All numerical methods, including the FEM and FVM, ultimately result in a set of linear or nonlinear algebraic equations, relating the values of the dependent variables at the nodal points of the mesh. These algebraic equations can be linear or nonlinear in the nodal values of the primary variables, depending on whether the governing differential equations being solved are linear or nonlinear. When the algebraic equations are nonlinear, we linearize them using certain assumptions and techniques, such as the Picard method or Newton’s method.

The equations governing flows of Newtonian viscous incompressible fluids were reviewed in Chapter 2. The equations are revisited here, in the Cartesian component form, for the two-dimensional case (i.e., set $v_z=0$ and the derivatives with respect to $z$ to zero).

There are several topics that are considered to be “advanced” for this book. We will briefly discuss some (but not all) of these topics to make the readers aware of the fact that the present coverage has precluded them, and then cover three topics in a greater detail.

Most engineering systems can be described, with the aid of the laws of physics and observations, in terms of algebraic, differential, and integral equations. In most problems of practical interest, these equations cannot be solved exactly, mostly because of irregular domains on which the equations are posed, variable coefficients in the equations, complicated boundary conditions, and the presence of nonlinearities. Approximate representation of differential and integral equations to obtain algebraic relations among quantities that characterize the system and implementation of the steps to obtain algebraic equations and their solution using computers constitute a numerical method.

General equations describing transport of momenta and energy by advection–diffusion was given in Chapter 2 (see, also, Example 4.2.3) and will not be repeated here. It is important to note that the entire finite volume formulation is based on local one-dimensional representation in each coordinate direction. The flux crossing a control volume surface is represented using a one-dimensional formulation.

In this chapter we will introduce the FEM as a technique of solving differential equations governing a single variable (or dependent variable). Once we understand how the method works, it can be extended to problems governed by coupled PDEs among several unknowns. In particular, equations governing steady-state heat transfer in one- and two-dimensional problems are used as the “model” equations to introduce the FEM.

Because this book is concerned with the numerical solutions to problems of heat transfer and fluid mechanics, it is useful to summarize the governing equations of these two fields, which are closely related. Subject areas as diverse as aerodynamics, biology, combustion, geology and geophysics, manufacturing, and meteorology can be studied using the equations governing heat transfer and fluid mechanics (for detailed discussion of the underlying physics and derivation of the equations