This paper focuses on a new error analysis and a recovering technique of frequently-used mixed FEMs for a dynamical incompressible magnetohydrodynamics (MHD) system. The methods use the standard inf-sup stable Taylor–Hood/MINI velocity-pressure space pairs to solve the Navier–Stokes equations and the Nédélec’s edge element for solving the magnetic field. We establish new and optimal error estimates. In particular, we prove that the method provides the optimal accuracy for the MINI element in \(L^2\)-norm and for the Taylor-Hood element in \(H^1\)-norm. The analysis is based on a modified Maxwell projection and the corresponding estimates in negative norms, while all the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order Nédélec’s edge approximation in analysis. In addition, at any given time step, we develop a simple recovery technique for numerical approximation to the magnetic field of one order higher accuracy in the spatial direction.

本文重点介绍了动态不可压缩磁流体动力学 (MHD) 系统的常用混合 FEM 的新误差分析和恢复技术。这些方法使用标准的 inf-sup 稳定 Taylor-Hood/MINI 速度-压力空间对来求解 Navier-Stokes 方程，并使用 Nédélec 的边缘元素来求解磁场。我们建立了新的和最优的误差估计。特别是，我们证明该方法为\(L^2\) -范数中的 MINI 元素和\(H^1\)中的 Taylor-Hood 元素提供了最佳精度-规范。该分析基于修正的麦克斯韦投影和相应的负范数估计，而由于系统的强耦合和分析中低阶内德莱克边缘近似的污染，现有的所有分析都不是最优的。此外，在任何给定的时间步长，我们开发了一种简单的恢复技术，用于在空间方向上对磁场进行数值近似，精度更高一个数量级。