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New tetrahedral finite elements are described for the analysis of vector electromagnetic fields. They are singular elements that are more accurate than conventional, polynomial-based elements near sharp edges and corners. They are hierarchical, meaning that elements of different orders (accuracies) are available and may be placed together in the same mesh without violating continuity of the field. The elements are formed from a series of scalar singular elements by taking the gradient, and are therefore called gradient singular. Results using the new elements in p-adaptive analysis, orders 1 to 3, demonstrate that when they are used in place of conventional elements the scattering parameters are as much as 10 times more accurate for the same number of unknowns.