A decoupled, energy-stable and charge-conservative finite element scheme for the incompressible inductionless MHD equations

In this paper, we propose a fully discrete finite element scheme for the incompressible inductionless magnetohydrodynamics (MHD) equations, featuring decoupling, linearization, unconditional energy stability and charge conservation. Based on the Lagrange multiplier approach, the scheme introduces a positive regularization parameter θ to reformulate the inductionless MHD system, preserving the original energy structure and ensuring the uniqueness of the solution. Velocity and pressure are decoupled through the stabilized Gauge-Uzawa approach in the Navier-Stokes equations, while divergence-conforming finite elements are used to preserve the physical constraints of current density and electric potential. We also theoretically establish the unique solvability at each time step and unconditional energy stability of the proposed scheme. Extensive numerical experiments further validate its accuracy and robustness, demonstrating that the velocity, pressure, current density, and electric potential all achieve the expected spatial and temporal convergence rates while strictly maintaining charge conservation.