The Finite Element Method in Electromagnetics, 3rd Edition

CHAPTER 3ONE-DIMENSIONAL FINITE ELEMENT ANALYSIS

In Chapter 2 we introduced the finite element method using a simple example. In this chapter we first follow the basic steps of the method to treat a general one-dimensional boundary-value problem. We then consider some special cases to illustrate application of the method. Finally, we introduce the concept of higher-order elements and their use in obtaining increased accuracy in the finite element solution. We note that the finite element method has not been widely used in one-dimensional electromagnetic problems because few of these problems result in a model complicated enough to require a numerical solution. Nevertheless, because of its simplicity, the one-dimensional problem is ideal for demonstrating formulation of the finite element method.

3.1 BOUNDARY-VALUE PROBLEM

The boundary-value problem to be considered is defined by the differential equation

(3.1)

where ϕ is the unknown function, α and β are known parameters or functions associated with the physical properties of the solution domain, and f is a known source or excitation function. The standard one-dimensional Laplace, Poisson, and Helmholtz equations are special forms of (3.1).

For this example, we assume that boundary conditions for ϕ are given by

(3.2)

and ...