Electromagnetics

Abstract

In this chapter we briefly study finite elements for electromagnetic applications. We start off by recapitulating Maxwell’s equations and look at some special cases of these, including the time harmonic and steady state case. Without further ado we then discretize the time harmonic electric wave equation using Nédélec edge elements. The computer implementation of the resulting finite element method is discussed in detail, and a simple application involving scattering from a perfectly conducting cylinder is studied numerically. Next, we introduce the magnetostatic potential equation as model problem. Using this equation we study the basic properties of the curl-curl operator ∇ × ∇ ×, which frequently occurs in electromagnetic problems. In connection to this we also discuss the Helmholtz decomposition and its importance for characterizing the Hilbert space H( curl; Ω). The concept of a gauge is also discussed. For the mathematical analysis we reuse the theory of saddle-point problems and prove existence and uniqueness of the solution as well as derive both a priori and a posteriori error estimates.