Differential and integral equation solvers based on generalized moments and partitions of unity

Methods to solve the both differential and integral equations have relied on the following procedure: develop the appropriate function spaces, and then use these spaces to create the appropriate weak form of the equation. Typically, these function spaces have relied on polynomials and are tightly coupled to the underlying description of the geometry. In the finite element community, there has been a concerted effort to develop methods that extend this reach, i.e., develop methods that rely on a distribution of points as opposed to meshes and the use of non-polynomial basis functions within the framework of partitions of unity. The challenges to applying these concepts directly to the analysis of vector electromagnetics problems has been overcome only recently, for both differential and integral equation based solvers. This paper will briefly describe the relevant theorems that make this possible and illustrate the applicability of these solvers with some relevant examples.