High-order finite elements of complete and incomplete bases in electromagnetic-field computation

Function spaces of high-order differential-forms-based finite elements of complete bases are analysed and compared with those of incomplete bases. Elements of complete and incomplete bases differ only in the null space of the differential operator. They model the same range space of the differential operator. A comparison of l-form 2nd order hierarchical elements (hierarchical edge elements) of complete and incomplete bases is carried out through the example of an eddy-current problem. It is found that there is no significant difference in the accuracy of the results given by the two kinds of elements, as long as the ratio mesh size/skin depth is sufficiently small. However, elements of complete bases are much more expensive to use