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arXiv · 2111.11693

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

Xujing Li·Lingxiao Li
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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.

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Xujing Li, Lingxiao Li. 2021-11-23. A Conservative Finite Element Solver for MHD Kinematics equations: Vector Potential method and Constraint Preconditioning. https://doi.org/10.1016/j.jcp.2022.111416

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