Iterative solution of large-scale scattering problems in computational electromagnetics with the multilevel fast multipole algorithm (MLFMA) requires strong preconditioners, especially for the electric-field integral equation (EFIE) formulation. Incomplete LU (ILU) preconditioners are widely used and available in several solver packages. However, they lack robustness due to potential instability problems. In this study, we consider various ILU-class preconditioners and investigate the parameters that render them safely applicable to common surface integral formulations without increasing the O(n log n) complexity of MLFMA. We conclude that the no-fill ILU(0) preconditioner is an optimal choice for the combined-field integral equation (CFIE). For EFIE, we establish the need to resort to methods depending on drop tolerance and apply pivoting for problems with high condition estimate. We propose a strategy for the selection of the parameters so that the preconditioner can be used as a black-box method. Robustness and efficiency of the employed preconditioners are demonstrated over several test problems. Introduction.A popular approach in studying wave scattering phenomena in computational electromagnetics (CEM) is to solve discretized surface integral equations, which give rise to large, dense, complex systems in the form of A · x = b. Two kinds of surface integral equations are commonly used. The electric-field integral equation (EFIE) can be used for both open and closed geometries, but it results in poorly conditioned systems, especially when the geometry is large in terms of the wavelength. On the other hand, the combined-field integral equation (CFIE) produces well-conditioned systems, but it is applicable to closed geometries only [21].For the solution of such dense systems, direct methods based on Gaussian elimination are still widely used due to their robustness [34]. However, the large problem sizes confronted in computational electromagnetics prohibit the use of these methods which have O(n 2 ) memory and O(n 3 ) computational complexity for n unknowns. On the other hand, by making use of the multilevel fast multipole algorithm (MLFMA) [12], the dense matrix-vector products required at least once in each step of the iterative solvers can be performed in O(n log n) time and by storing only the sparse near-field matrix elements, rendering these solvers very attractive for large problems.However, the iterative solver may not converge, or convergence may require too many iterations. We need to have a suitable preconditioner to reach convergence in

Abstract-We report fast and accurate simulations of metamaterial structures constructed with large numbers of unit cells containing split-ring resonators and thin wires. Scattering problems involving various metamaterial walls are formulated rigorously using the electric-field integral equation, discretized with the Rao-WiltonGlisson basis functions. Resulting dense matrix equations are solved iteratively, where the matrix-vector multiplications are performed efficiently with the multilevel fast multipole algorithm. For rapid solutions at resonance frequencies, convergence of the iterations is accelerated by using robust preconditioning techniques, such as the sparse-approximate-inverse preconditioner. Without resorting to homogenization approximations and periodicity assumptions, we are able to obtain accurate solutions of realistic metamaterial problems discretized with millions of unknowns.

We present rigorous solutions of electromagnetics problems involving 3-D dielectric photonic crystals (PhCs). Problems are formulated with recently developed surface integral equations and solved iteratively using the multilevel fast multipole algorithm (MLFMA). For efficient solutions, iterations are accelerated via robust Schur-complement preconditioners. We show that complicated PhC structures can be analyzed with unprecedented efficiency and accuracy by an effective solver based on the combined tangential formulation, MLFMA, and Schur-complement preconditioners.Index Terms-Multilevel fast multipole algorithm (MLFMA), photonic crystals (PhCs), Schur-complement preconditioners, surface integral equations (SIEs).

Abstract-We present an iterative inner-outer scheme for the efficient solution of large-scale electromagnetics problems involving perfectlyconducting objects formulated with surface integral equations. Problems are solved by employing the multilevel fast multipole algorithm (MLFMA) on parallel computer systems.In order to construct a robust preconditioner, we develop an approximate MLFMA (AMLFMA) by systematically increasing the efficiency of the ordinary MLFMA. Using a flexible outer solver, iterative MLFMA solutions are accelerated via an inner iterative solver, employing AMLFMA and serving as a preconditioner to the outer solver. The resulting implementation is tested on various electromagnetics problems involving both open and closed conductors. We show that the processing time decreases significantly using the proposed method, compared to the solutions obtained with conventional preconditioners in the literature.