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

TL;DR

This paper introduces a new divergence-free finite element method for 3D steady MHD kinematics equations, ensuring magnetic and current divergence-free conditions using vector potentials and H(div) elements, with verified numerical robustness.

Contribution

The paper presents a novel mixed finite element scheme that guarantees divergence-free magnetic and current fields in steady MHD kinematics using vector potentials and specialized elements.

Findings

The scheme satisfies divergence-free conditions for magnetic induction and current density.

Numerical experiments confirm the scheme's convergence and divergence-free properties.

The preconditioned iterative solver is robust and efficient.

Abstract

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.