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

Abstract

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.