The complex bi-conjugate gradient solver applied to large electromagnetic scattering problems, computational costs, and cost scalings

The complex bi-conjugate gradient iterative method is applied to an isoparametric boundary integral equation formulation for frequency-domain electromagnetic scattering problems. It is demonstrated to work well on large and geometrically complex examples, including a 20 wavelength slender dipole, the NASA almond, and a resonant cavity. On such problems, with asymmetric curvilinear irregular meshes and nontrivial geometries, the number of iterations required seems to increase rather more than linearly with body size, indicating an overall ~sixth power cost scaling. This scaling is essentially as for direct methods, but with costs still a small fraction of the direct approach. A method is proposed for the selection of a termination condition designed to avoid seeking the approximate answer too precisely; it typically permits a further halving of costs