  This paper is devoted to the design and analysis of some structure-preserving
finite element schemes for the magnetohydrodynamics (MHD) system. The main
feature of the method is that it naturally preserves the important Gauss law,
namely $\nabla\cdot\boldsymbol{B}=0$. In contrast to most existing approaches
that eliminate the electrical field variable $\boldsymbol{E}$ and give a direct
discretization of the magnetic field, our new approach discretizes the electric
field $\boldsymbol{E}$ by N\'{e}d\'{e}lec type edge elements for
$H(\mathrm{curl})$, while the magnetic field $\boldsymbol{B}$ by Raviart-Thomas
type face elements for $H(\mathrm{div})$. As a result, the divergence-free
condition on the magnetic field holds exactly on the discrete level. For this
new finite element method, an energy stability estimate can be naturally
established in an analogous way as in the continuous case. Furthermore,
well-posedness is rigorously established in the paper for both the Picard and
Newton linearization of the fully nonlinear systems by using the Brezzi theory
for both the continuous and discrete cases. This well-posedness naturally leads
to robust (and optimal) preconditioners for the linearized systems.
