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In this paper,the monolithic multigrid method is investigated for reduced magnetohydrodynamic equations.We propose a diagonal Braess-Sarazin smoother for the finite element discrete system and prove the uniform convergence of the MMG method with respect to mesh sizes.A multigrid-preconditioned FGMRES method is proposed to solve the magnetohydrodynamic equations.It turns out to be robust for relatively large physical parameters.By extensive numerical experiments,we demonstrate the optimality of the monolithic multigrid method with respect to the number of degrees of freedom.