An efficient, preconditioned, high-order solver for scattering by two-dimensional inhomogeneous media

We consider the problem of evaluating the scattering of TE polarized electromagnetic waves by two-dimensional penetrable inhomogeneities: building upon previous work [IEEE Trans. Antennas Propag. 48 (2000) 1862] we present a practical and general fast integral equation algorithm for this problem. The contributions introduced in this text include: (1) a preconditioner that significantly reduces the number of iterations required by the algorithm in the treatment of electrically large scatterers, (2) a new radial integration scheme based on Chebyshev polynomial approximation, which gives rise to increased accuracy, efficiency and stability, and (3) an efficient and stable method for the evaluation of scaled high-order Bessel functions, which extends the capabilities of the method to arbitrarily high frequencies. These enhancements give rise to an algorithm that is much more accurate and efficient than its previous counterpart, and that allows for treatment of much larger problems than permitted by the previous approach. In one test case, for example, the present algorithm results in far-field errors of 8.9×10−13 in a 2.12s calculation (on a 1.7 GHz PC) whereas the original algorithm gave rise to far-field errors of 1.1×10−8 in 88.91s on a 400 MHz PC. In another example, the present algorithm evaluates accurately the scattering by a cylinder of acoustical size κR=256, which is of the order of 20 times larger (400 times larger in square wavelengths) than the largest scatterers that could be treated by the previous approach. Yielding, at worst, third-order far field accuracy (or substantially better, for smooth scatterers) in fast computing times (O(NlogN) operations for an N point mesh) even for discontinuous and complex refractive index distributions (possibly containing severe geometric singularities such as corners and cusps), the proposed approach is the highest-order O(NlogN) solver in existence for the problem under consideration.