## 1. Introduction

[2] The analysis of the electromagnetic interaction with penetrable inhomogeneous materials is a problem with direct impact to a number of areas of research including medicine, geo-remote sensing, and stealth technology. Due to the necessity of volume discretization, the analysis of such problems has predominately been solved via partial differential equation based solvers that result in sparse linear systems such as the finite element method [*Jin*, 1993; *Volakis et al.*, 1998] or the finite difference time domain [*Taflove and Hagness*, 2000]. With the advent of fast solvers such as the fast multipole method [*Coifman et al.*, 1993], integral equation based solvers are becoming a competitive alternative [*Lu and Chew*, 2000]. In fact, when analyzing the electromagnetic interaction of thin material coatings or shells, volume integral equation methods combined with fast solution algorithms can be more efficient than finite-methods or surface integral equation approaches [*Lu and Chew*, 2000; *Lu*, 2001].

[3] The focus of this paper is the development of a high-order solution of the volume integral equation for the electromagnetic scattering of penetrable inhomogeneous dielectric objects. The proposed scheme leads to exponential convergence in both the radar cross section (RCS) and the volume current density. Previously, a high-order solution method based on the Locally Corrected Nsytröm (LCN) method [*Canino et al.*, 1998] has been proposed for the scattering of two-dimensional inhomogeneous material volumes [*Liu and Gedney*, 2000, 2001]. More recently, a method of moment analogy to the LCN method via a quadrature-point-based discretization scheme has been proposed [*Gedney*, 2001, 2003] for the scattering by PEC surfaces. This method is extended herein to the high-order solution of the volume integral equation method.

[4] Point-based discretization schemes have been previously proposed for volume integral equation methods [e.g., *Jin et al.*, 1989; *Papagiannakis*, 1997; *De Doncker*, 2001]. The scheme presented by *De Doncker* [2001] is based on a potential formulation. The method employs parametric elements for accurate modeling of curved surfaces, arbitrary order basis functions chosen to be Lagrange interpolation polynomials, and a point-based sampling at the interpolation points. In contrast, the proposed scheme is based on the solution of the electric field integration equation (EFIE), and the point matching is done at the appropriate quadrature points. The quadrature point sampling is equivalent to a Galerkin scheme with a fixed-point integration of the outer integral, thus leading to exponential convergence [*Gedney*, 2001, 2003]. In the proposed formulation, inhomogeneous dielectrics with continuous profiles are also conveniently and efficiently modeled within a cell. This simplifies discretization, and provides a robust solution method for inhomogeneous materials. Finally, it is shown that when analyzing thin materials the volume integral equation (VIE) approach can be more efficient than a surface integral equation (SIE) solution approach [*Poggio and Miller*, 1973; *Mautz and Harrington*, 1979; *Sheng et al.*, 1998]. Namely, fewer unknowns can be required by a VIE approach than a SIE approach for the same level of accuracy.