Introduction

Finite element methods have been successful in electromagnetics largely because the conventional field equations permit numerous different reformulations. These bring the electromagnetic field within the scope of numerical methods that rely on high-order local approximations while permitting comparative laxity with respect to boundary and interface conditions. After a very brief review of electromagnetic theory as it is relevant to finite elements, this chapter describes the projective and variational reformulations that lead directly to finite element methods, the distinctions between various types of boundary conditions, and the sometimes confusing terminology attached to them.

Maxwell's equations

Electromagnetic field problems occupy a relatively favourable position in engineering and physics in that their governing laws can be expressed very concisely by a single set of four equations. These evolved through the efforts of several well-known scientists, mainly during the nineteenth century, and were cast in their now accepted form as differential equations by Maxwell. There also exists an equivalent integral form.

Differential relations

The variables that the Maxwell equations relate are the following set of five vectors and one scalar:

Each of these may be a function of three space coordinates x, y, z and the time t. The four Maxwell equations in differential form are usually written as follows:

To these differential relations are added the constitutive relations

describing the macroscopic properties of the medium being dealt with in terms of its permittivity ∈, permeability µ and conductivity σ.