A fully discrete finite element method for unsteady magnetohydrodynamic flow in porous media

Abstract

This article explores the unsteady magnetohydrodynamic (MHD) model within the framework of porous media flow. This model consists of the Brinkman–Forchheimer equations and Maxwell equations in the porous media domain, which are coupled by the Lorentz force. We propose and analyze a numerical discretization method for MHD porous model. The second-order backward difference formula is utilized for temporal derivative terms, and the mixed finite element method is employed for spatial discretization. Rigorous proofs of stability and uniqueness are provided for the numerical solutions. We establish optimal $L^2$-error estimates for the velocity and magnetic induction without imposing constraints on the relationship between the time step and mesh size. Finally, several three-dimensional numerical experiments are performed to illustrate the features of the proposed numerical method and validate the theoretical findings. To our knowledge, this is the first error analysis and simulation to address unsteady MHD flow through porous media.