Abstract: Magnetorheological finishing of first-order discontinuous optical elements is one of the difficult problems that restrict the development of high-precision and high-efficiency optical manufacturing in China. In this paper, a fluid dynamics analysis method for magnetorheological finishing concerning the first order discontinuous optical elements is presented. Firstly, reasonable assumptions are made for the flow field under the practical condition of magnetorheological finishing. Secondly, based on the micro-element hydrodynamic equation, a flow field analysis method suitable for the first order discontinuous surface shape is established. Finally, a numerical calculation method for the flow field governing equation is established based on the finite difference method and numerical iteration method. Through numerical simulation for cutting distance ranging over 1~18 mm, it is found that the pressure distribution pattern of the first-order discontinuous surface obtained by the proposed method is correct, and the discontinuous pressure drop generated by the proposed method is consistent with that of the experimental observation.