The Finite Element Method in Electromagnetics, 3rd Edition

CHAPTER 8VECTOR FINITE ELEMENTS

In Chapters 5 and 7 we encountered several serious problems when the usual node-based finite elements, obtained by interpolating the nodal values, were employed to represent vector electric or magnetic fields. First, we observed the occurrence of nonphysical or so-called spurious solutions, which are generally attributed to lack of enforcement of the divergence condition. Second, we noted the inconvenience of imposing boundary conditions at material interfaces as well as at conducting surfaces. Third, we noted the difficulty in treating conducting and dielectric edges and corners due to field singularities associated with these structures. These were all discussed, and among them the last outweighs the others for its lack of a general cure. Even for the first two, the current status of the treatment is not entirely satisfactory. Therefore, it is necessary to explore other possibilities or other approaches, beyond simple modifications, to take the finite element analysis of electromagnetics to a new era.

Fortunately, a revolutionary approach was discovered in the 1980s. This approach uses vector basis or vector elements that assign degrees of freedom to the edges rather than to the nodes of the elements. For this reason, they are also called edge elements. Although these types of elements were described by Whitney [1] as early as 55 years ago, their use and importance in electromagnetics were not realized until in the 1980s. In the early 1980s, ...