Multiscale electromagnetic computations using a hierarchical multilevel fast multipole algorithm

A hierarchical multilevel fast multipole method (H-MLFMM) is proposed herein to accelerate the solutions of surface integral equation methods. The proposed algorithm is particularly suitable for solutions of wideband and multiscale electromagnetic problems. As documented in Zhao and Chew (2000) that the multilevel fast multipole method (MLFMM) achieves O(N log N) computational complexity in the fixed mesh size scenario, hk = cst, where h is the mesh size and k is the corresponding wave number, for problems discretized under conventional mesh density. However, its performance deteriorates drastically for overly dense meshes where the couplings between different groups are dominated by evanescent waves or circuit physics. In the H-MLFMM algorithm, two different types of basis functions are proposed to address these two different natures of physics corresponding to the electrical size of the elements. Specifically, for the propagating wave couplings, the plane wave basis function adopted by MLFMM are effective and they are inherited by H-MLFMM. Whereas in the circuit physics and for the evanescent waves, H-MLFMM employs the so-called skeleton basis. Moreover, the proposed H-MLFMM unifies the procedures to account for the couplings using these two distinct types of basis functions. O(N) complexity is observed for both memory and CPU time from a set of numerical examples with fixed mesh sizes. Numerical results are included to demonstrate that H-MLFMM is error controllable and robust for a wide range of applications.