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:本文主要针对一类混合有限元方法在标准不可压Navier-Stokes方程和N\'ed\'elec边界元在磁场方面的应用中进行了新的误差分析。该方法在过去几十年中已广泛用于各种数值模拟,并且由于系统的强耦合和低阶N\'ed\'elec边界逼近在分析中的污染,现有的分析并不是最优的。我们根据新的修改后的Maxwell投影建立了新的最优误差估计。特别地,我们证明了基于通常使用的泰勒-胡德/最低阶N\'ed\'elec边界元的方法是有效的,该方法为数值速度提供了二阶精度。文中还呈现了凸多边形和非凸多边形环境下的两个问题的数值实例。数值结果证实了我们的理论分析。