:

\[L \phi = \delta_1 \alpha = \star_0^{-1} d_0^\top \star_1 \alpha\]

(the codifferential of $\alpha$), then set $\alpha_{\text{exact}} = d_0 \phi$.

On a closed surface $L$ is singular; the Tikhonov regularisation $(L + \varepsilon I) \phi = \delta_1 \alpha$ is used.

Step 2: Coexact component

Solve the dual Poisson equation for $\psi$:

\[\tilde{L} \psi = d_1 \alpha\]

where $\tilde{L} = \star_2^{-1} d_1 \star_1^{-1} d_1^\top \star_2$ is the Laplacian on 2-forms (equivalently, on dual 0-cochains).

Set $\alpha_{\text{coexact}} = \delta_2 (\star_2 \psi) = \star_1^{-1} d_1^\top \star_2 \psi$.

Step 3: Harmonic residual

\[h = \alpha - \alpha_{\text{exact}} - \alpha_{\text{coexact}}\]

For a genus-0 surface (sphere) $h \approx 0$ (up to numerical error). For a torus $h$ captures the two independent topological cycles.

API

# Full decomposition
result = hodge_decompose_1form(mesh, geom, dec, α; reg=1e-10)
# result.exact       Vector{T}  — α_exact  = d0 φ
# result.coexact     Vector{T}  — α_coexact = δ₂(⋆₂ ψ)
# result.harmonic    Vector{T}  — h = α - α_exact - α_coexact
# result.phi         Vector{T}  — scalar potential φ
# result.psi         Vector{T}  — stream potential ψ

# Individual components
α_exact,   φ = exact_component_1form(mesh, geom, dec, α)
α_coexact, ψ = coexact_component_1form(mesh, geom, dec, α)
h              = harmonic_component_1form(mesh, geom, dec, α)

Diagnostics

Reconstruction residual

\[r = \frac{\|\alpha - \alpha_{\text{exact}} - \alpha_{\text{coexact}} - h\|_{\star_1}}{\|\alpha\|_{\star_1}}\]

where the $\star_1$-norm is $\|\cdot\|_{\star_1}^2 = \alpha^\top \star_1 \alpha$.

r = hodge_decomposition_residual(mesh, geom, dec, α, result)
# r ≈ 0 (machine precision) by definition

Orthogonality check

The three components should be mutually orthogonal in $L^2$:

\[\langle \alpha_{\text{exact}},\, \alpha_{\text{coexact}} \rangle_{\star_1} \approx 0\]

\[\langle \alpha_{\text{exact}},\, h \rangle_{\star_1} \approx 0\]

\[\langle \alpha_{\text{coexact}},\, h \rangle_{\star_1} \approx 0\]

ips = hodge_inner_products(mesh, geom, dec, result)
# ips.exact_coexact   ≈ 0
# ips.exact_harmonic  ≈ 0
# ips.coexact_harmonic ≈ 0

Physical interpretation

ComponentGeometryPhysics
$\alpha_{\text{exact}} = d\phi$Gradient field, curl-freeConservative force, electric field
$\alpha_{\text{coexact}} = \delta\psi$Divergence-free, no potentialMagnetic field, solenoidal flow
$h$ (harmonic)Both curl-free and divergence-freeTopological obstruction, DC current on a ring

Example: sphere (genus 0)

For a gradient 1-form $\alpha = d_0 u$ (exact by construction):

  • \[\alpha_{\text{exact}} \approx \alpha\]

  • \[\alpha_{\text{coexact}} \approx 0\]

  • \[h \approx 0\]

u = [p[3] for p in mesh.points]   # height function z
α = dec.d0 * u                    # exact 1-form

result = hodge_decompose_1form(mesh, geom, dec, α)
println(norm(result.coexact))    # → ~1e-12
println(norm(result.harmonic))   # → ~1e-12

Example: torus (genus 1)

On the torus there exist two linearly independent harmonic 1-forms dual to the two fundamental cycles (meridional circle and longitudinal circle).

For a non-exact 1-form on the torus:

# Build a 1-form that winds around the torus once (not exact)
α = build_winding_oneform(mesh)   # custom test form

result = hodge_decompose_1form(mesh, geom, dec, α)
println(norm(result.harmonic))   # → significant — the harmonic component
                                   # captures the topological winding

The harmonic part has the same $L^2$ norm independent of the particular representative chosen in the cohomology class.

Potential recovery

The scalar potential $\phi$ satisfies:

\[\nabla_\Gamma \phi = \alpha_{\text{exact}}\]

and the stream potential $\psi$ satisfies:

\[\mathrm{rot}_\Gamma \psi \approx \alpha_{\text{coexact}}\]

These potentials can be used to integrate trajectories, compute stream lines, or reconstruct the original 1-form:

φ = result.phi   # scalar potential (vertex field)
ψ = result.psi   # stream function (vertex field)

References

  • Desbrun, M., Kanso, E., & Tong, Y. (2008). Discrete differential forms for computational modeling. In Discrete Differential Geometry, pp. 287–324.
  • Gu, X., & Yau, S.T. (2003). Global conformal surface parameterization. Proc. Eurographics Symposium on Geometry Processing.
  • Hirani, A.N. (2003). Discrete exterior calculus. Ph.D. thesis, Caltech.

See also