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

Abstract

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 H, 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.