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.

翻译：我们为理想的压缩磁流动力等式提出了相容的有限元素空间分解。磁场在分流和弯曲相容的空格中分别被考虑,导致强度或弱度保持零振荡状态。等式在空间中分解,使动能、内能和磁能之间的转移具有一致性,导致总能量的保存。我们还讨论进一步调整离散性,以进一步实现磁热力保护。最后,我们描述了维持零振荡和节能特性的磁场方程式的新的运输稳定方法,包括一种也保持磁性高度特性的方法。用稳定的状态和磁振动试验案例,从数字上证实这些方法的保全和改进的稳定性特性。