The variational multiscale element free Galerkin method for three-dimensional steady magnetohydrodynamics duct flows

Magnetohydrodynamics (MHD) has extensive applications in diverse fields, making the study of three-dimensional (3D) MHD problems crucial. For MHD flows, when the Hartmann ($Ha$) number is large, leading to a convection-dominated regime where convection terms overcome diffusion. In such scenarios, standard Galerkin methods fail to suppress non-physical oscillations in solutions, as they lack inherent stabilization mechanisms for strong convection. This paper introduces the variational multiscale element free Galerkin (VMEFG) method to solve 3D steady MHD equations. The VMEFG method inherits the advantage of the element free Galerkin (EFG) method in avoiding the complex meshing process, which is particularly challenging for complex 3D problems. Moreover, compared with the EFG method, it shows enhanced stability in dealing with convection-dominant problems and can automatically generate stabilized parameters, outperforming other stabilization techniques. To verify the numerical stability and accuracy of the VMEFG method, several numerical experiments on various domains including cubic, annular cubic, spherical, and annular spherical domains were conducted and compared with EFG solutions and existing literature results. The results indicate that the VMEFG method can effectively avoid numerical oscillations and maintain stability for 3D MHD problems at high $Ha$ number, providing a reliable and efficient solution for such problems.