Radio Science, Volume22, Number 6, Pages 929-934,November1987 Some recentdevelopments in iterative techniques for solving electromagnetic boundary value problems C. H. Chan and R. Mittra Department of Electrical & Computer Engineering, Universityof Illinois, Urbana (Received December 1. 1986; revised March 12, 1987; accepted March 16, 1987) The objective of this paper is to present a brief reviewof recent develop- ments in a class of iterative techniques for solving electromagnetic boundary value problems. The iterative algorithm discussed herein is basedon the minimization of the norm of the boundary condition error on a bodywhose scattering properties are of interest. It is shown that different choices of variational functions in the minimization process lead to different iteration schemes whose convergence properties can vary considerably. To illustrate this point, three different variational functions are chosen to demonstrate their applications in solving three different typesof electromagnetic boun- dary value problems. Some observations on the useof iterativetechniques for solving scattering problems with multiple incidence angles are included in the paper. 1. INTRODUCTION Iterative approaches to solving electromagnetic problems have received considerable attention in recent years, due primarily to the fact that they enableone to investigate problems that require deal- ing with a large number of unknowns, and are conse- quently intractable using conventional matrix methods. Some examples of these problems include scattering from electrically large and inhomogeneous bodies that are too complex to be treated with asymptotic techniques, e.g., the geometricaltheory of diffraction (GTD); determination of propagation and coupling characteristicsof multiconductor printed circuit transmission lines; and, calculation of the equivalent circuits of microstrip discontinuity prob- lems. During the last few years, an iterative technique called the conjugate gradientmethod(CGM) has been employed by a number of authors [Sarkar et al., 1981; van den Berg, 1984, 1985; Sultan and Mittra, 1985; Peterson and Mittra, 1985a, b; Ray and Mittra, 1984: Hurst and Mittra, 1984; Mittra and Chan, 1985] to solvea wide variety of electromagnetic problems requiring up to several thousand unk- nowns. A lucid exposition of the conjugate gradient Copyright1987 by •he American Geophysical Union. Paper number 7S0312. 0048-6604/87/007 S -0312508.00 method, and one of its variants based on concepts derived from the spectral iterative technique, has been providedby van den Berg [1984, 1985] and some useful interpretations of variousiterationalgo- rithms have been suggested by Mittra and Chart [1985]. In this paper, weconsider three representative problems, viz., an interdigital transducer in a mul- tilayered structure [vandenBerg et al., 1985], a fre- quency selectivesurface [Rubin and Bertoni, 1983] and a large flat plate with impedance loading, to illustrate the application of three iterative schemes. Some conclusions drawn on the basis of the numeri- cal study of the above problems are included in the paper. 2. ITERATIVE ALGORITHMS The key to successful application of an iteration scheme is to develop,as a first step, an expression for an appropriate functional that can be minimized to extract the numerical solution to the original opera- tor equation. It is essential that the functional be chosen such that it enables the development of an iteration procedure that can be guaranteed to con- verge. The development of such a functional for an operator equation is described by [van den Berg, 1984, 1985; Mittra and Chan, 1985] and is summar- ized here. Consider the operatorequation LX = Y (1) 929