A Preconditioning Approach for the Domain Decomposition Simulation of High-Speed Circuits

Domain decomposition methods (DDMs) provide a powerful discretization framework for analyzing the electromagnetic (EM) phenomena in complex high-speed circuits. However, even with recent advancements in DDMs, achieving iterative convergence remains challenging for sophisticated structures. To facilitate the convergence of DDMs, we develop a preconditioner in this work, which is constructed by forming the approximate inverse of the DDM system matrix. The subdomains are recursively grouped into a multilevel tree, and randomized matrix compressions are adapted to handle their couplings. The approximate inverse preconditioner is then constituted by the inverses of the subdomain matrices with low-rank updates that account for their couplings. Through simulations of practical circuit examples, the strengths of the preconditioner are demonstrated by comparing it to a standard DDM solver. Importantly, a weakly convergent scenario is identified for DDMs when circuit signals propagate parallel to the subdomain interfaces. This situation, which significantly hinders computational efficiency and causes stability issues, can be effectively addressed through the proposed preconditioning approach. Finally, a novel integration is introduced to combine our method with a recently developed subdomain compression scheme.