Two-Level Newton Iterative Method for the 2D/3D Stationary Incompressible Magnetohydrodynamics
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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 \(m\) Newton iterations on a coarse mesh with size \(H\) and then computing the Stokes and Maxwell problems on a fine mesh with size \(h\ll H\). 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 \(h=O(H^2)\). 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.
KeywordsTwo-level method Newton iteration Finite element method Stationary incompressible magnetohydrodynamics
Mathematical Subject Classification35Q30 65M60 65N30 76D05
The authors sincerely thank the reviewers and editor for their valuable comments and suggestions which led to a large improvement of the paper. The research was supported by the National Natural Science Foundation of China (Grant Nos.: 11271298, 11362021).
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