Development of a modified, hybrid nodal-integral/finite-element method for the steady-state convection-diffusion equation

A modified, hybrid nodal-integral/finite-element method (NI-FEM) is developed to solve the two- and three-dimensional, steady-state, convection-diffusion equation. The hybrid NI-FEM method takes advantage of the high efficiency of the conventional nodal-integral method (NIM) and the flexible mesh generation of the finite element method (FEM), which is applicable to arbitrary geometries. In this method, the computational domain is discretized into, for 2D problems, three-node triangular elements and four-node rectangular elements. The rectangular elements are used to discretize the interior region and regions adjacent to horizontal or vertical boundaries, while the triangular elements are used to discretize regions along boundaries that are not parallel to any axis. The conventional NIM is used to develop difference equations for the transverse-averaged variables on the interface between two adjacent rectangular elements. The FEM is applied to solve the node temperatures of triangular elements. On the interface between two different kinds of elements, the transverse-averaged variable for the rectangular element is obtained by averaging the node values of its adjacent triangular element(s), while the heat flux for the triangular element is calculated using the transverse-averaged variables of its adjacent rectangular element. The hybrid NI-FEM uses a matrix formulation for the entire domain rather than an iterative procedure, detailed derivations of which are presented in the paper. Using the developed NI-FEM, two convection-diffusion test problems are solved, and the numerical results are compared to the exact manufactured solutions to demonstrate the accuracy and efficiency of the method.