Abstract In this paper, we consider the nonstationary magnetohydrodynamic coupled heat equation through the well-known Boussinesq approximation. The Crank-Nicolson extrapolation scheme is used for time derivative terms, and the mixed finite method is used for spatial discretization. We employ the Taylor-Hood finite elements to approximate Navier-Stokes equations, Nedelec edge element for the magnetic induction and the equal order Lagrange elements for the thermal equation. This fully discrete scheme only needs to solve a linear system at each time step, and the system is unique solvable. We prove the proposed scheme is unconditionally energy stable. Under a weak regularity hypothesis on the exact solution, we present optimal error estimates for the velocity, magnetic induction and temperature. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.

中文翻译：

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热耦合不可压缩磁流体动力系统的Crank-Nicolson外推全离散格式的收敛性分析

摘要 在本文中，我们通过著名的Boussinesq 近似来考虑非平稳磁流体动力耦合热方程。Crank-Nicolson 外推方案用于时间导数项，混合有限方法用于空间离散化。我们使用 Taylor-Hood 有限元来逼近 Navier-Stokes 方程，使用 Nedelec 边缘元素来逼近磁感应，使用等阶拉格朗日元素来逼近热方程。这种完全离散的方案只需要在每个时间步求解一个线性系统，并且该系统是唯一可解的。我们证明了所提出的方案是无条件的能量稳定的。在精确解的弱规律性假设下，我们提出了速度、磁感应强度和温度的最佳误差估计。最后，