  We develop structure-preserving finite element methods for the
incompressible, resistive Hall magnetohydrodynamics (MHD) equations. These
equations incorporate the Hall current term in Ohm's law and provide a more
appropriate description of fully ionized plasmas than the standard MHD
equations on length scales close to or smaller than the ion skin depth. We
introduce a stationary discrete variational formulation of Hall MHD that
enforces the magnetic Gauss's law exactly (up to solver tolerances) and prove
the well-posedness and convergence of a Picard linearization. For the transient
problem, we present time discretizations that preserve the energy and magnetic
and hybrid helicity precisely in the ideal limit for two types of boundary
conditions. Additionally, we present an augmented Lagrangian preconditioning
technique for both the stationary and transient cases. We confirm our findings
with several numerical experiments.

Numerical Analysis;Numerical Analysis;Computational Physics