## 1. Introduction

[2] The volume-surface integral equation (VSIE) [*Sarkar and Arvas*, 1989; *Lu and Chew*, 2000] for the electric field is applied to solve radiation and scattering problems for composite metallic and dielectric objects. The VSIE involves a system of two coupled equations for a dielectric volume and a perfectly electrically conducting (PEC) surface. The VSIE formulation allows full-wave three-dimensional analysis of, e.g., microstrip antennas taking into account the finite-size effects of the dielectric substrate and a ground plane. The VSIE is also suitable for modeling of scattering by metal objects with dielectric coating. Due to its volume integral equation part the VSIE can easily handle inhomogeneous dielectric objects, which are otherwise very inefficient to treat with a surface integral equation (SIE).

[3] The VSIE has been solved previously with the standard method of moments (MoM) [*Sarkar and Arvas*, 1989; *Lu and Chew*, 2000] as well as with its accelerated versions, including the multilevel fast multipole method (MLFMM) [*Lu*, 2001b], the adaptive integral method (AIM) [*Ewe et al.*, 2004], and the precorrected fast Fourier transform method (P-FFT) [*Nie et al.*, 2005]. All these methods utilize low-order basis functions, e.g., pulse, rooftop, or RWG, to discretize the integral equation. In this paper we solve the VSIE using higher-order basis functions defined on higher-order curvilinear volume and surface elements. Unlike the accelerated solvers, which reduce the complexity of the standard MoM but leave the number of low-order basis functions unchanged, higher order basis functions reduce the number of unknowns, since they generally require much fewer unknowns per wavelength to achieve the same accuracy. Here we utilize higher-order hierarchical Legendre basis functions [*Jørgensen et al.*, 2004b], which have previously been applied to the analysis of metallic objects in free space [*Jørgensen et al.*, 2002] and in layered media [*Jørgensen et al.*, 2004a] by surface integral equations, and to the analysis of dielectric objects by volume integral equations [*Kim et al.*, 2004]. Being near-orthogonal these functions allow a low condition number of the MoM matrix to be achieved. Moreover, we show here that the higher-order Legendre basis functions are able to explicitly enforce the continuity condition for the surface charge density on a PEC surface and the normal component of the electric flux density in the dielectric at those parts where the PEC surface is in contact with the dielectric material. Explicit enforcement of the continuity condition not only reduces the number of unknowns but also improves the convergence of the solution. The exact Mie series expansion for a dielectrically coated PEC sphere is used to validate the presented technique. Numerical results for more complex electrically large scatterers are also given and compared with the results obtained by other numerical methods.