An Adaptive Meshless Method Based on the Residual‐Based a Posteriori Error Estimation for Magnetohydrodynamics Problem at Very High Hartmann Number

ABSTRACT This paper presents an adaptive element‐free Galerkin (EFG) method with streamline upwind Petrov‐Galerkin (AEFG‐SUPG) stabilization for solving steady magnetohydrodynamic (MHD) flows in insulated ducts with varying cross‐sections at very high Hartmann number. The governing equations are first decoupled into convection‐diffusion form via variable transformations, and discretized using the EFG method with SUPG (EFG‐SUPG) stabilization to deal with numerical oscillations induced by strong convection dominance. To further improve accuracy and computational efficiency, an adaptive algorithm is developed, which incorporates a posteriori error estimation based on the background integration cell residual into the EFG‐SUPG framework. This adaptive strategy enables node refinement in critical regions, remarkably reducing the total number of nodes compared to traditional uniform refinement methods. Four numerical cases involving square, circular, and arbitrary duct geometries with different magnetic field orientations are analyzed for Hartmann numbers ranging from to . The results highlight that the proposed method successfully resolves boundary layers and suppresses spurious oscillations even at . Compared with existing adaptive meshless methods, it demonstrates superior computational efficiency and solution stability, showcasing its effectiveness in handling very high Hartmann number MHD flow problems in complex duct structures.