Solution to the Low-Frequency Breakdown Problem in Computational Electromagnetics

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In this chapter, the low frequency breakdown problem in Computational Electromagnetics is studied. The root cause of this problem is analyzed and is shown to be attributable to finite machine precision. A solution is presented, which is applicable to both partial-differential-equation- and integral-equation-based methods. In this solution, the original full-wave system of equations is rigorously solved at an arbitrary frequency including zero frequency. It does not make use of low-frequency quasi-static approximations, and is equally rigorous at electrodynamic frequencies. A fast method is also introduced to speed up the low-frequency computation in a reduced system of O(1). Moreover, this chapter demonstrates, both theoretically and numerically, that although the problem is termed low-frequency breakdown, the solution at breakdown frequencies can be a full-wave solution for which static and quasi-static assumptions are invalid. This is especially true in multi-scale problems that span many orders of magnitude difference in geometrical scales. The methods presented in this chapter bypass the barrier posed by finite machine precision, and provide a rigorous solution to Maxwell’s equations as a continuous function of frequency from electrodynamic frequencies all the way down to DC. It can also shed light on other unsolved research problems, the root cause of which is finite machine precision.