Optimal error estimates of a decoupled finite element scheme for the unsteady inductionless MHD equations

This article focuses on a new and optimal error analysis of a decoupled finite element scheme for the inductionless magnetohydrodynamic (MHD) equations. The method uses the classical inf-sup stable Mini/Taylor-Hood (Mini/TH) finite element pairs to appropriate the velocity and pressure, and Raviart-Thomas (RT) face element to discretize the current density spatially, and the semi-implicit Euler scheme with an additional stabilized term and some delicate implicit-explicit handling for the coupling terms temporally. The method enjoys some impressive features that it is linear, decoupled, unconditional energy stable and charge-conservative. Due to the errors from the explicit handing of the coupling terms and the existence of the artificial stabilized term, and the contamination of the lower-order RT face discretization in the error analysis, the existing theoretical results are not unconditional and optimal. By utilizing the anti-symmetric structure of the coupling terms and the existence of the extra dissipative term, and the negative-norm estimate for the mixed Poisson projection, we establish the unconditional and optimal error estimates for all the variables. Numerical tests are presented to illustrate our theoretical findings.