  We present compatible finite element space discretizations for the ideal
compressible magnetohydrodynamic equations. The magnetic field is considered
both in div- and curl-conforming spaces, leading to a strongly or weakly
preserved zero-divergence condition, respectively. The equations are
discretized in space such that transfers between the kinetic, internal, and
magnetic energies are consistent, leading to a preserved total energy. We also
discuss further adjustments to the discretization required to additionally
achieve magnetic helicity preservation. Finally, we describe new transport
stabilization methods for the magnetic field equation which maintain the
zero-divergence and energy conservation properties, including one method which
also preserves magnetic helicity. The methods' preservation and improved
stability properties are confirmed numerically using a steady state and a
magnetic dynamo test case.
