## 1. Introduction

[2] Computation of electromagnetic (EM) wave scattering from non-penetrable targets has been an active research topic for many years. It can be argued that the surface integral equation (SIE) methods are best in addressing such an application. Particularly, in recent years, there are many fast and efficient integral equation methods that extend the SIEs to EM wave problems with thousands of wavelength in dimensions. Among them, we mention the multilevel fast multipole method (MLFMM) [*Rokhlin*, 1985; *Coifman et al.*, 1993; *Rokhlin*, 1990; *Song and Chew*, 1995; *Lu and Chew*, 1994], which reduces the memory and CPU complexities to *O*(*N*) and *O*(*N log N*), respectively, though a strict decomposition of the Green's function is required. Unfortunately, FMM, or it's multilevel version MLFMM, suffers the “sub-wavelength breakdown problem” [*Dembart and Yip*, 1998] and it is heavily kernel dependent. Other methods such as the pre-corrected fast Fourier transform (pFFT) [*Phillips and White*, 1997] and the adaptive integral method (AIM) [*Bleszynski et al.*, 1996; *Wang et al.*, 1998] accelerate the matrix-vector multiplications by substituting the current basis functions of the original problem via the new equivalent current sources that reside on a rigid grid, thus facilitates the use of FFT algorithm.

[3] Although we have witnessed significant advancements of fast integral equation methods in recent years, most of them mainly address the issue with the speed of the matrix-vector multiplications. Nonetheless, the overall success still relies on the availability of a robust and effective preconditioner for the integral equation methods. Even though, there are many substantial developments in this regard [*Peng et al.*, 2011], the existence of a preconditioner that guarantees the convergence in the iterative matrix solution process remains largely elusive. Direct solvers for integral equation methods are another important and interesting branch, they are sometimes favored over their iterative counterparts, especially in solving ill-conditioned matrix equations. Moreover, they often exhibit high efficiency in multiple right-hand-sides (RHSs) owing to the small constant in front of the complexity asymptotic when dealing with small or moderate electrical size problems. However, the conventional direct solver, based on the LU factorization, scales as*O*(*N*^{2}), *O*(*N*^{3}) for memory consumption and the factorization time, respectively. The inherent high complexities of the conventional LU direct solvers severely limit their application to solve practical EM problems. To circumvent these difficulties, several fast direct solvers have been proposed in the literature. In *Shaeffer* [2008], the author reported solving an one-million unknown problem using MultiLevel Adaptive Cross Approximation (ML-ACA) algorithm. Also, in*Adams et al.* [2008], a local-global solution method separates the radiating current from the non-radiating counterpart and reported to achieve*O*(*N*^{1.3}) complexity in terms of memory consumption for electrically large problems. Additionally, *Heldring et al.* [2007] discussed a compressed block decomposition (CBD) method and demonstrated a complexity of *O*(*N*^{1.5}) for the memory consumption. Another work conducted in *Winebrand and Boag* [2007] and *Boag* [2007]adopts the non-uniform grid (NG) based matrix compression method, it introduces a non-redundant coarse spherical non-uniform sampling grid to effectively skeletonize the coupling process and compress the matrix using Schur's complement.*Chai and Jiao* [2011] claimed to find the ^{2} representation of the inverse of the dense matrix in an error-controllable manner and reported a linear complexity for both CPU time and memory consumption. However, we disagree with the complexity analyses presented in*Chai and Jiao* [2011] and remain unconvinced of the performance reported. Moreover, one of the recently published works, *Li et al.* [2012] shares some similarities with the proposed algorithm in this paper. It also seeks for a unique mapping matrix for each group to represent the coupling.

[4] In this paper, a fast direct solver, based largely on the algorithm outlined in *Greengard et al.* [2009] and *Martinsson and Rokhlin* [2005], is presented to solve SIE matrix equations for electrodynamic applications. This algorithm utilizes a low rank decomposition of the off-diagonal coupling blocks of the dense matrices [*Cheng et al.*, 2005]. Moreover, a multilevel version in-conjunction with a Huygens' surface to account for couplings between well-separated groups is also discussed in detail. Although, we believe that the algorithm will not alter the complexities of matrix solutions in SIEs (in the worst case scenario for electrically large problems), the proposed algorithm can be very efficient for many practical numerical examples. Particularly, during the process of*h-refinement*, where the discretization size decreases to improve the accuracy, the complexities observed are *O*(*N*) and (*N*^{1.5}) for memory and CPU time, respectively. The reported complexities agree well with the theoretical predictions in *Martinsson and Rokhlin* [2005]for smooth integral kernels on general two-dimensional surfaces. Several numerical results are included to validate the algorithm. Additionally, numerical experiments are conducted for fixed mesh size scenario, where the frequency increases, as well as for fixed frequency, while the mesh size decreases (the*h*-refinement) [*Wei et al.*, 2011].