A Conservative Finite Element Solver for MHD Kinematics equations: Vector Potential method and Constraint Preconditioning

TL;DR

This paper introduces a conservative finite element solver for 3D steady MHD kinematics equations using vector potential and current density, ensuring divergence-free constraints and magnetic helicity preservation, with an efficient preconditioning approach.

Contribution

It presents a novel finite element method that guarantees divergence-free solutions and magnetic helicity conservation, along with a specialized preconditioner for solving the resulting linear systems.

Findings

The solver preserves divergence-free constraints and magnetic helicity.

Numerical experiments show robust convergence and efficiency.

The preconditioner improves solver performance for ill-conditioned systems.

Abstract

A new conservative finite element solver for the three-dimensional steady magnetohydrodynamic (MHD) kinematics equations is presented.The solver utilizes magnetic vector potential and current density as solution variables, which are discretized by H(curl)-conforming edge-element and H(div)-conforming face element respectively. As a result, the divergence-free constraints of discrete current density and magnetic induction are both satisfied. Moreover the solutions also preserve the total magnetic helicity. The generated linear algebraic equation is a typical dual saddle-point problem that is ill-conditioned and indefinite. To efficiently solve it, we develop a block preconditioner based on constraint preconditioning framework and devise a preconditioned FGMRES solver. Numerical experiments verify the conservative properties, the convergence rate of the discrete solutions and the robustness of the preconditioner.