On the Preconditioning of the Algebraic Linear Systems Arising from the Discretization of the EFIE

The integral equations involved in electromagnetics are discretized by subdomain or entire domain-type Method of Moments (MoM). These eventually result into complex algebraic linear systems, whose size depends on both the meshing fineness and the electrical dimensions of the structures at hand. So the overall performance of the numerical method is dramatically affected by the availability of efficient matrix solvers. Though at times we can still think of employing direct factorization, as far as the structures are small or the mesh is coarse, this approach is useless in practice, as the size of the discretized problem grows. Then the only concrete alternative is the use of Krylov Subspace methods. However, when dealing with systems produced by discretization of EFIE by MoM, Krylov methods are known to converge very slowly or not at all. This behavior is caused by unfavorably spectral properties of the impedance matrix Z. Therefore, it is crucial to use an iterative method in conjunction with an efficient preconditioner. In technical literature, researches relevant preconditioning have been mainly directed toward the design of algebraic preconditioners as AINV, SPAI and ILUT [1]. In this work, we consider a simple pre-conditioner based on the skew Hermitian component S of the Z impedance matrix. This choice is justified by the observation that S is expected to dominate (in some sense) the Hermitian part H of Z causing a localization of the spectrum of the preconditioned system around the point (1; 0) of the complex plane. It is well-known that this condition causes an enhancement of the convergence rate of Krylov Subspace Methods.