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 $\Omega \subset \R^3$ with a connected boundary $\Gamma$, 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.

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