Compatibility Relations and the Finite-Element Formulation of Electromagnetic Field Problems

When computing an electromagnet ic field using the f ini te element method it is possible that, although Maxwell’s equations are discret ized accurately, highly inaccurate computational results are obtained. In those cases it can easily be s h o w n that (some of) the electromagnetic compatibi l i ty relations (field properties that follow from Maxwell’s equat ions) are not satisfied. The divergence condition on the fluxes, for instance, follows d i rec t ly from the field equations but not necessarily from their discret ized counterparts. This necessi ta tes inclusion of the compat ibi l i ty relations in the finite-element formulation of the field problem. A survey is given of all electromagnetic compatibi l i ty relations. tion will be generalized to the use of the compatibility relations for electromagnetic fields. It will also be shown that the use of divergence-free edge elements, which is advocated by some authors (see [4] and the references contained in it) for satisfying some of these compatibility relations, as well as the use of face elements, introduces the possibility of violating additional relations of the compatibility type. The importance of including the electromagnetic compatibility relations explicitly in the finite-element formulation of the problem is stressed. The analysis of the compatibility relations is carried out for methods for computing time-domain (transient) electromagnetic fields. The analysis of methods for timeharmonic electromagnetic as well as for static electric or magnetic field problems runs along similar lines and leads to similar conclusions.