Fast Integral Equation Solution by Multilevel Green's Function Interpolation Combined With Multilevel Fast Multipole Method

Abstract

A fast wideband integral equation (IE) solver combining the multilevel interpolatory fast Fourier transform accelerated approach (MLIPFFT) with the multilevel fast multipole method (MLFMM) is discussed. On electrically fine levels within an oct-tree multilevel structure, coupling computations are performed by MLIPFFT. This method is based on a 3D Lagrange factorization of the pertinent Green's functions with a smooth approximation error in space and it does not suffer a low frequency breakdown as known from MLFMM. For high frequency integral equation problems, MLIPFFT has decreased computational efficiency as the Nyquist theorem requires increasing numbers of samples in 3 dimensions. Due to a transition from the interpolation point based MLIPFFT source/receive formulation towards an appropriate <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -space representation at a certain level within the oct-tree, the high frequency efficient MLFMM can be employed for coarse levels. The hybrid algorithm is hence well suited for fast wideband integral equation solutions. Both, mixed-potential and direct-field formulations are considered. Furthermore, a method for MLIPFFT extrapolation error reduction based on fine level interpolation domain spreading is introduced. In several numerical examples, the performance of the proposed algorithm is demonstrated.