Hierarchical Divergence Conforming Bases for Tip Singularities in Quadrilateral Cells

Electromagnetic scattering from targets such as thin conducting plates induce singular currents and charges at sharp edges and sharp tips. In this article, a hierarchical family of divergence-conforming singular basis functions are presented for modeling the singularities associated with current and charge density at tips. These new basis functions are used to increment existing edge-singular bases so that on cells that contain a singular tip where two singular edges join together, the final base combines a hierarchical polynomial representation with linearly independent singular terms that incorporate general exponents that may be adjusted for the specific wedge angle of interest and for the specific angle at the tip. Several variations on the tip functions are proposed.