problems in computational electromagnetics may be classified mainly into four categories: (1) those based on the It is commonly believed that the divergence equations in the Maxwell equations are ''redundant'' for transient and time-vector potentials; (2) those based on the first-order curl harmonic problems, therefore most of the numerical methods in equations; (3) those based on the second-order curl-curl computational electromagnetics solve only two first-order curl equations; (4) those based on the Helmholtz equations.
equations or the second-order curl-curl equations. This misconcep-
The potential approach is widely used in the computation is the true origin of spurious modes and inaccurate solutions tion of static fields and eddy currents. It also can be used in computational electromagnetics. By studying the div-curl system this paper clarifies that the first-order Maxwell equations are not for time-harmonic problems; see, e.g., Boyse et al. [4]. The ''overdetermined,'' and the divergence equations must always be potential approach does not give rise to spurious modes, included to maintain the ellipticity of the system in the space dosince the divergence-free equations are automatically satismain, to guarantee the uniqueness of the solution and the accuracy fied by introducing the vector potentials. It also makes of the numerical methods, and to eliminate the infinitely degenerate material discontinuities easy to model. However, this apeigenvalue. This paper shows that the common derivation and usage of the second-order curl-curl equations are incorrect and that proach involves difficulties related to the appropriate gaugthe solution of Helmholtz equations needs the divergence condition ing method and the loss of accuracy and continuity (in to be enforced on an associated part of the boundary. The div-curl homogeneous media) of the calculated field intensity from method and the least-squares method introduced in this paper prothe potentials by the numerical differentiation. vide rigorous derivation of the equivalent second-order Maxwell The most widely used numerical method for the solution equations and their boundary conditions. The node-based leastsquares finite element method (LSFEM) is recommended for solving of time-dependent electromagnetic problems has been the the first-order full Maxwell equations directly. Examples of the nufinite-difference time-domain (FD-TD) scheme developed merical solutions by LSFEM are given to demonstrate that the by Yee [70] and extensively utilized and refined by Taflove LSFEM is free of spurious solutions. แฎ 1996 Academic Press, Inc.
and Umashankar [62], Kunz and Luebbers [32], as well as others. In the Yee scheme, only two Maxwell curl equations are solved. Some other time-domain methods are also 104