NODAL-BASED FINITE-ELEMENT MODELING OF MAXWELL EQUATIONS

Weak forms are derived for Maxwell's equations which are suitable for implementation on conventional C0 elements with scalar bases. The governing equations are expressed in terms of general vector and scalar potentials for E over arrow pointing right. Gauge theory is invoked to close the system and dictates the continuity requirements for the potentials at material interfaces as well as the blend of boundary conditions at exterior boundaries. Two specific gauges are presented, both of which lead to Helmholtz weak forms which are parasite-free and enjoy simple, physically meaningful boundary conditions. The extended weak form introduced by Lynch and Paulsen along with associated boundary conditions, is recovered in greater generality from the first gauge considered, where the vector potential is discontinuous at material interfaces and when the scalar potential vanishes. The second and preferred gauge allows the use of continuous vector and scalar potentials at the expense of introducing coupling among the two potentials. A general and numerically efficient procedure for enforcing the jump discontinuities on the normal components of vector fields at dielectric interfaces and boundary conditions on curved surfaces is given in the Appendix.