Analysis of a time filtered finite element method for the unsteady inductionless MHD equations

This paper studies a time filtered finite element method for the unsteady inductionless magnetohydrodynamic (MHD) equations. The method uses the semi-implicit backward Euler scheme with a time filter in time and adopts the standard inf-sup stable fluid pairs to discretize the velocity and pressure, and the inf-sup stable face-volume elements for solving the current density and electric potential in space. Since the time filter for the velocity is added as a separate post-processing step, the scheme can be easily incorporated into the existing backward Euler code and improves the time accuracy from first order to second order. The unique solvability, unconditional energy stability, and charge conservativeness of the scheme are also proven. In terms of the energy arguments, we establish the error estimates for the velocity, current density, and electric potential. Numerical experiments are performed to verify the theoretical analysis.