A novel divergence-free Finite Element Method for the MHD Kinematics equations using Vector-potential
This work addresses the need for conservative and efficient numerical methods for MHD kinematics, which is relevant to computational magnetohydrodynamics.
The authors propose a mixed finite element method for 3D steady MHD kinematics equations that enforces divergence-free conditions for both current density and magnetic induction using a vector potential and H(div)-conforming elements, with a block preconditioner for efficient solving. Numerical experiments verify divergence-free properties, convergence rates, and robustness.
We propose a new mixed finite element method for the three-dimensional steady magnetohydrodynamic (MHD) kinematics equations for which the velocity of the fluid is given. Although prescribing the velocity field leads to a simpler model than full MHD equations, its conservative and efficient numerical methods are still active research topic. The distinctive feature of our discrete scheme is that the divergence-free conditions for current density and magnetic induction are both satisfied. To reach this goal, we use magnetic vector potential to represent magnetic induction and resort to H(div)-conforming element to discretize the current density. We develop an preconditioned iterative solver based on a block preconditioner for the algebraic systems arising from the discretization. Several numerical experiments are implemented to verify the divergence-free properties, the convergence rate of the finite element scheme and the robustness of the preconditioner.