Review of low frequency computational electromagnetics

Summary form only given: This talk will review the history of low-frequency computational electromagnetic (CEM). First, we start with the use of loop-tree decomposition to stabilize the low-frequency CEM. This method, though giving rise to the bounded condition numbers for the matrix system, still yields slow convergence of the equations. Then the basis-rearrangement method is proposed to give rise to faster convergence solution. Next, we review the development of fast algorithms for low-frequency CEM. For this, many methods exist, such as wavelets, Barnes-Hut algorithm, Appel algorithm, approximate cross approximation, matrix-decomposition method, and various matrix compression methods. We will review the pros and cons of these methods. Finally, we will review the mixed-form fast multipole algorithm that works seamlessly from static to microwave. We shall also review how these computations are done in layered media. Variational formulas for accelerating the convergence of capacitance calculations will also be reviewed.