In dealing with partial differential equations, it is useful to differentiate between several types. In particular, we classify partial differential equations of second order as elliptic, hyperbolic, and parabolic. Both the theoretical and numerical treatment differ considerably for the three types. For example, in contrast with the case of ordinary differential equations where either initial or boundary conditions can be specified, here the type of equation determines whether initial, boundary, or initial-boundary conditions should be imposed.

The most important application of the finite element method is to the numerical solution of elliptic partial differential equations. Nevertheless, it is important to understand the differences between the three types of equations. In addition, we present some elementary properties of the various types of equations. Our discussion will show that for differential equations of elliptic type, we need to specify boundary conditions and not initial conditions.

There are two main approaches to the numerical solution of elliptic problems: finite difference methods and variational methods. The finite element method belongs to the second category. Although finite element methods are particularly effective for problems with complicated geometry, finite difference methods are often employed for simple problems, primarily because they are simpler to use. We include a short and elementary discussion of them in this chapter.