Stabilized solution of the 3-D MHD flow problem with FEM-BEM coupling approach

In this study, we obtain the numerical solution of the different problem configurations of the 3-D Magneto-hydrodynamic (MHD) flow. We solve the MHD flow inside an unbounded conducting domain or around a conducting solid problems which require to consider both MHD equations and the Laplace equation on the connected domains with the coupled boundary conditions. As a numerical procedure we will propose the BEM (Boundary Element Method) and FEM (Finite Element Method) coupling approach with stabilization in order to eliminate the instabilities due to the numeric formulation. We consider the formulations of the MHD equations on the spherical or cubic domains inside an unbounded domain as a first problem and also on the annular spherical or annular cubic region around a conducting solid for the second problem with stabilized FEM. Then obtained discretized equations are combined with BEM formulation of the Laplace equations just only on the boundary. Finally, the BEM is used in order to obtain the rest of the unknown values on the Laplace domain using the boundary values obtained from the coupled formulation. Different values of the problem parameters and configurations are tested, and obtained solutions are displayed in terms of the figures as the 2-D slices of the 3-D solutions.