New error analysis of charge-conservative finite element methods for stationary inductionless MHD equations

In this paper, we present a new error analysis of a class of charge-conservative finite element methods for stationary inductionless magnetohydrodynamics (MHD) equations. The methods use the standard inf-sup stable Mini/Taylor-Hood pairs to discretize the velocity and pressure, and the Raviart-Thomas face element for solving the current density. Due to the strong coupling of the system and the pollution of the lower-order Raviart-Thomas face approximation in analysis, the existing analysis is not optimal. In terms of a mixed Poisson projection and the corresponding estimate in negative norms, we establish new and optimal error estimates. In particular, we prove that the method with the lowest-order Raviart-Thomas face element and Mini element provides the optimal accuracy for the velocity in L-2-norm, and the method with the lowest-order Raviart-Thomas face element and P-2-P(1)Taylor-Hood element supplies the optimal accuracy for the velocity in H-1-norm and the pressure in L-2-norm. Furthermore, we propose a simple recovery technique to obtain a new numerical current density of one order higher accuracy by re-solving a mixed Poisson equation. Numerical results are provided to verify the theoretical analysis.