The paper focuses on a new error analysis of a class of mixed FEMs for stationary incompressible magnetohydrodynamics with the standard inf-sup stable velocity-pressure space pairs to Navier-Stokes equations and the N\'ed\'elec's edge element for the magnetic field. The methods have been widely used in various numerical simulations in the last several decades, while the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order N\'ed\'elec's edge approximation in analysis. In terms of a newly modified Maxwell projection we establish new and optimal error estimates. In particular, we prove that the method based on the commonly-used Taylor-Hood/lowest-order N\'ed\'elec's edge element is efficient and the method provides the second-order accuracy for numerical velocity. Two numerical examples for the problem in both convex and nonconvex polygonal domains are presented. Numerical results confirm our theoretical analysis.

翻译：本文聚焦于对一类混合有限元方法的新误差分析，该方法用于稳态不可压缩磁流体动力学问题，采用标准满足inf-sup稳定条件的速度-压力空间对（针对Navier-Stokes方程）以及用于磁场求解的Nédélec边单元。这类方法在过去几十年中被广泛应用于各类数值模拟，然而现有分析因系统的强耦合性以及低阶Nédélec边单元近似在分析中的干扰而未达到最优性。基于一种新修正的Maxwell投影，我们建立了新的最优误差估计。特别地，我们证明了基于常用Taylor-Hood/最低阶Nédélec边单元的方法是高效的，且该方法对数值速度实现了二阶精度。文中给出了凸多边形域和非凸多边形域内问题的两个数值算例，数值结果验证了我们的理论分析。