Unconditionally optimal convergence analysis of second-order BDF Galerkin finite element scheme for a hybrid MHD system

Abstract

In this paper, a second-order backward differentiation formula (BDF) scheme for a hybrid MHD system is considered. Being different with the steady and nonstationary MHD equations, the hybrid MHD system is coupled by the time-dependent Navier-Stokes equations and the steady Maxwell equations. By using the standard extrapolation technique for the nonlinear terms, the proposed BDF scheme is a semi-implicit scheme. Furthermore, this scheme is a decoupled scheme such that the magnetic field and the velocity can be solved independently at the same time as discrete level. A rigorous error analysis is done and we prove the unconditionally optimal second-order convergence rate $\mathcal O(h^{2}+({\Delta } t)^{2})$ in L2 norm for approximations of the magnetic field and the velocity, where h and Δt are the grid mesh and the time step, respectively. Finally, the numerical results are displayed to illustrate the theoretical results.