## 1. Introduction

[2] With the aid of fast solvers, the integral equation approach has been widely used in solving problems of electromagnetic scattering from large and complex structures. It has been reported that a multilevel fast multipole algorithm has been applied to calculate the radar cross section of a model airplane in X-band (with 9.99 million unknowns) [*Song and Chew*, 2000]. Though there are some researches on fast matrix inversions, most fast algorithms developed so far are based on iterative solvers. It is known that for iterative solvers, the major computer resource usages are (1) time and memory needed to perform each matrix-vector multiplication, and (2) time needed for multiple iterations for a convergence solution. For solving an integral equation, the fast algorithms are mainly designed for the reduction of the computational complexity of a matrix-vector multiplication. For example, the multilevel fast multipole algorithm has been successfully applied to reduce the computational complexity from *O*(*N*^{2}) to *O*(*N*log *N*) for a matrix-vector multiplication in solving electromagnetic scattering from 3-D complex conducting objects [*Song and Chew*, 1995, 2000; *Rokhlin*, 1990; *Coifman et al*., 1993]. This leaves the number of iterations an important part for overall solution time when multiple right-hand sides are considered. For some structures, the matrix may be poorly conditioned and convergence can be very slow. The convergence rate depends on a number of factors, such as the iterative solvers used, the target characteristics, and the equations used to model the physical problem. For example, in the integral equation approach, the combined field integral equation (CFIE) convergences much faster than that of the electric field integral equation (EFIE) for conducting scatters [*Song and Chew*, 1995] as well as for homogeneous dielectric scatters [*Harrington*, 1989]. Various efforts have been made to increase the convergence rate in the iterative solution of a matrix equation. These include the use of CFIE, better conditioned formulations, and preconditioning. As a result, if a problem allows a choice of different operator equations, one criterion to make a selection is to consider the rates of convergence for the operators.

[3] To calculate the EM scattering from 3-D coated structures, both the surface integral equation (SIE) and the hybrid volume-surface integral equation (VSIE) approaches can be applied given that the coating material is piece-wise homogeneous (required by SIE approach). An intuitive comparison may conclude that the SIE may need less number of unknowns than that of the VSIE. This is true when the dielectric material region has a large fractional volume. However, for thin material coatings of one or two layers (one tenth of a wavelength per layer), we can show that the number of unknowns needed by SIE and VSIE are of the same order. It is known that for the modeling of large-scale problems, the multilevel fast multipole algorithm can be applied to both SIE and VSIE to reduce the computational complexity to the same order. Under this circumstance, a key factor to determine which algorithm to be used will depend on which one needs less iteration numbers to obtain a convergent solution. Because the integral operator in the VIE is less singular than that in the SIE, it is expected that the VSIE approach will need less number of iterations compared with the SIE approach for scattering problems with material coating. In this paper, we present some numerical results to compare the convergence rate of SIE versus VSIE for solving a scattering problem. In the following, we will first present a brief formulation of the two algorithms in frequency domain with time factor exp(−*i*ω*t*), followed by some numerical examples to compare their performances in solving scattering problems. Based on the examples, we conclude that for thin material coating, the VSIE algorithm has significant advantage over the SIE approach if iterative solvers are used.