Our explorations on the methods of finite differences and finite elements have come to an end. In Chapter 1, it was intended that the reader recognize the analogy between these two methods in one dimension. In fact, such an analogy exists for linear problems in all multidimensional geometries as long as the grid configurations are structured. In structured grids, with adjustments of the temporal parameters in generalized Galerkin methods and both temporal and convection diffusion parameters in generalized Petrov-Galerkin methods, the analogy between finite difference methods (FDM) and finite element methods (FEM) can be shown to exist also.

Traditionally, FEM equations are developed in unstructured grids as well as in structured grids. The FEM equations written in unstructured grids have global nodes irregularly connected around the entire domain, thus resulting in a large sparse matrix system, but the data management can be handled efficiently by using the element-by-element (EBS) assembly as discussed in Sections 10.3.2 and 11.5. FDM equations cannot be written in unstructured grids unless through FVM formulations. Thus, the FDM equations written only in structured grids cannot be directly compared with FEM equations written in general unstructured grids. Thus, the notion of FEM being more complicated, requiring more computer time than FDM, is an unfortunate comparison. For fair comparisons, FEM equations must be written in structured grids as in FDM.

In unstructured adaptive methods (Chapter 19), our assessments as to the merits and demerits of FDM versus FEM will be faced with a new challenge. This is because adaptive methods are instrumental in resolving many problems of numerical difficulties such as in shock waves and turbulence, making the fair comparison between FDM and FEM difficult.