The frequency band where the solution to Maxwell's equations is unknown —A challenge facing the analysis of multiscale problems and its solution

The problems encountered in the system-level analysis and design of integrated circuits (IC) are multiscaled in nature, the geometric scales of which range from centimeter to nanometer. For such multiscale problems, in this paper, we demonstrate, both theoretically and numerically, that there exists a band of frequencies in which the solution to Maxwell's equations is unknown. This is because in this band, traditional fullwave solvers break down due to finite machine precision while static and quasistatic solvers are invalid, i.e. the solution at the breakdown frequencies of traditional fullwave solvers is a fullwave solution. As a result, the prevailing approach for broadband IC design, which stitches the results from a fullwave solver at high frequencies with those from static/quasi-static solvers at low frequencies, can be totally incorrect when applied in a multiscale setting. Consequently, to sustain the continued scaling of integrated circuits, it is important to find the solution to the original fullwave Maxwell's equation in a complete electromagnetic spectrum, whether high or low. Such a universal solution has been developed in [1]. In this work, in addition to demonstrating the existence of a frequency band where the solution of Maxwell's equations is unknown from existing electromagnetic solvers, we present a fast algorithm to find the unknown solution, thus addressing a major challenge in multiscale analysis.