## 1. Introduction

[2] Electromagnetic integral equations are often discretized using the method of moments (MoM) [*Mahachoklertwattana et al.*, 2008; *Ye and Jin*, 2008; *Kim et al.*, 2004; *Rao et al.*, 1982], one of the most widespread and well established techniques for electromagnetic problems. However, the matrix associated with the resulting linear systems is usually large and dense for electrically large targets for electromagnetic scattering problems [*Mahachoklertwattana et al.*, 2008]. It is basically impractical to solve electric field integral-equation (EFIE) matrix equations using direct methods because their memory requirement and computational complexity are *O*(*N*^{2}) and *O*(*N*^{3}), respectively, where *N* is the number of unknowns. Recently, a number of fast methods have been developed for accelerating the iterative solution of electromagnetic integral equations discretized using MoM. Most of them are based on the multilevel subdomain decomposition, requiring a computational complexity with the order of *N*log*N*. One of the most popular techniques is the fast multipole method (FMM) [*He et al.*, 2001; *Ergül and Gürel*, 2007, 2009; *Chew et al.*, 2001; *Pan et al.*, 2001; *Malas and Gurel*, 2009; *Oo et al.*, 2002, 2004; *Rodriguez et al.*, 2008], which has been widely used to solve very large electromagnetic problems due to its excellent computational efficiency.

[3] Another fast method is the multilevel simply sparse method (MLSSM) [*Canning and Rogovin*, 2002; *Zhu et al.*, 2005; *Cheng et al.*, 2008; *Adams et al.*, 2006; *Xu et al.*, 2010; *Xu*, 2009]. The single level of SSM is presented by *Canning and Rogovin* [2003], whereas the MLSSM is shown by *Canning and Rogovin* [2002], *Zhu et al.* [2005], *Cheng et al.* [2008], *Adams et al.* [2006], *Xu et al.* [2010], and *Xu* [2009]. This method can be implemented easily, with a critical advantage that it can be programmed on top of any existing computer codes. The adaptive cross approximation (ACA) [*Zhao et al.*, 2005] was used to fill up the far-field matrix of the conventional MLSSM, which is more efficient than the direct filling. However, both the memory and computation complexities of the ACA are of the order of *O*(*N*^{4/3}log*N*) for moderate-size problems that are formulated using the surface integral equation approach [*Zhao et al.*, 2005], which are higher than that of the MLFMA.

[4] The MLSSM introduced by *Canning and Rogovin* [2002, 2003], *Adams et al.* [2006], *Xu et al.* [2010], and *Xu* [2009] is applied to the direct solution of the matrix equation, whereas the method introduced by *Zhu et al.* [2005] and *Cheng et al.* [2008] is applied to the iterative solution of the matrix equation. An improved MLSSM presented in this paper is an iterative solution for general 3-D electromagnetic problems. Numerical results show that when compared to the ACA, the MLFMA gives a much more efficient far-field matrix filling operation for the MLSSM. Hence, the MLFMA can be used to fill up the far-field matrix of MLSSM efficiently.

[5] This paper is organized as follows. Section 2 gives a brief introduction to the MLFMA and the electric field integral equations (EFIE) for electromagnetic scattering problems. Section 3 describes the improved MLSSM in detail. Section 4 gives some numerical examples to demonstrate the accuracy and computation efficiency of our approach.