A fully discrete finite element method for a constrained transport model of the incompressible MHD equations

¹ Henan Academy of Big Data, Zhengzhou University, Zhengzhou 450052, P.R. China
² School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, P.R. China
³ Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P.R. China
⁴ College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, P.R. China

^* Corresponding author: shymath@126.com

Received: 27 October 2022
Accepted: 21 July 2023

Abstract

In this paper, we propose and analyze a fully discrete finite element method for a constrained transport (CT) model of the incompressible magnetohydrodynamic (MHD) equations. The spatial discretization is based on mixed finite elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, the magnetic field and magnetic vector potential are discretized by H(curl)-conforming edge element. The time marching is combining a backward Euler scheme and some subtle implicit-explicit treatments for nonlinear and coupling terms. With these treatments, the fully discrete scheme is linear in the implementation and the computation of the magnetic vector potential is decoupled from the whole coupled system. The most attractive feature of this scheme that it can yield the exactly divergence-free magnetic field and current density on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic field and magnetic vector potential are further demonstrated under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme.