## 1. Introduction

[2] The demand for reliable and effective methods for predicting the scattering and propagation of electromagnetic waves arises both at the fundamental scientific level and in technological application. Accurate predictions of the wave interaction with its environment, its transmitting antenna, a sensor, or a remotely sensed target, is essential in assessing the feasibility or success of the potential applications in areas such as communications, remote sensing and imaging, and so on. Robust theoretical and numerical methods are not merely cost-effective alternatives or supplements to experimental methods, but are primarily design tools for the ever expanding array of devices and sensors of modern electrical engineering.

[3] The growth of widely available and powerful computer resources has had a profound effect on the development and applications of numerical methods in this area. If an integral equation with a Green's function kernel (such as the electric field integral equation) is used to formulate the scattering problem, the method of moments, first popularized by *Harrington* [1968], produces a finite system of linear algebraic equations for the coefficients of the basis functions that have been selected to represent the desired surface current. A variety of applications is collected by *Miller et al.* [1992]. Alternatively, direct discretization of Maxwell's equations leads to methods such as the finite difference time domain (FDTD) method [*Yee*, 1966] or the finite element method (FEM) [*Silvester and Ferrari*, 1990]. The greatest success of both integral and differential approaches has been in the low- to intermediate- (or resonance) wavelength regime (where the scatterer dimensions are one to several wavelengths in size); the computational cost limits the size of scattering problem that may be effectively solved. The text [*Peterson et al.*, 1998] surveys these methods and their effective implementation.

[4] However, there is a more profound question about the accuracy of numerical methods as the wavelength dimensions increase: A dense system of linear equations of hundreds of thousands of variables cannot avoid the possibility or indeed likelihood of ill-conditioning that substantially degrades the accuracy of any numerical solution computed by direct or iterative means (even with preconditioning).

[5] The mathematical foundations for error estimation in numerical solutions of the integral equations that arise in electromagnetics has been carefully discussed by *Hsaio and Kleinman* [1997] and *Colton and Kress* [1992], and while progress has been made in obtaining reliable bounds for smooth closed scatterers (such as the sphere), open surfaces (with edges) and nonsmooth surfaces are rather more problematic.

[6] This recognition provides one of the enduring imperatives for analytical methods and solutions in electromagnetics: reliable benchmark solutions enable direct assessment of the accuracy of numerical methods in every aspect of choices of surface discretization, basis functions, and numerical algorithm to solve the discretized system (direct or iterative).

[7] Another enduring reason for the study of analytical solutions to Maxwell's equations is the identification of dominant scattering mechanisms: solutions of canonical problems chosen to highlight features, such as edges or cavity-backed apertures, provide reliable quantitative predictions and identify those target features that must be carefully addressed by purely numerical general purpose codes for other noncanonical objects, of arbitrary shape. The classic text by *Bowman et al.* [1987] provides an excellent survey of known scattering results for a variety of mainly closed canonical surfaces (such as the sphere).

[8] Related to analytic methods is the development of more accurate “semianalytical” or “analytical numerical” methods that are applicable to a wider class of scatterers than the idealized canonical shapes. The method of regularization (MoR) described in this article falls into this class: it addresses the key difficulty of error estimation encountered with the usual integral equations of electromagnetics by analytically transforming them to second-kind matrix systems that are intrinsically well conditioned and have a firm basis for error estimation in Fredholm theory. This rigorous approach to scattering was extensively developed by V. P. Shestopalov and coworkers in Kharkov (Ukraine) starting in the mid-1970s, in both electromagnetic and acoustic contexts [*Radin and Shestopalov*, 1973; *Vinogradov and Shestopalov*, 1977; *Vinogradov*, 1978].

[9] The roots of the method may be traced to closely related regularization methods for potential theory problems [*Sneddon*, 1966; *Vinogradov et al.*, 2001]. The underlying guiding principle is to transform from first kind integral equations to second kind equations, by a process of analytical regularization. If we represent the integral equation in operator form as **AJ** = **E**, where the unknowns **J** (typically surface currents) are to be determined from a known excitation **E**, we seek a splitting of the operator **A** as the sum **A** = **A**_{0} + **A**_{1}, where the operator **A**_{0} is explicitly invertible (by some means). Then the equation can be transformed to (**I** + **H**)**J** = **A**_{0}^{−1}**E**, where **H** = **A**_{0}^{−1}**A**_{1}. Provided **H** has the correct property that makes this a second kind equation (**H** must be a compact operator in an appropriate function space), this transformed equation is the desired second kind regularized form (a compact perturbation of the identity **I**). Second kind functional equations have some intrinsic advantages over those of first kind, and there is a well developed literature and standard numerical techniques [*Atkinson*, 1997] for their solution. Moreover, Fredholm theory can be applied, the numerical convergence properties are excellent and the numerical error can be reliably estimated [*Kantorovich and Akilov*, 1974]. However reliable, if somewhat less easily applied, methods have been developed based upon coercive bilinear forms by E. P. Stephan, W. L. Wendland, and M. Costabel [see, e.g., *Carstensen and Stephan*, 1996]. The generic idea, of transformation to second kind equations, has been widely used in many areas of applied mathematics; for example it was used by *Jones* [1981] to establish the existence and uniqueness of the solution to the integral equation for the thin wire antenna (with exact kernel, rather than reduced kernel). An exposition of the method for scattering problems is given by *Vinogradov et al.* [2002].

[10] The validation of putative methods for calculating scattering from complex objects incorporating edges and cavity-backed apertures, which themselves may enclose a variety of other scatterers, depends entirely upon comparison with the results of other proven approaches, whether analytical, computational or experimental.

[11] This paper surveys some recent developments in analytical or analytical: numerical methods that illustrate their continuing importance in electromagnetic theory and its applications. Some recently developed analytical solutions for elliptically shaped cavities are described in section 2. The mathematical background that motivates the development of rigorous analytical-numerical methods such as MoR is described in section 3. Sections 4 and 5 explain the application of the MoR to canonical problems for spherical structures and spheroidal structures, by choosing the simplest representative example: the dipole excited spherical or spheroidal cavity. Sections 6 and 7 treat noncanonical problems for two classes of arbitrarily shaped structures: two-dimensional structures and bodies of revolution. Sections 8 and 9 briefly describe other applications and approaches to regularization methods.