It is commonly believed that the divergence equations in the Maxwell e
quations are ''redundant'' for transient and time-harmonic problems, t
herefore most of the numerical methods in computational electromagneti
cs solve only two first-order curl equations or the second-order curl-
curl equations. This misconception is the true origin of spurious mode
s and inaccurate solutions in computational electromagnetics. By study
ing the div-curl system this paper clarifies that the first-order Maxw
ell equations are not ''overdetermined,'' and the divergence equations
must always be included to maintain the ellipticity of the system in
the space domain, to guarantee the uniqueness of the solution and the
accuracy of the numerical methods, and to eliminate the infinitely deg
enerate eigenvalue. This paper shows that the common derivation and us
age of the second-order curl-curl equations are incorrect and that the
solution of Helmholtz equations needs the divergence condition to be
enforced on an associated part of the boundary. The div-curl method an
d the least-squares method introduced in this paper provide rigorous d
erivation of the equivalent second-order Maxwell equations and their b
oundary conditions. The node-based least-squares finite element method
(LSFEM) is recommended for solving the first-order full Maxwell equat
ions directly. Examples of the numerical solutions by LSFEM are given
to demonstrate that the LSFEM is free of spurious solutions. (C) 1996
Academic Press. Inc.