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SUMMARYWe consider the Galerkin finite element method (FEM) for the incompressible magnetohydrodynamic (MHD) equations in two dimension. The domain is discretized into a set of regular triangular elements and the finite-dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual-free bubble functions. To find the bubble part of the solution, a two-level FEM with a stabilizing subgrid of a single node is described and its application to the MHD equations is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems including the MHD cavity flow and the MHD flow over a step. The results show that the proper choice of the subgrid node is crucial to get stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Furthermore, the approximate solutions obtained show the well-known characteristics of the MHD flow.

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a b s t r a c tNumerical solutions are given for magnetohydrodynamic (MHD) pipe flow under the influence of a transverse magnetic field when the outside medium is also electrically conducting. Convection-diffusion-type MHD equations for inside the pipe are coupled with the Laplace equation defined in the exterior region, and the continuity requirements for the induced magnetic fields are also coupled on the pipe wall. The most general problem of a conducting pipe wall with thickness, which also has magnetic induction generated by the effect of an external magnetic field, is also solved. The dual reciprocity boundary element method (DRBEM) is applied directly to the whole coupled equations with coupled boundary conditions at the pipe wall. Discretization with constant boundary elements is restricted to only the boundary of the pipe due to the regularity conditions at infinity. This eliminates the need for assuming an artificial boundary far away from the pipe, and then discretizing the region below it. Thus, the computational efficiency of the proposed numerical procedure lies in the solving of small sized systems, as compared to domain discretization methods. Computations are carried out for several values of the Reynolds number Re, the magnetic pressure Rh of the fluid, and the magnetic Reynolds numbers Rm 1 and Rm 2 of the fluid and the outside medium, respectively. Exact solution of the problem of MHD pipe flow in an insulating medium validates the results of the numerical procedure.

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