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In this paper we study finite element method for three-dimensionalincompressible resistive magnetohydrodynamic equations, in which the velocity,the current density, and the magnetic induction are divergence-free. It isdesirable that the discrete solutions should also satisfy divergence-freeconditions exactly especially for the momentum equations. Inspired byconstrained transport method,we devise a new stable mixed finite element methodthat can achieve the goal. We also prove the well-posedness of the discretesolutions. To solve the resulting linear algebraic equations, we propose aGMRES solver with an augmented Lagrangian block preconditioner. By numericalexperiments, we verify the theoretical results and demonstrate thequasi-optimality of the discrete solver with respect to the number of degreesof freedom

A constrained transport divergence-free finite element method for Incompressible MHD equations