A fast direct matrix solver for surface integral equation methods for electromagnetic wave scattering from non-penetrable targets

[1] The implementation details of a fast direct solver is described herein for solving dense matrix equations from the application of surface integral equation methods for electromagnetic field scatterings from non-penetrable targets. The proposed algorithm exploits the smoothness of the far field and computes a low rank decomposition of the off-diagonal coupling blocks of the matrices through a set of skeletonization processes. Moreover, an artificial surface (the Huygens' surface) is introduced for each clustering group to efficiently account for the couplings between well-separated groups. Furthermore, a recursive multilevel version of the algorithm is presented. Although asymptotically the algorithm would not alter the bleak outlook of the complexity of the worst case scenario, O(N ³) for required CPU time where N denotes the number of unknowns, for electrically large electromagnetic (EM) problems; through numerical examples, we found that the proposed multilevel direct solver can scale as good as O(N ^1.3) in memory consumption and O(N ^1.8) in CPU time for moderate-sized EM problems. Note that our conclusions are drawn based on a few sample examples that we have conducted and should not be taken as a true complexity analysis for general electrodynamic applications. However, for the fixed frequency (h-refinement) scenario, where the discretization size decreases, the computational complexities observed agree well with the theoretical predictions. Namely, the algorithm exhibits O(N) and O(N ^1.5) complexities for memory consumption and CPU time, respectively.