Harmonic potentials in the de Rham complex

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.