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For the representation of a divergence-free vector field defined on a bounded simply connected domain of $\mathbb{R}^3 $ with a smooth boundary by its curl and its normal component on the boundary, a mixed formulation also involving a vector potential is proposed. The vector fields are discretized with help of curved finite elements that are conforming in $H({\operatorname{div}})$ or $H({\operatorname{curl}})$. A discrete gauge condition is proposed to assume the uniqueness of the stream function. Optimal error estimates are derived, and a numerical experiment is presented.

Discrete Vector Potential Representation of a Divergence-Free Vector Field in Three-Dimensional Domains: Numerical Analysis of a Model Problem