O(N) iterative and O(NlogN) direct volume integral equation solvers for large-scale electrodynamic analysis

State-of-the-art volume integral equation (VIE) solvers for solving electrically large problems are iterative solvers with the complexity of each matrix-vector multiplication being O(NlogN), where N is matrix size. In this work, we reduce this complexity to strictly O(N) irrespective of electrical size. Furthermore, we develop a fast inversion based direct VIE solver of O(NlogN) complexity, which is also independent of electrical size. Numerical experiments have demonstrated the complexity, accuracy, and efficiency of the proposed new VIE solvers. Very large-scale VIE system matrices involving millions of unknowns have been directly solved in fast CPU time and modest memory consumption on a single core running at 3 GHz.