Harmonic potentials in the de Rham complex

Representing vector fields by potentials can be a challenging task in domains with cavities or tunnels, due to the presence of harmonic fields which are both irrotational and solenoidal but may have no scalar or vector potentials. For harmonic fields normal to the boundary, which exist in domains with cavities, the standard approach is to construct scalar potentials by solving Laplace’s equation with Dirichlet boundary conditions fitted to the closed surfaces surrounding the domain’s cavities. For harmonic fields tangent to the boundary, which exist in domains with tunnels, a similar method was lacking. In this article we present a construction of vector potentials obtained by solving curl-curl problems with inhomogeneous tangent boundary conditions fitted to closed curves looping around the tunnels. Just as the cavity surfaces represent a basis for the 2-chain homology group, these tunnel curves represent a basis for the 1-chain homology group and the corresponding vector potentials yield a basis for the tangent harmonic fields. In our analysis the linear independence of the harmonic fields is established by considering their fluxes through a collection of reciprocal surfaces. These surfaces, whose boundaries lie on the boundary of the domain and which are in intersection duality with the tunnel curves, represent a basis for the relative 2-chain homology group modulo the boundary: their existence in general domains follows from the Poincaré-Lefschetz duality. Applied to structure-preserving finite elements, our method also provides an exact geometric parametrization of the discrete harmonic fields in terms of discrete potentials.

💡 Research Summary

The paper addresses the long‑standing problem of representing divergence‑free and curl‑free (harmonic) vector fields in three‑dimensional domains that contain cavities or tunnels. In such non‑simply‑connected regions the de Rham complex admits non‑trivial harmonic spaces: H¹(Ω) consists of fields tangent to the boundary (normal trace zero) and H²(Ω) consists of fields normal to the boundary. Their dimensions equal the first and second Betti numbers β₁ and β₂, i.e. the numbers of independent tunnels and cavities. While scalar potentials for normal harmonic fields (associated with cavities) are well understood via Laplace problems on closed cavity surfaces, an analogous construction for tangent harmonic fields (associated with tunnels) has been missing.

The authors propose a systematic construction of vector potentials for the tangent harmonic fields. The key geometric ingredients are:

1. A set of closed curves Γ₁,…,Γ_{β₁} lying on the boundary ∂Ω, each encircling a distinct tunnel and generating the first homology group H₁(Ω).
2. For each Γ_j a “reciprocal surface” Σ_j, a relative 2‑chain whose boundary lies on ∂Ω and which intersects Γ_j transversally with intersection number one. The existence of such surfaces follows from the Poincaré‑Lefschetz duality.

Using these objects the authors define linear flux functionals Φ_j(v)=∫_{Σ_j} v·n dS. The collection {Φ_j} separates elements of H¹(Ω) and will later be used to prove linear independence.

The construction of a vector potential A_j proceeds in two steps. First, a boundary lift ϕ_j is built: a scalar function λ_j defined on Γ_j (with prescribed jump) is extended to a tangential vector field on ∂Ω and then into Ω as a curl‑free field whose tangential trace matches λ_j on Γ_j. This field satisfies curl ϕ_j=0 but does not yet belong to H¹(Ω) because it may have non‑zero curl in the interior.

Second, a correction field ψ_j ∈ H(curl;Ω)∩H₀(div;Ω) is obtained by solving the homogeneous curl‑curl problem

 curl curl ψ_j = –curl curl ϕ_j in Ω, n×ψ_j = 0 on ∂Ω.

Standard variational theory guarantees a unique solution. The final potential A_j = ϕ_j + ψ_j satisfies curl A_j ∈ H¹(Ω) and has the prescribed tangential boundary data.

The authors then prove that the set {h_j = curl A_j} forms a basis of the tangent harmonic space H¹(Ω). By evaluating the fluxes Φ_k(h_j) they obtain the identity Φ_k(h_j)=δ_{kj}, which follows from the intersection duality of Γ_j and Σ_k. Hence the h_j are linearly independent and span a β₁‑dimensional subspace, which by the de Rham theorem must be the whole H¹(Ω).

After establishing the continuous theory, the paper turns to discretization within the Finite Element Exterior Calculus (FEEC) framework. Using structure‑preserving finite element families (Whitney 1‑forms, higher‑order Nédélec elements), the authors construct discrete analogues of the Γ_j, Σ_j, the lifted boundary potentials, and the curl‑curl correction. The resulting discrete vector potentials A_h^j yield discrete harmonic fields curl A_h^j that form a basis of the discrete harmonic space H_h¹, whose dimension also equals β₁. The discrete flux matrix remains (up to numerical error) the identity, confirming linear independence at the algebraic level.

The paper concludes by emphasizing that the method provides an exact geometric parametrization of both continuous and discrete tangent harmonic fields, without resorting to artificial cuts or non‑smooth scalar potentials. This advances the treatment of topologically non‑trivial domains in electromagnetic, fluid‑dynamic, and plasma‑physics simulations, where preserving the underlying de Rham structure is crucial for stability and physical fidelity. The approach is also readily extensible to other dimensions, to mixed boundary conditions, and potentially to nonlinear problems where harmonic components must be isolated.