We investigate a fully discrete finite element scheme for the three-dimensional incompressible magnetohydrodynamic problem based on magnetic vector potential formulation that was introduced in Hiptmair et al. (MMMAS 28:659–695, 2018). The formulation enjoys the novel feature that it can always produce an exactly divergence-free magnetic induction discretized solution. Using a mixed finite element approach, we discretize the model by the fully discrete semi-implicit Euler scheme with the velocity and the pressure approximated by stable MINI finite elements and the magnetic vector potential by Nédélec edge elements. Under a reasonable regularity hypothesis for the exact solution, error estimates for the velocity and the magnetic vector potential are rigorously established. Finally, several numerical experiments are presented to illustrate the convergence properties of the numerical scheme.

我们研究了基于 Hiptmair 等人中引入的磁矢量势公式的三维不可压缩磁流体动力学问题的完全离散有限元方案。（MMMAS 28：659-695，2018 年）。该公式具有新颖的特点，即它始终可以产生完全无发散的磁感应离散解。使用混合有限元方法，我们通过完全离散的半隐式欧​​拉方案离散模型，速度和压力由稳定的 MINI 有限元近似，磁矢量势由 Nédélec 边缘元近似。在精确解的合理规律性假设下，严格建立了速度和磁矢量势的误差估计。最后，