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We present compatible finite element space discretizations for the idealcompressible magnetohydrodynamic equations. The magnetic field is consideredboth in div- and curl-conforming spaces, leading to a strongly or weaklypreserved zero-divergence condition, respectively. The equations arediscretized in space such that transfers between the kinetic, internal, andmagnetic energies are consistent, leading to a preserved total energy. We alsodiscuss further adjustments to the discretization required to additionallyachieve magnetic helicity preservation. Finally, we describe new transportstabilization methods for the magnetic field equation which maintain thezero-divergence and energy conservation properties, including one method whichalso preserves magnetic helicity. The methods' preservation and improvedstability properties are confirmed numerically using a steady state and amagnetic dynamo test case.

Structure preserving transport stabilized compatible finite element methods for magnetohydrodynamics