A multilevel fast direct solver for EM scattering from quasi-planar objects

A fast multilevel direct solver for electromagnetic scattering from quasi-planar objects is presented. The solver relies on the compression of the off-diagonal blocks in the impedance matrix, which describe interactions between distinct domains. The compression is performed in a number of steps. First, the scatterer is decomposed into sub-domains using a multilevel quad-tree hierarchical subdivision. Then, the field radiated by each sub-domain, onto the rest of the scatterer, is determined using a Non-uniform sampling grid approach. Subsequently, a rank-revealing QR decomposition is applied to the grid matrix to find the current basis functions that actually contribute to the radiated field. At the same time, the employed decomposition singles out the most important grid points (called grid skeleton), from which the field on the observation domain can be reconstructed. Finally, compression of the local interacting currents and fields inside each sub-domain is performed using the Schur's complement method. The non-interacting currents are solved locally, whereas interacting currents and grid skeletons are repeatedly aggregated with neighboring sections in a multilevel process. The resulting compressed system of equations is solved directly. The algorithm is analyzed for performance and stability. It is shown that approximately O(N^1.5) complexity is attained for the matrix compression and approximately O(N) for each right-hand-side solution, N being the number of unknowns. The analytical complexity estimates are supported by the results of a numerical case study.