THE ORIGIN OF SPURIOUS SOLUTIONS IN COMPUTATIONAL ELECTROMAGNETICS

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.