Detailed CVFEM algorithm for three dimensional advection-diffusion problems

The Control Volume Finite Element Method (CVFEM) combines the geometric flexibility of the Finite Element Method (FEM) with the physical intuition of the Control Volume Method (CVM). These two features of the CVFEM make it a very powerful tool for solving heat and fluid flow problems within complex domain geometries. In solving problems in the two-dimensional domains the development of the CVFEM has been well documented. For the three-dimensional problems, while there is extensive reporting on the details of the numerical approximation, there is relatively sparse information on important issues related to data structure and interpolation. Here, in the context of a general 3D advection-diffusion problem, a step-by-step derivation of the CVFEM is provided. Significant emphasis is placed on clearly defining an appropriate geometric data structure and detailing the key elements required to arrive at the final discrete equation. The performance and operation of the resulting 3D CVFEM scheme is highlighted by comparing its predictions against existing analytical solutions.