Hierarchical Divergence Conforming Bases for Edge Singularities in Quadrilateral Cells

Abstract

Singular divergence-conforming bases have been proposed for the solution of integral equations although they have seen only occasional use in practical applications. The existing singular bases are not hierarchical, which prevents their use in adaptive p-refinement applications. In this paper, a new family of singular hierarchical basis functions is proposed for quadrilateral cells. These functions model the singularities associated with current and charge density at edges and are more convenient for modeling such singularities than triangular bases of the same kind. The basis functions are of the additive kind and combine a hierarchical polynomial representation on quadrilaterals with linearly independent singular terms that incorporate general exponents that may be adjusted for the specific wedge angle of interest. Moreover, the added singular basis functions are computed on the fly. On the basis of various reported numerical results, this paper also illustrates the difficulties, the advantages, the accuracy, and the cost of using such bases in the method of moment solutions of integral equations.