Calculation of modes in a microstrip line with no zero or spurious modes

It can be seen that nodal finite element basis functions impose too high a level of continuity on the electromagnetic fields. The fields are forced to be continuous over the whole region, which of course is not physically correct as Maxwell´s equations demand only that the tangential component of the electric/magnetic field intensity is continuous at a dielectric/magnetic material interface. A number of authors show that the correct choice of finite element basis functions for solving Maxwell´s equation are `edge elements´. `Edge elements´ have eliminated the occurrence of spurious modes in microstrip calculations, however, they have introduced `zero modes´. `Zero modes´ are also non-physical modes (they are contained in the kernel of the Curl operator), however, they can be eliminated immediately without recourse to any post processing technique. However, the occurrence of zero modes would imply a weakness in the original algorithm. It can be seen that though `edge element´ basis functions satisfy the divergence condition in the interior of each element, they do not necessarily satisfy the condition on inter-element edges. This Joly et al. (see Proc. 10 International Conf. Comput. Methods in App. Sciences and Engineering, INRIA, p.433-44, 1992) suggest to be the source of `zero modes´. A novel finite element formulation, similar to that of Joly et al. is presented which eliminates both spurious and zero modes in microstrip calculations