A generalized basis function framework for integral and differential equations in electromagnetics

The development of appropriate basis functions and their application to the solution of integral and differential equation has been a topic of continuous research for the past two decades. Hierarchial basis functions that provide h-, p-, and hp- convergence is now the state of the art. However, all these basis function spaces have been designed such that they closely rely on the definitions of the underlying tesselation. Recent efforts have focused on the development of function spaces that remove this tight inter-relationship. Examples of this from the finite element community are the meshless methods-EFG, Point Clouds, Generalized finite elements (GFEM), etc. In this paper, we will present a unified framework for both integral and differential equations that build upon the GFEM framework and is specialized to vector functions. The paper (and the conference presentation) will will briefly describe the development of a class of basis functions for both differential and integral equations, and demonstrate some of the advantages offered by these classes of basis functions. These include seamlessly mixing different types and orders of basis functions as well as its flexibility in the handling non-conformal discretization of scatterers.