Schur complement preconditioners for surface integral-equation formulations of dielectric problems solved with the multilevel fast multipole algorithm
Surface integral-equation methods accelerated with the multilevel fast multipole algorithm (MLFMA) provide a suitable mechanism for electromagnetic analysis of real-life dielectric problems. Unlike the perfect-electric-conductor case, discretizations of surface formulations of dielectric problems yield 2 × 2 partitioned linear systems. Among various surface formulations, the combined tangential formulation (CTF) is the closest to the category of first-kind integral equations, and hence it yields the most accurate results, particularly when the dielectric constant is high and/or the dielectric problem involves sharp edges and corners. However, matrix equations of CTF are highly ill-conditioned, and their iterative solutions require powerful preconditioners for convergence. Second-kind surface integral-equation formulations yield better conditioned systems, but their conditionings significantly degrade when real-life problems include high dielectric constants. In this paper, for the first time in the context of surface integral-equation methods of dielectric objects, we propose Schur complement preconditioners to increase their robustness and efficiency. First, we approximate the dense system matrix by a sparse near-field matrix, which is formed naturally by MLFMA. The Schur complement preconditioning requires approximate solutions of systems involving the (1,1) partition and the Schur complement. We approximate the inverse of the (1,1) partition with a sparse approximate inverse (SAI) based on the Frobenius norm minimization. For the Schur complement, we first approximate it via incomplete sparse matrix-matrix multiplications, and then we generate its approximate inverse with the same SAI technique. Numerical experiments on sphere, lens, and photonic crystal problems demonstrate the effectiveness of the proposed preconditioners. In particular, the results for the photonic crystal problem, which has both surface singularity and a high dielectric constant, shows that accurate CTF solutions for such problems can be obtained even faster than with second-kind integral equation formulations, with the acceleration provided by the proposed Schur complement preconditioners.