Discrete Vector Potentials for Nonsimply Connected Three-Dimensional Domains

In this paper, we focus on the representation of a divergence-free vector field, defined, on a connected nonsimply connected domain Ω ⊂ R3 with a connected boundary Γ, by its curl and its normal component on the boundary. The considered problem is discretized with H(curl)and H(div)-conforming finite elements. In order to ensure the uniqueness of the vector potential, we propose a spanning tree methodology to identify the independent edges. The topological features of the domain under consideration are analyzed here by means of the homology groups of first and second order.