Solution of the electric field integral equation at very low frequencies

We discuss the solution of Maxwell´s equations from static to microwave frequencies. First, we review the fast multipole method and the multilevel fast multipole algorithm we have developed for electrodynamic problems. Next, we discuss the problem encountered when electromagnetic problems are solved at very low frequencies. For such problems, we develop a new method to precondition the matrix equation resulting from applying the method of moments (MOM) to the EFIE. This preconditioning method is based on first applying the loop tree or loop-star decomposition of the currents to arrive at a Helmholtz decomposition of the unknown currents. However, the MOM matrix thus obtained cannot be solved efficiently by iterative solvers due to the required large number of iterations. We propose a permutation of the loop-tree or loop-star currents by a connection matrix, to arrive at a current basis that yields a MOM matrix that can be solved efficiently by iterative solvers