In this paper we study finite element method for three-dimensional incompressible resistive magnetohydrodynamic equations, in which the velocity, the current density, and the magnetic induction are divergence-free. It is desirable that the discrete solutions should also satisfy divergence-free conditions exactly especially for the momentum equations. Inspired by constrained transport method, we devise a new stable mixed finite element method that can achieve the goal. We also prove the well-posedness of the discrete solutions. To solve the resulting linear algebraic equations, we propose a GMRES solver with an augmented Lagrangian block preconditioner. By numerical experiments, we verify the theoretical results and demonstrate the quasi-optimality of the discrete solver with respect to the number of degrees of freedom. A comparison with other discretization using lid driven cavity is also given.

在本文中，我们研究了三维不可压缩电阻磁流体动力学方程的有限元方法，其中速度，电流密度和磁感应无散度。理想的是，离散解也应准确地满足无散度条件，尤其是对于动量方程式。受约束运输方法的启发，我们设计了一种新的稳定的混合有限元方法，可以达到目标。我们还证明了离散解的适定性。为了解决由此产生的线性代数方程，我们提出了带有增强拉格朗日块预处理器的GMRES求解器。通过数值实验，我们验证了理论结果，并证明了离散求解器相对于自由度数的拟最优性。