An H²-Matrix-Based Integral-Equation Solver of Reduced Complexity and Controlled Accuracy for Solving Electrodynamic Problems

Using anH2matrix as the mathematical framework, we compactly represent a dense system matrix by a reduced set of parameters, thus enabling a significant reduction in computational complexity. The error bound of theH2-matrix-based representation of an electrodynamic problem was derived. We show that exponential convergence with respect to the number of interpolation points can be achieved irrespective of the electric size. In addition, we show that a direct application ofH2-matrix-based techniques to electrodynamic problems would result in a complexity greater thanO(N), withNbeing the matrix size, due to the need of increasing the rank when ascending an inverted tree in order to keep a constant order of accuracy. A rank function was hence developed to maintain the same order of accuracy in a wide range of electric sizes without compromising computational complexity. With this rank function, we demonstrate that given a range of electric sizes which lead to a range ofN, the dense system ofO(N2) parameters can be compactly stored inO(N) units, and the dense matrix-vector multiplication can be performed inO(N) operations. Moreover, the same order of accuracy can be kept across this range. The method is kernel independent, and hence is suitable for any integral-equation-based formulation. In addition, it is applicable to arbitrary structures. Numerical experiments from small electric sizes to 64 wavelengths have demonstrated the performance of the proposed method.