An efficient two-step algorithm for the stationary incompressible magnetohydrodynamic equations

A new highly efficient two-step algorithm for the stationary incompressible magnetohydrodynamic equations is studied in this paper. The algorithm uses a lower order finite element pair (i.e., P1b−P1−P1) to compute an initial approximation, that is using the Mini-element (i.e., P1b−P1) to approximate the velocity and pressure and P1 element to approximate the magnetic field, then applies a higher order finite element pair (i.e., P2−P1−P2) to solve a linear system on the same mesh. Furthermore, the convergence analyses of standard Galerkin finite element method and the two-step algorithm are addressed. Lastly, numerical experiments are presented to verify both the theory and the efficiency of the algorithm.