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Most authors are very clear that it's very dangerous to just use $\mathrm{H}(curl)$ conforming edge elements, which are divergence free, to satisfy $\mathrm{div}(\mathbf{B})=0$ and implement this condition in no other way. I just don't understand why. Could anyone explain in which case this could fail or point me to some good source explaining in which case $\mathrm{H}(curl)$ conforming Nedelec Elements would fail to satisfy $\mathrm{div}(\mathbf{B})=0$ and why?

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One issue (and this is mentioned by Mur in the first paper you linked in the comments above) is the fact that, while these edge functions provide tangential field continuity across interfaces and zero divergence within each element, they also allow for discontinuities in normal fields across interfaces. This behavior is non-physical, giving rise to artificial surface charges which produce errors in finite element solutions.

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