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Harmonic potentials in the de Rham complex

Martin Campos Pinto, Julian Owezarek

TL;DR

This work tackles the challenge of representing tangent harmonic vector fields in 3D domains with cavities and tunnels by constructing vector potentials through curl-curl problems with tangent boundary conditions tied to tunnel curves. It leverages Poincaré-Lefschetz duality to pair tunnel curves with reciprocal surfaces, yielding a linearly independent basis for the tangent harmonic space $\mathfrak H^1$ and an exact geometric parametrization compatible with structure-preserving FEEC discretizations. The authors provide a two-part construction that lifts boundary data via a boundary potential and adds a correction to enforce homogenous curl-curl behavior, with a duality between the resulting flux functionals $\Phi_i$ and the harmonic fields. The framework extends naturally to discrete settings, enabling robust, topology-aware computation of tangent harmonic potentials in finite element exterior calculus while preserving geometric and topological structure.

Abstract

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.

Harmonic potentials in the de Rham complex

TL;DR

This work tackles the challenge of representing tangent harmonic vector fields in 3D domains with cavities and tunnels by constructing vector potentials through curl-curl problems with tangent boundary conditions tied to tunnel curves. It leverages Poincaré-Lefschetz duality to pair tunnel curves with reciprocal surfaces, yielding a linearly independent basis for the tangent harmonic space and an exact geometric parametrization compatible with structure-preserving FEEC discretizations. The authors provide a two-part construction that lifts boundary data via a boundary potential and adds a correction to enforce homogenous curl-curl behavior, with a duality between the resulting flux functionals and the harmonic fields. The framework extends naturally to discrete settings, enabling robust, topology-aware computation of tangent harmonic potentials in finite element exterior calculus while preserving geometric and topological structure.

Abstract

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.

Paper Structure

This paper contains 20 sections, 14 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Homology groups and harmonic function spaces
  3. Harmonic fields in the Hilbert de Rham complex
  4. Singular chains and homology groups
  5. The de Rham theorem for the $L^2$ de Rham complex
  6. Scalar and vector potentials
  7. Scalar potentials for the normal harmonic fields
  8. Construction of vector potentials for tangent harmonic fields
  9. Tunnel curves and reciprocal surfaces
  10. The surface flux functionals
  11. Construction by decomposition
  12. Lifted boundary potentials
  13. Correction potentials
  14. Main result and invariance principles
  15. Structure-preserving discrete harmonic potentials
  16. ...and 5 more sections

Key Result

Theorem 2.1

For $k = 0, \dots 3$, the following relation holds:

Figures (2)

  • Figure 1: Example of a non-contractible domain $\Omega$ with two cavities ($\beta_2 = 2$) and two tunnels ($\beta_1 = 2$): one that goes through the domain, and another one that is also a cavity and circles around the first tunnel. The domain boundary has three connected components which are closed surfaces (in orange, only partially drawn for the sake of visibility): $S_0$ is the "outer" boundary which bounds the unbounded component of $\mathbb{R}^3 \setminus \Omega$, $S_1$ bounds the contractible cavity, and $S_2$ bounds the inner tunnel. The last two represent independent classes of the 2-chain homology group $\mathcal{H}_2(\overline{\Omega})$, while the first one is homologous to $-(S_1+S_2)$. Finally, the closed curves $\Gamma_i \subset \partial \Omega$, $i = 1, 2$, (in light blue) represent two independent classes of the 1-chain homology group $\mathcal{H}_1(\overline{\Omega})$. They admit reciprocal surfaces $\Sigma_i$ as described in Assumption \ref{['as:Gamma']}, visualized by their boundary curves $\tilde{\Gamma}_{i,a} \subset \partial \Omega$, drawn in red.
  • Figure 2: Illustration of a tunnel curve $\Gamma_i$ and its neighborhood $X_i$, parametrized with a piecewise smooth homeomorphism $T_i$ defined on the (straight) torus $\mathcal{O}$, periodic along the variable $\theta$.

Theorems & Definitions (35)

  • Remark 2.1
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Definition 3.2
  • ...and 25 more