Anisotropic adaptive finite element method for magnetohydrodynamic flow at high Hartmann numbers

This paper presents an anisotropic adaptive finite element method (FEM) to solve the governing equations of steady magnetohydrodynamic (MHD) duct flow. A residual error estimator is presented for the standard FEM, and two-sided bounds on the error independent of the aspect ratio of meshes are provided. Based on the Zienkiewicz-Zhu estimates, a computable anisotropic error indicator and an implement anisotropic adaptive refinement for the MHD problem are derived at different values of the Hartmann number. The most distinguishing feature of the method is that the layer information from some directions is captured well such that the number of mesh vertices is dramatically reduced for a given level of accuracy. Thus, this approach is more suitable for approximating the layer problem at high Hartmann numbers. Numerical results show efficiency of the algorithm.