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 Optimal $H^1$ error analysis of a fractional step finite element scheme for a hybrid MHD system Jingke Wu,Rong An,Yuan Li Keywords:Magnetohydrodynamics equations; Fractional step method; Finite element method; Error analysis Abstract: This paper presents a fractional step finite element scheme for a hybrid MHD system coupled by the nonstationary Navier-Stokes equations and the steady Maxwell equations, which can be viewed that the magnetic phenomena reach their steady state "infinitely" faster than the fluid hydrodynamics phenomena. The proposed fractional step scheme has the following features: the first one is that the proposed scheme is a decoupled scheme, which means the magnetic field and velocity field can be solved independently at the same time discrete level. The second one is that the nonlinearity and the divergence-free of the Navier-Stokes equations are splitted by introducing an intermediate velocity field. We focus on a rigorous error analysis and obtain the optimal $\H^1$ convergence order $\mathcal O(\Delta t+h)$ for the magnetic and the velocity under the time step condition $\Delta t=\mathcal O (h)$, where $h$ is the mesh size. Finally, numerical results are shown to illustrate the theoretical convergence analysis.