Two-Level Newton Iterative Method for the 2D/3D Stationary Incompressible Magnetohydrodynamics

Abstract

In this paper, Newton iteration and two-level finite element algorithm are combined for solving numerically the stationary incompressible magnetohydrodynamics (MHD) under a strong uniqueness condition. The method consists of solving the nonlinear MHD system by $$m$$m Newton iterations on a coarse mesh with size $$H$$H and then computing the Stokes and Maxwell problems on a fine mesh with size $$h\ll H$$h?H. The uniform stability and optimal error estimates of both Newton iterative method and two-level Newton iterative method are given. The error analysis shows that the two-level Newton iterative solution is of the same convergence order as the Newton iterative solution on a fine grid with $$h=O(H^2)$$h=O(H2). However, the two-level Newton iterative method for solving the stationary incompressible MHD equations is simpler and more efficient than Newton iterative one. Finally, the effectiveness of the two-level Newton iterative method is illustrated by several numerical investigations.