Two-level finite element variational multiscale method based on bubble functions for the steady incompressible MHD flow

In this paper, a finite element variational multiscale method (VMM) is developed for the stationary incompressible magnetohydrodynamics (MHD) problem and the corresponding stability and convergence are established. Our VMM is based on the polynomial bubble functions as a subgrid scale and the numerical implementation is based on the local Gauss integrations. Compared to the common VMM, our method does not introduce any extra variable. Furthermore, the two-level finite element VMM is also developed. Compared to the one-level VMM, the two-level VMM consists of solving a nonlinear MHD problem on a coarse mesh with mesh size H, and then a linearized magnetohydrodynamic problem based on the Oseen iteration on a fine mesh is solved with mesh size h (h << H). Stability and convergence of numerical approximations in two-level VMM are presented. Finally, some numerical examples are provided to demonstrate the effectiveness of the developed numerical algorithms.