Stabilization in 3‐D FEM and solution of the MHD equations

Abstract

In this study, we consider the numerical solution of convection‐diffusion typed equations defined in 3‐D domain using the finite element method (FEM) with the stabilized version in order to fill the gap in literature on this area in the sense of different applications. The formulation processed is performed step by step. For this purpose, we started with FEM formulation of the 3‐D Laplace equation. Then, the obtained formulation is extended to the convection‐diffusion equation with standard Galerkin FEM. In order to solve the stability problems for the convection dominated case, the most popular stabilized formulation known as the streamline upwind Petrov‐Galerkin (SUPG) type stabilization method is considered. As a further step, obtained stabilized formulation is used in the solutions of the MHD equations by transforming the coupled system of differential equations to the decoupled convection‐diffusion type equations defined on different 3‐D domains. For the comparison purpose, MHD equations are also formulated with finite volume method (FVM). Obtained numerical results are displayed in terms of a table and figures as the 2‐D slices of the 3‐D plots in order to visualize the effect and accuracy of the proposed numerical scheme over the different test problems.