Calderon preconditioned CFIE with MLFMM for acceleration

It is well known that the application of the method of moments to solving electromagnetic wave scattering from large PEC targets usually results in very ill-conditioned matrix equations. In fact, the condition number of the MoM matrices grows as O(1/(kh)²), where k is the wavenumber, and h is the discretization length. To circumvent this difficulty, in the recent literature there are quite a few proposed approaches. Roughly, we can divide them into two categories: the fast direct factorization methods and analytic preconditioners employing the Calderon identities. Among the fast direct factorization approaches, we mention the work by R. Adams et. al [1] and J. Shaeffer [2]. Among the analytic preconditioning strategies, we list the papers by R. Adams [3], Christiansen and Ńd́lec [4], and Buffa and Christiansen [5]. This paper closely follows the later approach, which is to precondition the MoM matrices using the Calderon formulas and hence reduce the iteration counts in Krylov iterative matrix solvers. We shall detail our progress in this paper, the successes, the remaining difficulties, and finally our ongoing efforts in making the proposed Calderon CFIE a fast and effective numerical method for solving electromagnetic wave scattering from large, closed PEC targets.