A multilevel hierarchical preconditioner for multiscale EM scattering

Integral equation schemes, both EFIE and MFIE, have become a powerful tool, particularly with the development of accelerated schemes such as the fast multipole method (FMM). Key to most such treatments is the requirement to solve matrix equations iteratively, which at their core involve matrixvector multiplications. Much of the cost of such solutions then depends on the number of iterations and the cost per iteration. The vast majority of integral equations implementations employ the simplest Rao-WiltonGlisson (RWG) basis functions on triangles. High order interpolatory bases have been developed which (in principle) offer improved accuracy for a given cost. Within the current developments, the next natural step beyond high order interpolatory methods is to arrange these bases hierarchically, as already experienced within the finite element community. Of themselves hierarchical bases offer little more than their high order interpolatory counterparts. However, as has been demonstrated in finite elements it is possible to employ this hierarchical structure to great effect in the reduction of the computational cost of the underlying iterative scheme via a multilevel Schwarz type pre-conditioner. In this paper we attempt to apply such hierarchical bases and their associated acceleration schemes to integral equations. The results suggest that 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. Unlike their finite element counterparts, equivalent p-MUS integral equation schemes appear to offer little gain, if any, over nonhierarchical schemes. On the other hand, for `low frequency´ problems, such as scattering of objects with sub-wavelength features (where