## 1. Introduction

[2] The integral equation approach is one of the methods traditionally employed for solving scattering and radiation problems involving dielectric materials [*Peterson et al.*, 1998]. While application of a surface integral equation to this kind of problems is restricted to objects composed by multiple homogeneous dielectric regions [*Goggans et al.*, 1994], the volume integral equation (VIE) can handle general inhomogeneous objects, for instance dielectric lenses. Owing to the three-dimensional nature of the problem, the number of unknowns *N* in the method of moment (MoM) solution of the VIE grows very fast with the size of the object. For the classical solution the required memory for the MoM matrix storage is *O*(*N*^{2}), the solution complexity is *O*(*N*^{3}) for a direct matrix solver, and *O*(*N*^{2}) for an iterative one. One way to reduce the complexity of the problem is to apply fast iterative solvers, such as fast multiple methods (FMM/MLFMM) [e.g., *Chen and Jin*, 1999; *Sertel and Volakis*, 2001; *Lu*, 2001] or conjugate-gradient algorithms combined with the fast Fourier transform (CG-FFT/BCG-FFT) [e.g., *Catedra et al.*, 1989; *Zhang et al.*, 2003; *Millard and Liu*, 2003], which do not compute and store the whole MoM matrix but only a matrix-vector product. These methods can reach the solution complexity and memory requirements of *O*(*N*log*N*), but they usually require a very fine discretization of the object and low-order basis functions (pulse [e.g., *Su*, 1993], rooftop [e.g., *Gan and Chew*, 1995], or volume RWG [e.g., *Lu*, 2003]) in order to be efficient.

[3] Another way to reduce the excessive CPU and memory requirements of the MoM solution for the VIE is to reduce the number of unknowns *N* in comparison to low-order methods. A considerable reduction of *N* can be accomplished by applying higher-order methods involving higher-order basis functions and higher-order geometry modeling. There are two classes of higher-order basis functions, interpolatory [*Graglia et al.*, 1997] and hierarchical [*Kolundžija and Popović*, 1993]. Hierarchical functions provide greater flexibility, enabling different expansion orders in different elements in the same mesh. The main bottleneck of existing hierarchical basis functions is the ill-conditioning of the MoM matrix which requires a direct solver to be employed [*Notaroš et al.*, 2001]. To apply an iterative solver, which is much more effective, the condition number of the MoM matrix must be improved. Recently, a new type of higher-order hierarchical basis functions was proposed by *Jørgensen et al.* [2002]. Calculated on the basis of appropriately modified Legendre polynomials, the new basis functions are near-orthogonal and therefore provide a low condition number of the MoM matrix. These functions have been successfully applied to surface integral equations for the analysis of metallic objects in free space [*Jørgensen et al.*, 2002] and in layered media [*Kim et al.*, 2002].

[4] In this paper, we apply the higher-order hierarchical Legendre basis functions by *Jørgensen et al.* [2002] to solve the electric field VIE. Numerical examples for scattering by a homogeneous dielectric sphere demonstrate excellent agreement between the results of our higher-order hierarchical MoM and the analytical Mie series solution. A very low condition number of the MoM matrix allows the use of an iterative matrix solver even for high expansion orders. A homogeneous dielectric cube serves as an example of a scatterer with corner singularities. The result with a very fine discretization and first-order basis functions is used as a reference. The memory requirements of the higher-order hierarchical MoM are compared with those of an existing multilevel fast multipole method (MLFMM) implementation with low-order basis functions [*Sertel and Volakis*, 2001]. Results for the scattering by a Luneburg lens are also presented.