Unconditional superconvergence analysis of low-order conforming mixed finite element method for time-dependent incompressible MHD equations

In this paper, we propose a backward Euler semi-implicit full discrete scheme for the dependent incompressible MHD equations and study the superconvergence behavior scheme. The spatial discretization is based on the bilinear-constant-bilinear elements velocity, pressure and magnetic fields, respectively, while the time discretization is based the first-order backward Euler scheme. Firstly, we prove a new high accuracy estimation related to the magnetic field, and prove the unconditional boundedness of numerical solutions in L infinity-norm by introducing a time-discrete auxiliary system. Then we derive the superclose estimates rigorously, which lead to the corresponding superconvergence results with assistance from interpolation post-processing techniques. In the end, we provide some numerical examples to verify the correctness of our theoretical analysis.