Two level penalty finite element methods for the stationary incompressible magnetohydrodynamics problem

In the paper we consider the penalty finite element method for the stationary incompressible magnetohydrodynamics (MHD) problem with a factor of penalty parameter. Stability and convergence of numerical solutions are established. Furthermore, two level penalty methods are also developed for the MHD problem. Our methods consist of solving a nonlinear MHD problem by the usual penalty method on a coarse mesh with mesh size HH, and then a linearized MHD problem based on the Stokes, Newton and Oseen iterations on a fine mesh is solved by the penalty method with mesh size h(h≪H), respectively. Stability and error estimates of numerical solutions in two level penalty methods are presented. Finally, some numerical tests are provided to demonstrate the effectiveness of the developed algorithms.