An optimal error estimate of the BDF-Galerkin FEM for the incompressible MHD system

In this article, a linearized backward differential formula type scheme for the numerical approximations of the magnetohydrodynamics equations is analyzed. The proposed fully discretized numerical scheme, which is comprised of a third order backward differential formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms and mixed finite element approximation for spatial discretization. Based on the temporal-spatial error splitting technique, we demonstrate the almost unconditional optimal error estimate in L-2 norm for the fluid velocity and the magnetic by getting boundedness of numerical solutions in certain norms. Finally, several numerical tests are carried out to verify the theoretical results. (C) 2022 Elsevier Inc. All rights reserved.