Convergence improvement for iterative solutions of the electric fields integral equation at very low frequencies

The matrix equations of the electric field integral equation (EFIE) at very low frequencies based on the loop-tree basis and the loop-star basis are transformed by a connection matrix which is a rearrangement of the basis. The new representation of the matrix equation can be solved by iterative solvers. It converges fast and no low frequency break-down occurs in the numerical computation. A method to perform the multiplication of the inverses of the connection matrix and its transpose with a vector for the tree basis with only O(N) floating-point operations is also developed. This work is incorporated into the low frequency multilevel fast multipole algorithm (LF-MLFMA) to solve large problems all the way from zero frequency to electrodynamic frequencies. The matrix transformation does not increase the computational complexity. The memory requirements and the number of floating-point operations still scale as O(N).