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We propose a new mixed finite element method for the three-dimensional steadymagnetohydrodynamic (MHD) kinematics equations for which the velocity of thefluid is given. Although prescribing the velocity field leads to a simplermodel than full MHD equations, its conservative and efficient numerical methodsare still active research topic. The distinctive feature of our discrete schemeis that the divergence-free conditions for current density and magneticinduction are both satisfied. To reach this goal, we use magnetic vectorpotential to represent magnetic induction and resort to H(div)-conformingelement to discretize the current density. We develop an preconditionediterative solver based on a block preconditioner for the algebraic systemsarising from the discretization. Several numerical experiments are implementedto verify the divergence-free properties, the convergence rate of the finiteelement scheme and the robustness of the preconditioner.

A novel divergence-free Finite Element Method for the MHD Kinematics equations using Vector-potential