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 from curl-curl problems with inhomogeneous boundary conditions fitted to closed curves looping around the tunnels. Just as the cavity surfaces represent a basis for the 2-chain homology group, the 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 guaranteed by 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 Poincare-Lefschetz duality. Applied to structure-preserving finite elements with commuting projections and standard compatibility properties on the boundaries, our approach provides an exact geometric parametrization of the discrete harmonic fields in terms of (strong) discrete potentials. An interesting by-product is a direct proof that the resulting discrete harmonic spaces have the correct dimensions, which does not rely on uniform stability properties for the commuting projections.