A p-MUS Preconditioner for the EFIE

This paper considers the solution of the electric field integral equation (EFIE) in electromagnetics. As with associated finite element methods, their solution relies upon the construction of conforming bases. While lowest order (RWG) spaces are near ubiquitous, their extension to higher order offers, potentially, a number of benefits in terms of accuracy and efficiency, which has been well documented in both finite elements and integral equation formulations. A further evolution of higher order conforming bases is the hierarchical basis. These have demonstrated considerable gains in efficiency in finite element applications. Such bases allow for the development of effective acceleration schemes, for instance, the multilevel Schwarz type preconditioner (p-MUS). An obvious question arises as to the applicability of such hierarchical bases and their associated acceleration schemes to integral equations. It is seen that the conclusions as to their efficacy depend strongly on the scattering regime. In particular, high-frequency problems (those where the wavelength is the principal determinant of mesh size) are shown to benefit little from hierarchical functions. On the other hand, for “low-frequency” problems (where geometry is the main determinant of mesh size), there are significant improvements in performance over corresponding interpolatory schemes.