In this chapter we introduce the basic quantities of electromagnetics, formulate their relations in terms of partial differential equations, and show how these equations can be solved via the finite element method. Emphasis is given to potential equations and to the Maxwell's equations, with particular interest in the time-harmonic field. We do not attempt to cover all interesting aspects of theoretical and computational electromagnetics: It is our goal to provide a sufficiently informative introduction that (a) should allow the reader to start solving practical problems and (b) prepare her/him for the study of more specialized literature. To mention just two books, [83] can be recommended to mathematically oriented readers who are especially interested in time-harmonic Maxwell's equations, and [102] addresses engineering audience.

Section 7.1 presents important basic facts about the macroscopic (continuous) model of the electromagnetic field, such as the four basic laws of electromagnetics, the Maxwell's equations in the integral and differential forms, media characteristics, basic properties of conductors, dielectrics and magnetic materials, and interface conditions. With an appropriate insight, many typical problems of electromagnetics can be formulated in terms of potentials and solved by means of the standard continuous finite elements. The scalar electric potential and the scalar and vector magnetic potentials are introduced in Section 7.2. The equations for the field vectors and the time-harmonic Maxwell's equations are derived in Section 7.3.

The rest of the chapter is devoted to the weak formulation and finite element analysis of the time-harmonic Maxwell's equations by means of edge elements. In Section 7.4 we define the Hilbert space if(curl), derive the weak formulation of the equations, show how

Partial Differential Equations and the Finite Element Method. By Pavel Solin Copyright © 2006 John Wiley & Sons, Inc.

various types of boundary conditions are incorporated into the sesquilinear weak form, and prove the existence and uniqueness of the weak solution in a simplified setting.

In Section 7.5 we perform the standard series of steps involved in the finite element method: We introduce the lowest-order Whitney element and the general higher-order edge element of Nedelec on the reference domain, use appropriate transformation to construct the basis functions in physical mesh elements, and transform the weak formulation of the Maxwell's equations to the reference domain. At the end the interpolation on higher-order nodal edge elements is discussed.

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