Harmonic potentials in the de Rham complex

TL;DR

This paper develops a method to construct vector potentials for harmonic fields in domains with cavities and tunnels, using solutions to curl-curl problems with boundary conditions, enabling precise representation of harmonic fields in complex geometries.

Contribution

It introduces a novel approach to construct vector potentials for tangent harmonic fields in domains with tunnels, filling a gap in existing methods for such geometries.

Findings

Provides a basis for tangent harmonic fields using tunnel curves.

Establishes linear independence via fluxes through reciprocal surfaces.

Enables exact geometric parametrization in structure-preserving finite elements.

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 Poincar\'e-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.