The fast multipole method (FMM) was developed by Rokhlin to solve acoustic scattering problems very efficiently. We have modified and adapted it to the second-kind-integral-equation formulation of electromagnetic scattering problems in two dimensions. The present implementation treats the exterior Dirichlet (TM) problem for two-dimensional closed conducting objects of arbitrary geometry. The FMM reduces the operation count for solving the second-kind integral equation (SKIE) from O(n 3) for Gaussian elimination to O(n 4/3) per conjugated-gradient iteration, where n is the number of sample points on the boundary of the scatterer. We also present a simple technique for accelerating convergence of the iterative method: "complexifying" k, the wavenumber. This has the effect of bounding the condition number of the discrete system; consequently, the operation count of the entire FMM (all iterations) becomes O(n 4/3). We present computational results for moderate values of ka, where a is the characteristic size of the scatterer. Comments

We present a simple but effective technique for accelerating the convergence of iterative methods in the solution of electromagnetic scattering problems described by a second-kind integral equation (SKIE). We call the technique "complexification and extrapolation," or simply "complexification." It is based on the mathematical principle of limiting absorption, and it alleviates the difficulties arising from the interior resonances of this SKIE, thus allowing the efficient solution of scattering from electrically large objects. The technique involves introducing an imaginary part to the real wavenumber and solving the problem, then repeating with a different imaginary part and extrapolating the solutions linearly back to the real axis. For higher-order extrapolations we use additional complex wavenumbers. We have tested the method on a number of closed two-dimensional conducting scatterers, using this SKIE discretized by Nystr6m's method and solved by the fast multipole method. We use a variant of the conjugate gradient (CG) method that we call the pseudoconjugate gradient (PCG) method. The PCG method as we employ it performs only 1.2 matrix vector products on average per iteration, as opposed to two for the standard CG. Complexification gives excellent results. Solutions are fast and accurate. The condition number of the discrete matrix is asymptotically bounded for a given problem as the number of points per wavelength increases. The empirical evidence we have gathered thus far also suggests that the condition number is essentially asymptotically bounded as the electrical size of the scatterer increases, holding the number of points per wavelength fixed. Thus the technique has great potential in the solution of scattering from electrically large objects. Note that the technique of complexification is not limited to the fast multipole method and should be of broad applicability in the numerical solution of scattering problems. 2 MURPHY ET AL.: ACCELERATION METHODS FOR THE ITERATIVE SOLUTIONand extrapolating back to the real axis. We refer to this technique as "complexification and extrapolation," or simply as "complexification." Complexification sidesteps difficulties arising from resonance for problems described by certain second-kind integral equations (SKIEs) and ensures bounded condition numbers for the discrete linear system to be solved.The numerical method that serves here as a platform for exploring complexification is an extremely efficient one known as the Fast Multipole Method (FMM). The FMM, first developed by Rokhlin, [1985a, 1990] for acoustic problems and later applied by us [see Engheta et al., 1992] to electromagnetic scattering problems, performs the matrix vector product directly without need to store the matrix. Both the operation count and the storage requirement of the FMM is 0(n4/3), where n is the number of unknowns or the size of the discrete linear system. This can be improved to O(n log n) [see ]. Small, bounded condition numbers ensure that the number of iterations required for converg...

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