As contemporary electronic devices continue to decrease in size and novel meta-materials and meta-surfaces find more applications, the demand for a method capable of addressing multi-scale and extreme-scale modeling and simulation becomes urgent. Unfortunately, the widely used vector wave equations encounter a significant challenge known as the "low-frequency breakdown" catastrophe. At low frequencies, corresponding to quasi-static and static scenarios, the decoupling of electric and magnetic fields renders the vector wave equations unable to account for Gauss's law, leading to the well-known low-frequency breakdown problem.

Our focus revolves around the development, implementation, validation, and application of a new formulation free of low-frequency breakdown. This innovative approach transcends the limitations of existing formulations, offering a versatile solution applicable across all frequencies — from static (dc) to microwave frequencies and beyond. This novel approach is employed in applications including the modeling of microwave integrated circuits, electrical machines and nonlinear magnetism, and wireless communication using intelligent reflecting surface (IRS) devices.

Key contributions include the discontinuous Galerkin time-domain (DGTD) methods with dynamic h- and p-adaptation, integral equation methods with phase extracted basis functions for electrically extra-large problems, and their hybridizations for multi-scale electromagnetic problems. Recent work includes a novel A-ϕ formulation with an implicit Coulomb gauge for wideband and multiscale simulation, and domain-decomposed formulations with non-conformal interfaces.