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.

翻译：我们为压抑性、耐抗性 Hall磁力动力学(MHD)方程式制定了结构保留有限元素的方法。这些方程式结合了Hall Hall当前术语在Ohm法律中的术语,比标准MHD等离子体在距离离离子皮肤深度近或小于离子皮肤深度的长度尺度上对全离子等离子体作了更适当的描述。我们采用了Hall MHD的固定离散变异配方,以精确地(直到解答器耐受力)执行磁高斯定律,并证明Picard线性化的稳妥性和趋同性。关于瞬时问题,我们提出了时间分解,在两种边界条件的理想限度内准确保存了能量和磁性和混合热性。此外,我们提出了一种强化的Lagrangian定型定型技术,用于固定和易变型情况。我们用若干数字实验证实了我们的调查结果。